Inverse Problem in Stochastic Approach to Modelling of Microstructural Parameters in Metallic Materials during Processing

Abstract

The need for a reliable prediction of the distribution of microstructural parameters in metallic materials during processing was the motivation for this work. The model describing the evolution of dislocation populations, which considers the stochastic aspects of occurring phenomena, was formulated. The validation of the presented model requires the application of proper parameters corresponding to the considered materials. These parameters have to be identified through the inverse analysis, which, on the other hand, uses optimization methods and requires the formulation of the appropriate objective function. In our case, where the model involves the stochastic parameters, it is a crucial task. Therefore, a specific form of the objective function for the inverse analysis was developed using a measure based on histograms. The elaborated original stochastic approach to modeling the phenomena occurring during the thermomechanical treatment of metals was validated on commercially pure copper and selected multiphase steel.

Through process stochastic model of hot strip rolling

“…Details of the identification procedure are described in [2,6]. Inverse algorithm developed in [11] was applied.…”

“…Description of the model. The model is discussed in details in [6] and only main equations are repeated here for a completeness of the presentation. The model origins from Kocks-Estrin-Mecking (KEM) approach [8,9] with the stochastic dynamic recrystallization term added.…”

“…where: t -time, ε -strain rate, a7 -coefficient accounting for the influence of the strain rate on the recovery, ∆t -time step, A1, A2 -coefficients responsible for hardening and recovery defined in [6] and in Fig. 1.…”

“…In Eq. 3 variable γ represents a mobile fraction of the recrystallized grain boundary and depends on the distribution of ξ(ti-1) in previous step, as it is explained in [6]. In the work [7] the model was further extended by including interpass times and static phenomena into simulations.…”

“…Our objective was to account for the random character of the recrystallization which occurs during deformation and after deformation at elevated temperatures and to transfer this randomness to equations describing evolution of the dislocation populations and the grain size. The idea of the model is presented in [6] and its identification for hot deformation processes is described in [2,7]. Since the model incorporates stochastic character of the phenomena, it may predict the distribution of parameters instead of their average values.…”