On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

Czyżewska, Natalia; Kusiak, J.; Morkisz, Paweł; Oprocha, Piotr; Pietrzyk, M.; Przybyłowicz, Paweł; Rauch, Ł.; Szeliga, Danuta

doi:10.1109/access.2022.3199006

Cited by 6 publications

(10 citation statements)

References 32 publications

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“…Besides the numerical solution for this equation, a detailed theoretical analysis of this equation in the case when . ε = 1 was performed in [51]. Classical approaches to the ordinary differential equations (ODE), presented in the literature e.g., [52,53], require some regularity of right-hand side function, usually Lipschitz condition, which is not satisfied by (2).…”

Section: Deterministic Modelmentioning

confidence: 99%

“…[51] also contains rigorous analysis of the error for the considered equations. It is also interesting that while [54] shows that simple DDEs in the form of ( 2) cannot be easily used in the modelling of biological or physical phenomena, a natural application to materials science is discussed in [51]. Specifically, it appears that admissible solutions of (2) for the case when .…”

Section: Deterministic Modelmentioning

confidence: 99%

“…As [51] had the rigorous error analysis for the Euler method performed and the stability of this method investigated, it could be confidently applied to a real industrial process. Numerical tests were conducted on a more complicated equation than (2), with strain rate, temperature, and other parameters defined as time dependent.…”

Section: Case Study-deterministic Internal Variable Modelmentioning

confidence: 99%

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A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

et al. 2023

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Modern construction materials, including steels, have to combine strength with good formability. In metallic materials, these features are obtained for heterogeneous multiphase microstructures. Design of such microstructures requires advanced numerical models. It has been shown in our earlier works that models based on stochastic internal variables meet this requirement. The focus of the present paper is on deterministic and stochastic approaches to modelling hot deformation of multiphase steels. The main aim was to survey recent advances in describing the evolution of dislocations and grain size accounting for the stochastic character of the recrystallization. To present a path leading to this objective, we reviewed several papers dedicated to the application of internal variables and statistical approaches to modelling recrystallization. Following this, the idea of the model with dislocation density and grain size being the stochastic internal variables is described. Experiments composed of hot compression of cylindrical samples are also included for better presentation of the utility of this approach. Firstly, an empirical data describing the loads as a function of time during compression and data needed to create histograms of the austenite grain size after the tests were collected. Using the measured data, identification and validation of the models were performed. To present possible applications of the model, it was used to produce a simulation imitating industrial hot-forming processes. Finally, calculations of the dislocation density and the grain size distribution were utilized as inputs in simulations of phase transformations during cooling. Distributions of the ferrite volume fraction and the ferrite grain size after cooling recapitulate the paper. This should give readers good overview on the application of collected equations in practice.

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Section: Deterministic Modelmentioning

confidence: 99%

Section: Deterministic Modelmentioning

confidence: 99%

Section: Case Study-deterministic Internal Variable Modelmentioning

confidence: 99%

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A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

et al. 2023

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“…e statistical approach to modeling dislocation density fields in the present work is based on the deterministic model. e authors' solution of this model is described in detail in earlier publications [19,21], and only the main equations are repeated here for the completeness of the paper.…”

Section: Stochastic Approach In Modeling Of Evolution Of Dislocation ...mentioning

confidence: 99%

“…e swarm's particle starting positions were chosen at random from the provided sets of bounds. For the objective function, we used the Bhattacharyya distance (21) and the scoring metric was Mean Average Percentage Error defined by equation (11). We have repeated the optimization procedure 10 times for each set.…”

Section: Problem Of Reconstructionmentioning

confidence: 99%

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

Klimczak

Oprocha

Kusiak

et al. 2022

Mathematical Problems in Engineering

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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.

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Stochastic model describing evolution of microstructural parameters during hot rolling of steel plates and strips

Szeliga

Czyżewska

Klimczak

et al. 2022

Archiv.Civ.Mech.Eng

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Enhancing strength-ductility synergy of materials has been for decades an objective of research on structural metallic materials. It has been shown by many researchers that significant improvement of this synergy can be obtained by tailoring heterogeneous multiphase microstructures. Since large gradients of properties in these microstructures cause a decrease of the local fracture resistance, the objective of research is to obtain smoother gradients of properties by control of the manufacturing process. Advanced material models are needed to design such microstructures with smooth gradients. These models should supply information about distributions of various microstructural features, instead of their average values. Models based on stochastic internal variables meet this requirement. Our objective was to account for the random character of the recrystallization and to transfer this randomness into equations describing the evolution of dislocations and grain size during hot deformation and during interpass times. The idea of this stochastic model is described in the paper. Experiments composed of uniaxial compression tests were performed to supply data for the identification and verification of the model in the hot deformation and static recrystallization parts. Histograms of the grain size were measured after hot deformation and at different times after the end of deformation. Identification and validation of the model were performed. The validated model, which predicts evolution of heterogeneous multiphase microstructure, is the main output of our work. The model was implemented in the finite element program for hot rolling of plates and sheets and simulations of these processes were performed. The model’s capability to compare and evaluate various rolling strategies are demonstrated in the paper.

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On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

Cited by 6 publications

References 32 publications

A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

A Comparative Study of Deterministic and Stochastic Models of Microstructure Evolution during Multi-Step Hot Deformation of Steels

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

Stochastic model describing evolution of microstructural parameters during hot rolling of steel plates and strips

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