Time and length scale issues in numerical modelling of dynamic recrystallization based on the multi space cellular automata method
Introduction
Thermo-mechanical treatment of processed metallic materials provides wide range of possibilities to control final microstructure morphology and in-use properties of products. Combination of deformation and varying process temperatures can be used to initiate and control three groups of phenomena responsible for the microstructure evolution. The texture evolution is the first one. The second group involves phase transformations in metallic materials both during heating and cooling stages and the last group is represented by thermally activated phenomena of recrystallization (static, dynamic and metadynamic). The later, has been experimentally and numerically investigated for many decades, increasing the level of understanding of interactions between processing conditions and microstructural changes. From the numerical point of view, progress in models complexity is directly related with the increasing resolution of experimental techniques providing knowledge on local heterogeneities, as well as, with the progress in computational recourses with multi core processors, computer clusters or efficient graphical processors. First numerical models for recrystallization were based on mean field approaches like simple closed form Avrami type equations. They provided general information on flow stress variations during recrystallization. Then more sophisticated models based on differential equations were developed. They may be considered as precursors of multiscale solutions, as information on micro scale features e.g. dislocation densities was used to evaluate macroscopic flow stress values. However, both approaches can provide quite general information on microstructure morphology e.g. the average grain sizes, or its distribution. Thus, again, more advanced full field models, which directly take into account microstructure morphology with all its features during modelling were proposed [1]. Phase Field (PF) [2], Level-Set (LS) [3], Monte Carlo (MC) [4] or Cellular Automata (CA) [5], [6] methods can be classified into this category.
Authors of the work investigated possibilities of application of the CA to simulate microstructure evolution during both static and dynamic recrystallization. The method provides a possibility to directly link physical mechanisms leading to particular phenomenon with the basic assumptions of the algorithm. Thus, capabilities of the CA approach have been investigated in the material science applications for almost 30 years.
First research works on practical application of the CA models to microstructure evolution simulations are related to the static recrystallization [7], [8], [9], [4]. This was followed by numerical investigation of dynamic recrystallization in [10] and flow stress behavior analysis with respect to initial grain size prior deformation. Development of new CA neighborhood types provided a possibility to obtain reliable grain shapes during and after growth of newly recrystallized grains [11], while incorporation of internal variable models gave the possibility to consider also hardening and recovery prior initiation of dynamic recrystallization [6], [12], [13]. Subsequent CA models, along with the wide range of applications of SEM/EBSD techniques, became more refined and introduced information on crystallographic aspects [14], [15], [16]. Then, the problem of solute drag effect was considered in [17] where authors used multiscale concept based on combination of cellular automata and finite element method to describe macro and micro scale deformation of copper. An important breakthrough in CA model development was related with possibility to deform the CA space during simulations [18], [19], [20]. This subject is still addressed in the scientific literature and more advanced approaches are being developed see e.g. [21], [22]. Recently, to increase predictive capabilities of cellular automata models other computational techniques are employed and interlinked e.g. crystal plasticity finite element method [23], mathematical statistics theory [24] or adaptive response surface method [25].
Thus, due advantages of the CA technique and recent progress, the method is often applied in many scientific institutes to investigate dynamic recrystallization in wide range of engineering metallic materials: stainless steels [26], C–Mn micro alloyed steel [27], titanium [23], copper [17], nickel based alloys [28].
As presented, the CA technique applications are broad, however, in most research works aspects related with the accuracy of the method in relation to time and length scales used within CA spaces during calculations seem to be neglected. Although, both parameters have significant influence on microstructure morphology as well as recrystallization kinetics. Thus, authors decided to develop a cellular automata model for dynamic recrystallization that will be insensitive to mentioned changes in the time step length and CA cell size. Initial works on the matter have already been published in [29], [30], where basic assumptions of the DRX model were proposed. Present paper is devoted to advanced model modifications that were required to reach the mentioned goal. Detailed study on model robustness is also presented to highlight negative influence of time step length and cell size on quality of obtained results, as such an investigation is missing in the scientific literature devoted to the CA technique.
Section snippets
The dynamic recrystallization cellular automata model
The cellular automata model for dynamic recrystallization is divided into four interconnected stages, including:
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generation of the initial microstructure,
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evaluation of average dislocation density due to hardening and recovery occurring during deformation,
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nucleation of new grains and,
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subsequent growth of recrystallized nuclei/grains.
