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1

The effect of model size and boundary conditions on the representativeness of digital material representation simulations of ferritic-pearlitic sample compression

Perzyński Konrad

Computer Methods in Materials Science

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2022

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Vol. 22, No. 2

59--66

EN

The main objective of this work is to investigate the representativeness of the digital material representation (DMR) models of ferritic-pearlitic steel generated by the hybrid cellular automata (CA) / Monte Carlo (MC) algorithm. Particular attention is focused on determining the effect of the size of the digital representation model on its representativeness under deformation conditions simulated with the finite element (FE) framework. In addition, the effect of periodic and non-periodic boundary conditions on the deformation behaviour of DMR models is analysed. A dedicated buffer zone approach applied the periodic boundary conditions on non-periodic finite element models. The results of equivalent stresses and strains and their average values are used to evaluate the differences between the models’ predictions.

2

The role of neighborhood density in the random cellular automata model of grain growth

Czarnecki Michał, Sitko Mateusz, Madej Łukasz

Computer Methods in Materials Science

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2021

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Vol. 21, No. 3

129--137

EN

The paper focuses on adapting the random cellular automata (RCA) method concept for the unconstrained grain growth simulation providing digital microstructure morphologies for subsequent multi-scale simulations. First, algorithms for the generation of initial RCA cells alignment are developed, and then the influence of cells density in the computational domain on grain growth is discussed. Three different approaches are proposed based on the regular, hexagonal, and random cells’ alignment in the former case. The importance of cellular automata (CA) cell neighborhood definition on grain growth model predictions is also highlighted. As a research outcome, random cellular automata model parameters that can replicate grain growth without artifacts are presented. It is identified that the acceptable microstructure morphology of the solid material is obtained when a mean number of RCA cells in the investigated neighborhood is higher than ten.

3

Cellular Automata-based computational library for development of digital material representation models of heterogenous microstructures

Boguń Konrad, Sitko Mateusz, Mojżeszko Mateusz, Madej Łukasz

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 2

335--349

EN

The development of an efficient numerical approach for the generation of a wide range of heterogeneous microstructures models with the application of the lean workflow concept is presented in the paper. First, the idea and implementation details of the developed cellular automata-based computational library allowing the development of digital material representation models within a workflow are presented in the paper. Such an approach provides the desired flexibility in the generation of various digital models of heterogenous microstructures. Therefore, the proposed library is mostly implemented within the object-oriented C + + programming language with the assumption of modularity. In this case, the main part of the application consists of classes and methods, which can be treated like base elements to be inherited and extended in other libraries. Each additional dynamic link library implements particular algorithms for the generation of specific microstructure features in the digital model within the unified data structures that allow the application of the workflow concept. The set of developed libraries and their assumptions are described as case studies to show the capabilities of the presented solution. Finally, examples of practical applications of the developed library in the full-field numerical simulations of complex material deformation are presented at the end of the paper.

4

Numerical Investigation of the Influence of Pulsed Laser Deposited Tin Thin Film Morphology on Deformation Inhomogeneities

Perzyński K., Zych D., Sitko M., Madej Ł.

Archives of Metallurgy and Materials

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2020

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Vol. 65, iss. 2

713--720

EN

In vestigation of influence of TiN thin film morphology on deformation inhomogeneities is an overall subject of the research. Numerical modelling approach that was selected for the study is based on the digital material representation concept, which gives an opportunity to directly replicate columnar microstructure morphology of an investigated thin film. Particular attention in this paper is put on the discussion of the influence of cellular automata neighbourhood on thin-film digital morphologies and their further deformation behaviour. Additionally, an evaluation of representativeness aspects of the digital models, in particular, the analysis of the influence of a number of columns, their dimensions and variations in their properties on the material behaviour during compression tests is also presented. The non-periodic boundary conditions are assumed during the investigation. Obtained data in the form of equivalent stress distributions as well as homogenized stress-strain curves from analyzed case studies are presented and discussed within the paper.

5

Numerical Investigation of an Explosive Welding of Ti/Cu Plates Based on a Meshfree Method

Mojżeszko M., Perzyński K., Sionkowski M., Paul H., Madej Ł.

