Mathematical morphology and uncertainty quantification applied to the study of dual phase steel formation

Dual Phase steels (DP steels) have shown high potential for automotive and other applications, due to their remarkable property combination between high strength and good formability. The mechanical properties of the material are strictly related with the spatial distribution of the two phases composing the steel, ferrite and martensite, and their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, thus one important industrial problem is to reduce the number of 2D sections needed to reconstruct or simulate in a realistic way the 3D geometry of the material. This reduction causes an increase of the uncertainty in the parameters estimates of suitable geometric models for the material. In this work we present an approach based on the definition of a germ-grain model which approximates the main geometric characteristics of the martensite. The parameters of the model are estimated on the basis of the morphological characteristics of the images of about 150 tomographic sections of a real sample, and plausibility regions for the estimated parameters are computed. The increase in uncertainty on the parameters estimates in presence of a reduced number of sections is then quantified in terms of increase in the volume of the corresponding plausibility regions. Even though the model still needs some improvements in the fitting with the real data, the overall procedure for uncertainty quantification that we have obtained can be generalized to other study cases and can be used by the industry to set up a suitable experimental plan to fit a model to the data with a desired accuracy.