Three-dimensional frontal cellular automata modeling of the grain refinement during severe plastic deformation of microalloyed steel

Computational Materials Science 102 (2015) 159–166

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Three-dimensional frontal cellular automata modeling of the grain refinement during severe plastic deformation of microalloyed steel Dmytro S. Svyetlichnyy, Krzysztof Muszka ⇑, Janusz Majta AGH University of Science and Technology, Mickiewicza 30, 30-059 Krakow, Poland

a r t i c l e

i n f o

Article history: Received 14 August 2014 Received in revised form 9 January 2015 Accepted 15 February 2015 Available online 9 March 2015 Keywords: Cellular automata Microstructure Grain refinement Crystallographic orientation Microalloyed steel

a b s t r a c t In the current work, a computer model based on three-dimensional Frontal Cellular Automata (FCA) for the simulation of grain refinement during multiaxial compression was developed. The strong grain refinement obtained in microalloyed steel through subdivision of the initial coarse-grained structure into dislocation substructure and subsequently into stabile UFG structure was simulated and analyzed by FCA. Proposed in the present study model is a step forward toward understanding deformation mechanisms occurring during metal forming processes with high energy accumulation. Conclusions regarding possibilities of proposed numerical tool were drawn basing upon qualitative and quantitative comparisons of the modeling and SEM/EBSD results. Results obtained with FCA-based model were compared with SEM/ EBSD results of real process and demonstrated a good agreement. A description of the model, results and conclusions are presented in the paper as well. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Mechanical and physical properties of metals are determined by their chemical composition and microstructure that, in turn, is a result of a range of strengthening mechanisms, as an effect of deformation/heat treatment. However, a grain refinement still remains one of the best ways to achieve the superior properties such as high strength, good toughness, as well as good high and low cycle fatigue performance. Therefore, over the last few decades, there has been a considerable interest paid to ultrafine-grained (UFG) and nanostructured metallic materials aimed at numerous applications and various branches of the industry. Nowadays, these materials can be fabricated both under laboratory and industrial conditions. The processing techniques can be divided into two major approaches: bottom-up and top-down. The first ones include electro-deposition or various compaction methods. The top-down techniques are more important from the point of view of industrial applications. In this case, UFG/nanostructured materials are produced from a coarsegrained initial state using Advanced Thermomechanical Processing (ATP) [1–3] or Severe Plastic Deformation (SPD) [4–7]. The ATP methodologies involve existing industrial processes (mostly rolling) using microstructural phenomena, e.g. in the case of steel, dynamic austenite recrystallization with subsequent phase transformation, dynamic strain-induced ferrite transformation, intercritical rolling, warm rolling, reverse transformation, and cold rolling with subse⇑ Corresponding author. Tel.: +48 12 6172908; fax: +48 12 6172576. E-mail address: [email protected] (K. Muszka). http://dx.doi.org/10.1016/j.commatsci.2015.02.034 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

quent annealing of martensitic starting microstructure. The SPD methods, in turn, as a main factor, use a large accumulative plastic strain at ambient temperature or warm deformation conditions. This group of techniques includes, first of all: Equal Channel Angular Pressing and Extrusion (ECAP, ECAE), Accumulative Roll Bonding (ARB), High Pressure Torsion (HPT) and multiaxisial compression. In these processing methods, applied severe plastic deformation causes a subdivision of the initial coarse-grained microstructure into dislocation cells and subgrains. At the beginning of the SPD processing, the microstructures are characterized by a high volume fraction of Low Angle Grain Boundaries (LABs). Then, due to the continuous dynamic recrystallization (recrystallization in situ) these LABs transform into more stable High Angle Grain Boundaries (HABs) [8–13]. Most of the SPD methods impose complex deformation modes that make the prediction of the microstructure evolution extremely difficult, mainly due to a high microstructural inhomogeneity that is introduced during multiple strain reversals. Thus, a concurrent effect of different elements of dislocations structure governs the microstructure evolution. Additionally, interactions between dislocations, LABs and HABs constantly change the overall hardening effect. Therefore, it is very difficult to determine a contribution of particular elements of the microstructure to the final material properties. So far, a computer modeling of microstructure evolution during SPD processes has been really difficult due to a lack of proper numerical tools that allow representing these high energy dislocation structures numerically. Recently, Digital Material Representation (DMR) has been proposed and widely used in computer