Archives of Metallurgy and Materials

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2020

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Vol. 65, iss. 2

707--711

EN

Development of a reliable numerical model capturing major physical mechanisms controlling explosive welding and considering properties of all process components i.e. base plate and flyer plate is the goal of the paper. To properly replicate materials behavior under these severe conditions a meshfree approach, namely Smooth Particle Hydrodynamics (SPH), was used to discretize the computational domain. The model is based on the Mie-Gruneisen shock equation of state applied to the Ti/Cu system as a case study. Examples of results in the form of velocity, equivalent stress, equivalent strain, and pressure fields are presented within the paper.

6

Controlling deformation inhomogeneity in the Accumulative Angular Drawing Process assisted by constitutive and multiscale numerical modelling

Lisiecka-Graca Paulina, Kwiecień Marcin, Madej Łukasz, Muszka Krzysztof, Majta Janusz, Wynne Bradley P.

Computer Methods in Materials Science

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2019

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Vol. 19, No. 3

113--121

EN

Ultrafine-grained structure was produced in the Accumulative Angular Drawing(AAD) process in which the complex strain path was applied. The microalloyed steel wire rods were produced using multi-pass wire drawing process where the high strain accumulation is used as a way to achieve much higher microstructure refinement level compared to the conventional wire drawing (WD) process. The wires after both AAD process and WD process were examined in order to assess mechanical properties and microstructure development. In order to evaluate the effects of complex deformation on microstructure development and mechanical properties of the drawn wires, a numerical model of the torsion tests were conducted using Abaqus software. The cyclic torsion tests were performed to study the effects of the applied hardening rule - described as a function of dislocations density and the accuracy of the prediction of material behaviour subjected to strain path change during AAD. It has been shown that an additional advantage of presented approach is its capability of tracking evolution of dislocation density during the deformation process. The initial dislocation densities used in the performed calculations were taken from the microstructural analysis using high resolution EBSD. During strain reversal, annihilation of the dislocations (Bauschinger effect) is a common phenomenon that leads to the decrease in dislocation density and affects the final strength. Finally, based on the proposed constitutive description, multiscale finite element modelling combined with Digital Material Representation (DMR) was used as a tool for prediction of the deformation and microstructure inhomogeneity in the drawn wires.

PL

W pracy przedstawiono badania wpływu złożonej ścieżki odkształcenia na rozdrobnienie mikrostruktury w drutach poddanych procesowi Kątowego Wielostopniowego Ciągnienia (AAD Accumulative Angular Drawing). Druty ze stali mikrostopowej zostały poddane procesowi wielostopniowego ciągnienia, w którym niejednorodna, silna akumulacja odkształcenia powoduje wystąpienie efektów rekrystalizacji in situ, co z kolei powoduje lokalny wzrost stopnia rozdrobnienia mikrostrutury w porównaniu z konwencjonalnym procesem ciągnienia (WD Wire Drawing). Druty wytworzone w procesach AAD oraz WD zostały poddane badaniom porównawczym własności mechanicznych oraz analizie mikrostrukturalnej. W celu zapewnienia poprawnej oceny wpływu złożonej ścieżki odkształcenia na rozwój mikrostruktury i własności mechaniczne w ciągnionych drutach, zastosowano symulacje numeryczne z wykorzystaniem komercyjnego pakietu Abaqus oraz modeli procesu skręcania. Symulacje procesu cyklicznego skręcania wykonano w celu oceny zdolności proponowanego modelu umocnienia odkształceniowego, opartego na zmianach gęstości dyslokacji, do symulacji zmiennej ścieżki odkształcenia podczas procesu AAD. Dodatkową zaletą zaproponowanego podejścia jest możliwość śledzenia rozwoju gęstości dyslokacji podczas procesu odkształcania. Początkowa gęstość dyslokacji wykorzystana w obliczeniach wyznaczona została z analizy mikrostrukturalnej z wykorzystaniem wysokorozdzielczej techniki EBSD. Podczas zmiany kierunku odkształcenia, proces anihilacji dyslokacji (efekt Bauschingera) jest częstym zjawiskiem prowadzącym do spadku gęstości dyslokacji, a w konsekwencji do obniżenia umocnienia odkształceniowego. W celu wyznaczenia niejednorodności mikrostrukturalnej oraz niejednorodności odkształcenia w drutach po procesie ciągnienia, zastosowano modelowanie wielkoskalowe w połączeniu z cyfrową reprezentacją obrazu (DMR).