Evaluation of the influence of time step length on quality obtained results
To the Authors knowledge, there is no information in scientific literature on negative effects of improper evaluation of time step length in the CA codes. On the other hand Authors believe that the issue is of paramount importance from the point of view of quality of obtained results.
The fundamental definition of the CA method assumes that the CA cell can change the state based on its previous state as well as based on previous states of its neighbors. At the same time due to the discrete
Evaluation of the CA cell size on obtained results
Another analysis, realized within the work, was focused on physical size of the microstructure. Authors checked how increasing CA space resolution – for the constant physical domain size – influences quality of obtained results. Thus, the analysis was focused on influence of physical size of CA cells on simulation results. For that reason different initial microstructures were generated as seen in Fig. 22. All microstructures contain the same number of grains equal to 20, and have the same
Conclusions
Evaluation of time step length and cell size influence on cellular automata models behavior was investigated with the dynamic recrystallization CA code as a case study. Description of the developed model with crucial modifications, which increase predictive capabilities, were presented first. Then series of analysis for different time step lengths and CA cell sizes were realized to highlight the importance of model robustness analysis. Based on the presented results it can be concluded that:
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The
Acknowledgment
Financial assistance of the NCN project no. 2012/04/M/ST8/00706 is acknowledged.
Mateusz Sitko, born on March 8, 1988 in Krakow. In 2011 he received his Engineering degree at AGH University of Science and Technology in Krakow at Faculty of Metallurgy and Materials Science in Applied IT. The research was on: Development of a tool for numerical analysis basis of crystal plasticity approach. In 2012 he defended his Master thesis at the Faculty of Metallurgy and Materials Science in Applied IT. The research was on: Development of the user friendly numerical system for analysis
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Mateusz Sitko, born on March 8, 1988 in Krakow. In 2011 he received his Engineering degree at AGH University of Science and Technology in Krakow at Faculty of Metallurgy and Materials Science in Applied IT. The research was on: Development of a tool for numerical analysis basis of crystal plasticity approach. In 2012 he defended his Master thesis at the Faculty of Metallurgy and Materials Science in Applied IT. The research was on: Development of the user friendly numerical system for analysis based on digital material representation approach. From 2012 he continues his education at Ph.D. studies at the Department of Applied Computer Science and Modelling in Metallurgy. His Ph.D. thesis is related with development of high efficient software dedicated to HPC platform to recrystallization simulation based on cellular automata method. He is a co-author of one book chapter, 12 articles in reviewed journals and international conference proceedings.
Maciej Pietrzyk, born on March 6, 1947 in Krakow. He received his MSc in 1970, PhD in 1975 and became a full professor in 1992. Currently working at the AGH University of Science and Technology in Krakow, Poland. His scientific interest focuses on application of numerical methods in metallurgy and materials science. Author of over 500 publications in scientific journals and conference proceedings, co-author of two books on Thermal-Mechanical Modelling of the Flat Rolling Process published by Springer-Verlag in 1991 and Mathematical and Physical Simulation of the Properties of Hot Rolled Products published by Elsevier in 1999. Author of the textbook on Application of numerical methods in metal forming published by AGH in 1992. Head of the Department of Applied Computational Science and Modelling since 1997.
Łukasz Madej, born on November 29, 1980 in Krakow. In 2003 he received his Engineering degree at the AGH University, Department of Physics and Nuclear Techniques in the field of the Solid State Physics. The research was on: Analytical and numerical solution of differential equations describing the behavior of the hot-deformed metals. In 2004 he defended his Master thesis at the Faculty of Metallurgy and Materials Science in the field of Information Technology in Material Science. The research was on: Inverse analysis and cellular automata method applied to determine the rheological parameters and modelling the development of microstructure of steel with different carbon contents. In 2007 he obtained a Ph.D. degree at the Faculty of Metals Engineering and Industrial Computer Science in the field of Applied Informatics. Ph.D. was awarded by the ECCOMAS the first prize for the Best PhD thesis in Europe in 2007. In 2015 he received ESAFORM Scientific Prize 2015 for outstanding contribution in the field of materials forming. He is also author and co-author of two books, 4 chapters in books, 45 publications in journals ranked in the Philadelphia list, 59 articles in other peer-reviewed journals and 70 publications in national and international conference proceedings.