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simulations of deformation processes, microstructure evolution and finally, prediction of the material properties [14]. The possibility of prediction of the microstructure evolution in the micro-, meso- and macro-scale is one of the most important problems in materials science. Cellular automata (CA) models [15] occupy the first place among the methods used in the computer modeling of microstructure evolution. CA are used for modeling of crystallization (solidification) [16–20], dynamic and static recrystallization [21–25], phase transformation [23,26–29], grain refinement [30–33], etc. CA-based models, which can be found in literature, usually are two-dimensional (2D). 2D CA models are simpler and faster, because they consist of fewer elements and connections, use more clear algorithms, and are much simpler for designing, implementation, as well as, more useful for visualization. The real microstructure, however, is not 2D in nature. Nevertheless, for simplification, there have been many numerical parameters introduced and effectively used in the microstructural analysis, e.g. the average grain size. Another parameter frequently used in the microstructural analysis describes the grain shape, which can be often represented by a residual strain or by the ratio of the grains volume to their surface area. A crystallographic orientation of grains and subgrains is the next feature that can be used to describe the presence and degree of the texture. One more subject of interest is the distribution of boundaries disorientation angle and its changes during the deformation. As mentioned above, the microstructure evolution is pointedly three-dimensional and results obtained by the 2D CA can never correspond to the real 3D processes. At least five main problems, partly unresolved, appear during the 2D CA modeling i.e.: kinetics of transformation, location of nuclei, grain growth rate, deformation of grains and crystallographic orientation. Some of them were investigated elsewhere in detail [20,29,33]. 3D CA are free of these problems, however, they are more complex and require much more time and memory for simulations. One of the possible modifications of the CA, known as Frontal Cellular Automata (FCA) [24], which allows for an algorithmical reduction of the calculation time, is used in this paper. Recently, Authors used FCA to predict grain refinement that was presented in publications [30–33]. Conception of FCA application was described in details in the first of these publications [30], then three models were developed, simulated and discussed [31]. The results presented in Fig. 6 in [31] do not take into account real process of deformation and the model chooses slip plains and slip directions arbitrary. Later, the effects of deformation were distributed on slip planes and slip directions according to the crystal plasticity theory, which was implemented into the model [32]. In publication [33] are summarized the results, that were previously obtained; and two processes – ARB process (accumulative rollbonding) and multiaxial compression (using MaxStrain deformation) are presented. ARB process is similar to flat rolling from the point of view of the state of deformation. It can be considered as a case of plane strain. This process was simulated for aluminum [33]. MAXStrain is one of the Mobile Conversion Units of Gleeble Systems, thermal–mechanical physical simulation and testing systems designed and manufactured by Dynamic Systems, Inc. The system restrains specimens lengthwise while allowing unlimited deformation in the other two dimensions, that is, a deformation process in the state of plane strain. A cycle is of two deformations, in which two axes are replaced. Elongation is replaced by compression, and vice versa. Then, the specimen after each cycle is back to its initial size, but the material accumulates strain. Some results of the computer simulation of two cycles MaxStrain deformation are presented in [33,39]. The first publication [33] considers an appearance of new boundaries only, without evolution of boundary disorientation. The second publication [39] considers the module of the initial