7

Considering influence of microstructure morphology of epoxy/glass composite on its behavior under deformation conditions - digital material representation case study

Madej L., Malinowski L., Perzynski K., Mojzeszko M., Wang J., Cios G., Czarnecki D., Bala P.

Archives of Civil and Mechanical Engineering

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2019

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Vol. 19, no. 4

1304--1315

EN

Digital Material Representation (DMR) concept in application to numerical investigation of the two different types of epoxy/glass composite morphologies under loading conditions is addressed within the paper. First, two algorithms for reconstruction of digital microstruc-tures based on metallography investigations are developed. Then, material properties of the investigated epoxy matrix and glass fillers are evaluated based on an in-situ tensile test as well as nano-indentation, respectively. At this stage a numerical investigation is also extended by series of experimental tensile tests to understand basic mechanisms occurring during deformation of the two different types of glass particles fillers. Finally, an example of practical application of the developed digital microstructure model for multi scale calcula-tions of the epoxy/glass composite under loading is presented.

8

Wykorzystanie koncepcji cyfrowej reprezentacji materiału w modelowaniu zachowania mikrostruktury materiału porowatego pod obciążeniem

Madej Ł., Legwand A., Perzyński K., Roskosz S., Chrapoński J., Cwajna J.

Hutnik, Wiadomości Hutnicze

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2017

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Vol. 84, nr 8

287--289

PL

Celem niniejszej pracy jest wykorzystanie koncepcji cyfrowej reprezentacji materiału podczas modelowania numerycznego zachowania się spieków metalicznych w warunkach obciążenia. Aby uzyskać informacje o mikrostrukturze analizowanego materiału porowatego wykonano badania techniką tomografii komputerowej. Tak uzyskane dane zostały wykorzystane do przygotowania trójwymiarowego modelu numerycznego, który w sposób bezpośredni uwzględnia niejednorodność mikrostruktury podczas dalszych symulacji. W ramach pracy opracowano program komputerowy, który przekształca dane pochodzące bezpośrednio z tomografu na binarny obraz trójwymiarowy, stanowiący cyfrową reprezentację materiału. Tak przygotowany model mikrostruktury został następnie poddany dyskretyzacji siatką elementów skończonych, i stanowi dane wejściowe do obliczeń numerycznych. Jako przykład testowy wykorzystania zaproponowanego podejścia, w pracy wybrano próbę swobodnego spęczania.Wyniki ilustrują wpływ porowatości na występowanie niejednorodności w uzyskiwanych rozkładach odkształceń czy naprężeń oraz umożliwiają precyzyjną analizę zamykania się porów w trakcie deformacji.

EN

The main goal of the present work is development of reliable numerical model of sintered metallic microstructures based on the digital material representation concept. To recreate morphology of the investigated porous material a computed tomography technique was used. To transfer the obtained measurement data into the digital form a series of numerical algorithms have been developed within the research. Obtained digital representation of microstructure morphology is then subjected to discretization procedure required for further numerical simulation of material behavior under loading conditions. As a case study a simple compression test was selected in the present work. Examples of obtained results, revealing various heterogeneities occurring during deformation are also presented within the paper.

9

Digital Material Representation Model of Porous Microstructure Based on 3D Reconstruction Algorithm

Madej Ł., Mojżeszko M., Chrapoński J., Roskosz S., Cwajna J.

Archives of Metallurgy and Materials

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2017

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Vol. 62, iss. 2A

563--569

EN

Development of the Digital Material Representation (DMR) model, based on 3D reconstruction algorithm and serial sectioning, is the main goal of the present paper. Details on the serial sectioning and image processing algorithms are presented first. Serial sectioning is realized on the basis of light microscopy (LM). Then concept of 3D reconstruction and developed algorithms are presented. Two approaches, based on shape coefficients and the flood fill algorithms, are developed to identify corresponding features on subsequent 2D images. Then, the interpolation algorithm to reconstruct 3D volume between 2D images is presented. Finally, obtained 3D model is an input for finite element mesh generation software for subsequent finite element calculations.

10

Digital/virtual microstructures in application to metals engineering – A review

Madej L.