microstructure taking into account the grain refinement. The initial microstructure was obtained in three stages: modeling the initial coarse microstructure – (a), the grain refinement according to required grain (subgrain) size distribution – (b) and fitting the disorientation boundary angle to the required distribution – (c). In the present work, the appearance of new boundaries and rotation of dislocation cells (subgrains and grains) are simulated simultaneously. Moreover, rotation rate is set in such a way that parameters of the simulated microstructure were close to data from EBSD study. That was the identification simulation. It allowed one to conclude that the rotation rate decreases from cycle to cycle, especially that it drops quickly in the first cycle. The objectives of the present work are experimental studies of the effect of the multiaxial compression on the grain refinement in microalloyed steel using MaxStain system [34] and an analysis of the evolution of the initial coarse microstructure into the ultrafine-grained structure with the use of three-dimensional FCA.

2. Experimental study of microstructure evolution during multiaxial compression In the present study, the MaxStrainÒ system [35] was used to produce an ultrafine-grained microstructure during multiaxial compression test. In this test, each deformation cycle consists of two compressive deformations, in which the longitudinal axis of the specimen is constrained so the deformation energy is accumulated only in the deformation zone between two anvils. After the first deformation pass, the specimen is rotated by 90° around its longitudinal axis; and the second deformation is applied. This cycle can be repeated many times, so severe accumulation of the deformation energy can be reached without a loss of the specimen’s integrity. In the current investigations, multiaxial compression was carried out on a microalloyed steel with the following basic chemical composition [wt%]: 0.07C–1.36Mn–0.27Si–0.067Nb– 0.073Ti–0.002B. This material is widely used in many branches of the industry (automotive, construction, pipeline, etc.) due to its high strength to ductility ratio resulting from a synergetic effect of various strengthening mechanisms. Additionally, precipitation and solid solution strengthened alloys can be processed by SPD much longer than pure metals (due to increased strain hardening rate [34,36]), what is clearly beneficial for the present investigations. Its initial as-received microstructure consisted of a polygonal ferrite structure with the average grain size of about 25 lm. Specimens for the MaxStrain testing were machined out of as-hot rolled plate in the rolling direction – they were 27 mm long and had square 10  10 mm cross-sections. During the deformation, total accumulative strains of 2, 5, 7, 10 and 20 were applied. As it was already mentioned, one cycle of MaxStrain deformation consists of two compressive deformations in the state of plane strain, in which two axes are replaced while the longitudinal axis is constrained. Tensile strains are replaced by compressive strains and vice versa. Then, the specimen after each cycle returns to the initial sizes, but the material accumulates strain. Each deformation was fulfilled with the strain e = 0.5, then, the total accumulative strain after one cycle was e = 1.0. The dimensions of the cross-section in the middle of the cycle became 16.48 mm  6.06 mm and after full one cycle they are back to 10 mm  10 mm. In the present work, specimens were deformed at the room temperature (TD = 20 °C), and then, after the deformation, annealed at the temperature of TA = 500 °C for tA = 1200 s in order to stabilize already obtained UFG microstructure. After the deformation and heat treatment, the specimens after the second, fifth, seventh, tenth, fifteenth and twentieth cycles were cut and their microstructure was studied using a Scanning

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Electron Microscope (SEM) equipped with the Electron Backscatter Diffraction (EBSD) detector. Examples of UFG microstructures obtained after the 2nd, 10th and 20th cycles of MaxStrain deformation are presented in Fig. 1 in the form of Euler maps acquired from SEM/EBSD analysis. In EBSD analysis, crystallographic orientation of grains is defined by the three Euler rotation angles. It is well seen in Fig. 1 that the fraction of the HABs increases with increasing SPD deformation. The results presented in Fig. 1 describe the evolution of UFG structure in a qualitative manner only – more details about LABs and HABs development during SPD straining can be found in Fig. 2, where distributions of the boundaries disorientation angle are presented. It can be observed that there is a sharp change in the distribution of grain boundaries around the disorientation angle of 20–25°. The most of the distributions of the boundaries disorientation angle are localized in two ranges, which can be separated about angle of 20–25°. In the initial stage of deformation, a strong decrease of the frequencies with increase of the disorientation angles, observed in the first part of the distribution, is a result of appearance of the subgrain structure and its subsequent evolution into the HABs structure, as a consequence of the grain refinement. The second part of the histogram, above 20–25° (though, in fact, it occupies whole range of the angles), represents the rest of the initial microstructure. The shape of the second part would be similar to the distribution of randomly orientated grains, but a small number of the grains make this dependence less regular and less smooth. The frequency of each class in Fig. 2 is the ratio of the number of boundaries with appropriate angle to the number of all possible places, where boundaries can be detected. Boundary class with disorientation angle below 3° is not shown because of its very high frequency and such substructures cannot be considered as a grain structure. Furthermore, comparison of LABs and HABs distributions after different number of MaxStrain deformation cycles can be an introduction to more detailed analysis. Further simulations of the grain refinement and changes of the disorientation angle fulfilled with the use of FCA, which are described in detail in Section 4, allow one to develop a numerical model of disorientation angle evolution during SPD. Two phenomena and two stages of the grain refinement process can be defined in the investigated microalloyed steel, which is characterized by a high stacking fault energy (SFE). The first phenomenon is a creation of dislocation cells, which evolve into subgrains and then, eventually, into fine stable grains characterized by the HABs. The second phenomenon is a rotation of existing dislocation cells, subgrains and grains under the deformation, which is a driving force for evolution of the HABs.

161

Fig. 2. Distribution of the grain boundaries disorientation angle during MaxStrain deformation.

However, neither microstructures (Fig. 1) nor distributions of boundaries disorientation angle (Fig. 2) give clear data for description of the analyzed process. On the other hand, division of the boundaries development process into two phenomena leads to definition of the main analyzed parameters – at least as far as the first phenomenon is concerned. General agreement to consider LABs separately from HABs does not allow one to analyze the process properly. That is why the overall number of boundaries was analyzed further, although, the boundaries with the disorientation angle between 2–3° and 20–25° can be considered as an alternative. Such numbers are presented in Fig. 3. It can be noted that most of the boundaries were formed in the first cycles of the MaxStrain deformation, and then, their number increases at much lower rate. It allows one to divide the grain refinement process into two stages. Formation of the cells-subgrains structures prevails in the first stage, while subgrains-grains rotations dominate in the second stage. After exceeding some threshold strain, there is probably no return of subdivided grain clusters back into the original coarse grains [37]. To summarize the experimental study, it should be stated that in the modeling procedure of the grain refinement during the multiaxial compression two phenomena and their role during UFG structure formation need to be considered.

Fig. 1. Microstructure with the grains boundaries after 2nd (a), 10th (b) and 20th (c) cycles of MaxStrain deformation. Black lines – HABs and light gray lines – LABs.

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Fig. 3. Changes of the relative number (frequency) of the boundaries during the SPD.

3. FCA model CA are considered as a very useful and universal tool for the modeling and simulation of microstructural phenomena. A general block-scheme of the system presented in Fig. 4 has been developed and presented elsewhere [33]. Bold lines mark the modules and proper connections that are used in the present study. The basis for this hierarchical system is FCA, while the tip is a set of considered processes. Microstructure evolution models are between them and interconnect them. The materials database completes the system. Frontal cellular automata and its computational advantages are described in detail elsewhere [24,33,29,38]. Additionally, detailed information about consideration of the deformation in FCA and space reorganization can be found in [27,38,39]. In the presented hierarchical model, the second level contains modules: ‘‘Initial microstructure’’ [39], ‘‘Crystallization’’ [20] ‘‘Recrystallization’’ [24,27], ‘‘Phase transformation’’ [29] and ‘‘Refinement’’ [33]. Technological processes that may be modeled by the third level of the system are the solidification in the continuous casting [20], the hot flat and shape rolling [27,38], the roll-bonding process and the multiaxial compression [33]. In the presented scheme, the role that FCA play, is twofold: at first, they serve as a tool for the creation process of the initial microstructure. Then, they are also responsible for modeling of all the microstructural changes observed during SPD.