Archives of Civil and Mechanical Engineering

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2017

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Vol. 17, no. 4

839--854

EN

Recent progress in application of digital/virtual microstructure models in the area of metals engineering science is presented within the paper. First, various approaches to digital reconstruction of microstructure morphology of investigated materials is presented. Possibilities of generation of both: exact replicas of morphology, as well as, synthetic microstructures are discussed. Advantages and limitations in the case of two and three-dimensional problems are highlighted in that section. Then, the state of the art in the evaluation of material properties at the microstructure scale is addressed. Various experimental techniques, characterized by different levels of complexity, which are capable of providing information on materials hardening behavior at the micro scale level are presented. Finally, possibilities of introduction of microstructure morphology with specific properties into the finite element solution are described. The work is complemented by series of practical applications of the digital/virtual microstructure models to show their capabilities and present directions for further development.

11

Numeryczny model pękania dla stali dwufazowych

Perzyński K., Madej Ł.

Hutnik, Wiadomości Hutnicze

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2016

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Vol. 83, nr 4

141--146

PL

Celem badań jest opracowanie modeli numerycznych odzwierciedlających mechanizmy pękania w stali dwufazowej Dual Phase (DP), bazujących na koncepcji cyfrowej reprezentacji materiału. Zaproponowane w pracy rozwiązanie wykorzystuje sprzęgnięcie dwóch podejść numerycznych szeroko stosowanych w symulacjach o charakterze komercyjnym: metody elementów skończonych i rozszerzonej metody elementów skończonych. Podejście takie umożliwia stosunkowo łatwe wykorzystanie modeli pękania kruchego oraz ciągliwego w praktycznych zastosowaniach ze względu na użycie standardowych rozwiązań stosowanych w aplikacjach numerycznych. Nowatorskie jest zastosowanie cyfrowej reprezentacji materiału i koncepcji modelowania wieloskalowego do opisu zachowania się materiału w niższych skalach wymiarowych. Opracowany model został zweryfikowany z wynikami pomiarów zrealizowanych podczas testu wywijania otworu (HE – Hole Expansion).

EN

Development of a robust model of failure for Dual Phase (DP) steels based on modern numerical approaches that take microstructure explicitly into account during simulation is the main aim of the paper. Modeling procedure is based on conventional numerical damage models, however combined with the DMR (Digital Material Representation) approach. As a result, a combined model of ductile and brittle fractures which occur in ferrite and martensite, respectively, was established. Ductile fracture is modelled by the Ductile Fracture criterion implemented within conventional FE model, while brittle one is predicted by more sophisticated eXtended Finite Element Method (XFEM). Proper data transfer protocols between these two methods were also proposed to create a hybrid numerical model. Developed solution was validated with experimental data from the hole expansion (HE) test.

12

Numerical analysis of data transfer quality in the 3D multi-scale uncoupled concurrent model connected with DMR

Szyndler J., Perzyński K., Madej Ł.

Computer Methods in Materials Science

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2016

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Vol. 16, No. 2

97--103

EN

Analysis of the quality of obtained results from a 3D multiscale concurrent finite element numerical model based on Digital Material Representation (DMR) concept is the main aim of the research. Particular attention is put on an influence of different number of data transfer nodes between subsequent scales on material behavior predicted by a micro DMR model. Conclusions are drawn based on results in the form of equivalent strain distribution, homogenized stress-strain curves and samples shape changes.

PL

Celem pracy jest analiza jakości uzyskanych wyników podczas stosowania współbieżnego trójwymiarowego modelu wieloskalowego, bazującego na kombinacji modeli elementów skończonych w skali makro i mikro. Szczególną uwagę poświecono wpływowi zróżnicowanej ilości węzłów przekazujących dane między skalami mikro i makro na zachowanie się materiału w skali mikro. Wyniki przedstawiono w formie rozkładu odkształceń, krzywej płynięcia materiału i zmian w kształtach próbek w skali mikro.

13

Numerical model of the nanoindentation test based on the digital material representation of the Ti/TiN multilayers

Perzyński K., Wiatr R., Madej Ł.

Materials Science Poland

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2015

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Vol. 33, No. 2

348--355

EN

The developed numerical model of a local nanoindentation test, based on the digital material representation (DMR) concept, has been presented within the paper. First, an efficient algorithm describing the pulsed laser deposition (PLD) process was proposed to realistically recreate the specific morphology of a nanolayered material in an explicit manner. The nanolayered Ti/TiN composite was selected for the investigation. Details of the developed cellular automata model of the PLD process were presented and discussed. Then, the Ti/TiN DMR was incorporated into the finite element software and numerical model of the nanoindentation test was established. Finally, examples of obtained results presenting capabilities of the proposed approach were highlighted.