The microstructural part of the model calculates the grain refinement during the cold deformation [30–33]. In the model two different CA are considered. The first one presents spatial discretization of representative volume. For example, cellular space of 300  300  300 cells can represent volume of 50  50  50 lm3. Every cell is of the same shape and size, and can change uniformly. They belong to the corresponding grain and have appropriate properties. Dislocation density however is not related to the cells. The grains are considered in the second type of CA model. Actually, every grain, subgrain, dislocation cell has own structure of CAs. Number of one-dimensional CAs depends on a number of main crystal slip systems. For example, there are 12 slip systems for materials with b.c.c structure. Correspondently, there are 12 1d CAs in the grain CA model. Number of cells in the 1d CA depends on resolution and grain size in appropriate direction. For every cell evolution of dislocation density is calculated, as it was described in detail elsewhere [31,33]. 1d CAs utilize information from the subroutine based on crystal plasticity approach and simulates the generation and growth of dislocation substructure. The active slip systems are determined for each grain (crystal) on the basis of crystallographic orientation. Low-angle boundaries (LAB) appear, when dislocation density reaches critical value. It means, that LAB divides the grain into several subregions, dislocation cells. They inherit CA model from parent grain; and they are considered further independently. Every new LAB divides old grain into two dislocation cells. Because slip rate is different for different slip directions, dislocation structure evolves at different rate. Cells with LABs state appear firstly in one direction that results in a formation of the first geometrically necessary boundaries. For the other slip directions, the LABs appear later. Such a structure with formed dislocation cells is a starting point for the simulation of further rotation of the dislocation cells, subgrains and grains during the straining. The dislocation motion on active slip systems and distortion of the crystal lattice are the main components of deformation of a single crystal. The material spin (rigid body spin) W of a polycrystalline material is defined as the skew-symmetric tensor obtained by decomposing the prescribed velocity gradient in a rotation and a deformation component. The material spin is the sum of the plastic spin and the lattice spin:

W ¼ Wp þ Wl

ð1Þ

here, Wp is the plastic spin and Wl is the lattice spin which represents macroscopic rotation, also called the constitutive spin or the rotation rate of the Mandel-frame. Crystal plasticity assumes that plastic spin could be related to the rate of slip in slip systems of the lattice:

Wp ¼

X

c_ s Psij

ð2Þ

s

Technological processes + material

Phase transformation

Recrystallization

Crystallization

Grain refinement

Initial microstructure

Cellular automata

Deformation of CA Reorganization of CA

Fig. 4. FCA-based model for simulation of microstructure evolution in technological processes [33].

where c_ s is the glide velocity for the active slip system s, and Psij is the skew part of the Schmid tensor defined by the slip plane and the slip direction. Details concerning the algorithm and some example results of the simulation of the creation process of dislocation cell structure can be found elsewhere [33]. However, those results are only qualitative, because neither sizes of the cells-subgrains-grains nor kinetics of the microstructure refinement process has not been considered there. 3.1. Formation of new boundaries As mentioned above in Section 2, the frequency (or the relative number) of all boundaries is chosen and analyzed as a main

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microstructural parameter. This parameter depends on the average grain (cell) size but also on the resolution of the EBSD map. The boundary length is proportional to the grain size. A digital representation of the boundary obtained from EBSD scan consists of vertical and horizontal line segments of the same length. Then, the digital representation is of a higher value then the real length. A factor of about 1.2–1.3 can compensate this difference. When a scale or resolution is changed, the frequency is changed as well. If one uses twice higher resolution, the number of boundaries increases twice, but number of all possible locations for boundaries arises fourfold, i.e. the frequency is inversely proportional to the resolution. On the other hand, the grain size measured in the number of the lines segments is inversely proportional to the frequency, which is approximately equal to: D = 1.25r/f, where D is the grain size, r is the distance between measured points and f is the frequency. For example, the frequency f  0.1 obtained after 20 cycles of MaxStrain deformation and presented in Fig. 1c corresponds to the average grain size of D  1.9 lm, although the real grain size is D > 2.1 lm (because not every LAB is accounted). Because the frequency depends on the resolution, this approach is not convenient for any computer model and another parameter should be chosen. Besides the frequency, average grain size, number of the grains on unit area of the cross-section and number of the grains in unit volume has been analyzed. Unfortunately, dependency of the average grain size on the strain (or the cycle number) was extremely nonlinear, especially for the first cycles of the multiaxial deformation. At the same time, however, the frequency and both numbers of the grains demonstrated less nonlinearity. Actually, all three dependencies on the strain gave very similar results. The number of the grains in unit volume was chosen as a basic model in FCA, because 3D simulations were expected. This model is presented in two forms. The first one is a dependence of the number of grains on the strain:

   n  e Ng ¼ N0 þ ðN1  N0 Þ 1  exp 

ec

DNg ¼ 1:69

N1  Ng

ec

8 Ng ð0Þ ¼ N0

ðe þ 0:25Þn1 De

ment. Then, all dislocations move to or from the LABs. A positive feedback switches on for the LAB cell and all its neighbors when the LAB cell appears. The dislocation density of the LAB cells increases because of this positive feedback that causes an increase of the strain hardening, while in all neighboring cells the dislocation density decreases very fast. 3D CA model considers all slip planes. Every slip system is considered as distinct 1D CA, and their algorithm is the same as described above. The structure in each slip direction evolves at different rate because of different slip rates for particular directions. The cells with the LAB state appear firstly in one 1D CA. It means appearance of the first geometrically necessary boundary. For the other slip systems 1D CA are decomposed into two 1D CA. In the present approach, instead of a critical value of the dislocation density, which determines appearing of the LAB, new boundaries are chosen from all 1D CA, which have the highest dislocation density. The number of new boundaries is determined by Eq. (5), because every new boundary defines appearance of a new dislocation cell, which further can evolve into a subgrain or a grain. 3.2. Rotation of the dislocation cells, subgrains and grains A rotation of crystals is defined by their crystallographic orientation and depends on the applied deformation. Grains that are divided by the new boundary are treated differently. They are rotated with a small angle of a randomly defined value. Unfortunately, rotation of the cells, subgrains and grains cannot be easily identified using experimental data. It requires several simulations in order to determine the dependence of the rotation angle on the crystal orientation and strain. Some simplifications have been made in simulations presented in the next section. At the moment, a dependency on the strain is considered only, while the consideration of the grain crystallographic orientation will be the object of further studies.

ð3Þ

where Ng – number of the grains (cells, subgrains) in unit volume, N0 and N1 – initial and final numbers of the grains, e – accumulative strain, ec – characteristic strain. Parameters identification gave following values: N0 = 0.00121 lm3, N1 = 0.18 lm3, ec = 14.3, n = 0.666. The number was presented in the form of differential equation and as increment of new grains as well:

dNg N1  Ng ¼ 1:69 ðe þ 0:25Þn1 de ec

163

ð4Þ

ð5Þ

The result recalculated into the appropriate frequency demonstrates a good agreement with experimental results and is presented in Fig. 3 by a dashed line. The model [33] is developed in 1D, 2D and 3D versions. Dislocation density changes in each cell of 1D CA with the strain and this is the product of two components: deterministic and stochastic. In the current model, the assumption should be underlined, that during the deformation, the evolution of the dislocation density is dependent on hardening and dynamic recovery. This is the deterministic component. The second component is the normal distribution with a small dispersion. During the deformation, when the dislocation density in the cell walls reaches a critical value, the cells change their state, and LABs appear. The cells in the LAB state influence on their neighborhood. Considering the deformation mechanism as a movement of dislocations, LABs act as the barriers for this move-