14

Development of the cellular automata framework dedicated for metallic materials microstructure evolution models

Rauch L., Madej L., Spytkowski P., Golab R.

Archives of Civil and Mechanical Engineering

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2015

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Vol. 15, no. 1

48--61

EN

The main goal of the paper is to design and implement a framework based on the cellular automata (CA) method, which is dedicated to numerical simulations of microstructure evolution in metallic materials under thermal and mechanical processing. Major assumptions and implementation details of the proposed solution involving classes containing dedicated fields and methods are discussed. Finally, the cellular automata framework (CAF) is tested for selected case studies supported by the Windows Workflow Foundation (WWF) approach. Particular attention is put on modelling simple grain growth, static recrystallization and phase transformation phenomena occurring at the microstructure level. Obtained results of simulations as well as performance characteristics are also presented in the paper. As a result, the CA framework, which supports design of complex algorithms with flexible data flow and reusable components is proposed.

15

Molecular models of polycrystalline and porous materials

Mrozek A.

Computer Methods in Materials Science

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2014

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Vol. 14, No. 1

27--36

EN

This paper is contribution to the special issue of the CMMS journal, devoted to modelling of the polycrystalline structures. Creation of the metallic polycrystalline and porous structures based on the molecular dynamics (MD) simulations are presented in this paper. The simple Morse potential, as well as, the more sophisticated Embedded Atom Method (EAM) are engaged to model atomic interactions. The presented methods of creation of the polycrystalline and porous molecular models are discussed and illustrated with proper numerical examples. The series of tensile tests and comparison of the mechanical properties between obtained polycrystalline structures is included, along with the description of the algorithm of the computation of the mechanical properties and the stress-strain relations. Additional tests are carried out with ideal Morse and EAM monocrystals in order to validate our molecular models and results. The simulations of creation polycrystalline and porous models are performed using the massively-parallel MD solver with NVT ensemble and tensile tests utilize so-called Non-Equilibrium Molecular Dynamics (NEMD).

PL

Tematyka pracy dotyczy metod tworzenia oraz badania własności mechanicznych atomowych modeli materiałów polikrystalicznych oraz porowatych przy wykorzystaniu Dynamiki Molekularnej. Przedstawionych zostało kilka różnych technik otrzymywania w/w struktur, m. in. kontrolowane schładzanie układu atomów oraz modyfikacja zasięgu oddziaływań międzyatomowych (funkcji potencjału atomowego). Każda metoda zilustrowana została przykładami, omówiony został również wpływ wybranych parametrów na wyniki symulacji numerycznej. W dalszej części artykułu opisano sposób wyznaczania własności mechanicznych modeli molekularnych: wyznaczania składowych tensora mikronaprężenia, krzywych naprężenie-odkształcenie i modułu Younga. Utworzone przedstawionymi metodami, zrównoważone i stabilne modele atomowe mogą posłużyć jako materiał wejściowy - rozwiązania początkowe - do dalszych symulacji wykorzystujących dynamikę lub statykę molekularną.

16

Effect of number of grains and boundary conditions on digital material representation deformation under plane strain

Szyndler J., Madej L.

Archives of Civil and Mechanical Engineering

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2014

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Vol. 14, no. 3

360--369

EN

The main goal of this paper is to investigate the digital material representation of a single-phase polycrystalline unit cell. Particular attention is put on the amount of grains in the microstructure, which can be considered as the representative volume element of a sample subjected to plastic deformation conditions. Additionally, the influences of the periodic and non-periodic boundary conditions on deformation behavior of the unit cells are evaluated. Possibility of the application of periodic boundary conditions on the non-periodic unit cells using a buffer zone approach is also discussed. Obtained results of equivalent strains and reaction force on case studies are presented.

17

Modelowanie numeryczne mechanizmów pękania w stalach DP z uwzględnieniem wpływu morfologii mikrostruktury

Perzyński K., Madej Ł.

Rudy i Metale Nieżelazne

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2013

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R. 58, nr 11

764--768

PL

Modelowanie mechanizmów pękania ciągliwego w stalach DP (ang. Dual Phase) wymaga utworzenia odpowiedniej cyfrowej reprezentacji morfologii badanej mikrostruktury. Tak utworzony cyfrowy model materiału jest zastosowany w niniejszej pracy do oceny wpływu ułamka fazy martenzytu na zachowanie się stali DP w warunkach odkształcenia plastycznego. W pierwszej części pracy omówiono metodę służącą do odtworzenia cyfrowej reprezentacji mikrostruktury stali DP. Oparta jest ona o algorytmy przetwarzania obrazu zgładu metalograficznego i zapewnia dokładne odwzorowanie badanej mikrostruktury. Utworzone cyfrowe mikrostruktury są następnie wykorzystywane jako dane wejściowe do numerycznego modelu procesu odkształcenia, który uwzględnia mechanizmy pękania ciągliwego. Model zastosowano do oceny wpływu ułamka fazy martenzytu na zachowanie się stali DP w warunkach odkształcenia plastycznego. Uzyskane wyniki obliczeń zaprezentowano w niniejszej pracy.

EN

The aim of this work is to develop a methodology for creating a digital microstructures representation of dual phase steels and incorporate them into a numerical model of ductile failure. In the first part of the paper an approach based on image processing algorithms for creating a digital representation of the microstructure is described. In the second part, obtained digital microstructures are used as input to the numerical model of deformation, which takes into account mechanisms of ductile fracture. The model is used to evaluate the impact of the martensite volume phase fraction on DP steel behavior under plastic deformation. The results of the calculations are presented in the paper.

18

Zastosowanie metody automatów komórkowych do opracowania cyfrowej reprezentacji wybranych cech mikrostruktury w oparciu o obrazy binarne jej składników

Opara J., Wrożyna A.

Prace Instytutu Metalurgii Żelaza

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2013

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T. 65, nr 4

2--7

PL

W ramach pracy stworzono program komputerowy, w którym zaimplementowano dwuwymiarowy automat komórkowy wraz z autorskim algorytmem do przetwarzania obrazów binarnych rzeczywistej mikrostruktury na jej cyfrową reprezentację. Szczegółowo opisano schemat dyskretyzacji badanego obszaru mikrostruktury. Zaprezentowano możliwości opracowanego rozwiązania.

EN

A computer program was built in this work, where a two-dimensional cellular automaton is implemented and including authors’ algorithm for conversion of binary images of the real microstructure to the digital microstructure representation. Discretization scheme of the investigated microstructure is explained in detail. The possibilities of the developed solution were presented.

19

Wykorzystanie metody Monte Carlo w zagadnieniach tworzenia cyfrowej reprezentacji materiału

Szyndler J., Madej Ł.

Hutnik, Wiadomości Hutnicze

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2013

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Vol. 80, nr 2

172--176

PL

W artykule zamieszczono wyniki badań związanych z opracowaniem i wykorzystaniem cyfrowej reprezentacji materiałów do analizy wpływu niejednorodności mikrostruktury na stan odkształcenia w materiałach polikrystalicznych w warunkach obciążenia. Przedstawiono wykorzystanie metody Monte Carlo do tworzenia statystycznie reprezentacyjnych mikrostruktur. Omówiono podstawy teoretyczne tej metody i jej przystosowanie do zagadnień związanych z generacją mikrostruktur materiałów polikrystalicznych oraz przedstawiono możliwość wykorzystania tak uzyskanej reprezentacji podczas symulacji odkształcenia metodą elementów skończonych.

EN

Recent progress in research related to application of digital material representation (DMR) approach for analysis of influence of microstructure heterogeneities on material processing is presented within this paper. The main focus is put on application of the Monte Carlo (MC) method for generation of the statistically representative digital microstructures. Fundamentals of the MC method as well as modifications of this approach that are necessary to deal with creation of digital representation of polycrystalline material are described. Based on the obtained morphology of digital microstructure an example of its application to modeling material deformation is also presented.

20

Two-dimensional hp-adaptive algorithm for continuous approximations of material data using space projection

Gurgul P., Sieniek M., Paszyński M., Madej Ł., Collier N.

Computer Science

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2013

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Vol. 14 (1)

97--112

EN

In this paper we utilize the concept of the L 2 and H 1 projections used to adaptively generate a continuous approximation of an input material data in the finie element (FE) base.This approximation, along with a corresponding FE mesh, can be used as material data for FE solvers. We begin with abrief theoretical background, followed by description of the hp -adaptive algorithm adopted here to improve gradually quality of the projections. We investigate also a few distinct sample problems, apply the aforementioned algorithms and conclude with numerical results evaluation.