4. Simulations and results Simulations were fulfilled in the cellular space of 300  300  300 cells that represents volume of 50  50  50 lm3. Periodic boundary conditions were applied for all faces of the space. The present study was divided into two stages. The initial microstructure with 247 randomly orientated grains was created at the first stage. Then, MaxStrain deformation cycles were simulated. Number of grains was determined with the algorithm described in Section 3.1 and by solving Eq. (5). At the beginning, the first two cycles (four compressive deformation steps) were simulated. The microstructures before, during and after the first cycle are presented in Fig. 5. The distributions of the boundaries disorientation angle are shown in Fig. 6. Different rotation rates were analyzed. As mentioned above, after appearance of a new boundary, two grains separated by this boundary begin to rotate at a random rate. This random rotation in the model is defined by Gauss distribution with expected value of m = 0 and standard deviation s. A value of an additional rotation obtained from the distribution for every grain is related to one deformation pass, then unit of such a rotation rate is degree per one deformation pass, [°]. An appearance of new boundaries is accompanied with an increase of the number of the LABs (Fig. 6a). A maximum of the frequency remains in the same place with low angles while a rotation rate is small enough. An increase of the number of the boundaries from a deformation to deformation leads to the increase of the maximal frequency; and some widening of that peak can be observed. Here, the process of the new boundaries appearance dominates above the cells’ rotations. Then, the number of new

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Fig. 5. Simulated microstructure: (a) – initial, (b) – after the first deformation, (c) – after the first cycle (two deformations).

Fig. 6. Distribution of the boundaries disorientation angle after two cycles of MaxStrain deformation and different rotation rate: (a) s = 1.2°, (b) s = 5.0°, (c) s = 10.0°.

boundaries decreases, the maximal frequency of LABs can remain almost the same (as after the forth deformation in Fig. 6a) or decreases, because rotations of the cell lead to the increase of the number of the boundaries with higher disorientation angles. It can be seen from the experimental data (Fig. 2) as well. For example, the highest frequency of LABs was obtained after the 5th cycle. The peak begins to drift to a higher value of disorientation angle (Fig. 6b and c) when more cycles are simulated or a higher rotation rate is set. The higher rotation rate is, the faster distribution approaches to the random one. Analyzing distributions and their changes during consequent deformation cycles, a significant decrease of the rotation rate should be noted, especially during the first deformation stages. The results for the first two cycles are presented in Fig. 7. The distributions calculated for the initial microstructure, after the first deformation and after the second cycle are presented in Fig. 7a. Here, the distribution after the second cycle obtained during experimental study is presented as well. Different scales are applied according to Section 3.1, because of different scale of experimental data and representative simulated cellular space; the difference between two and three dimensions is also taken into account. Angles over 22° were not analyzed, because they are related to the initial microstructure, although an increase of the frequencies of those angles from cycle to cycle should be noted. But more attention is paid to the low disorientation angles. Some difficulty in a fitting of the simulated distribution to the experimental one is connected with joint changes of the number of the grains and rotation rate. It leads to a situation when parameters of every

deformation influence the changes in the next deformation so every deformation inherits previous changes. The changes are not simply added, their connection is more complex. Moreover, the biggest drop of the rotation rate is received during the first deformation (about 5–7 times). Then, during the next 3–4 deformation passes, rotation rate decreases 2–3 times; and after the 2–3 cycles it remains on the low level, below 1° per cycle. Three-dimensional microstructure on cuboid’s surfaces with the boundaries disorientation angle after two cycles of MaxStrain deformation is presented in Fig. 7b. The two-dimensional representation of top cuboid’s surface is presented in Fig. 7c. It can be compared with the experimental representation in Fig. 1a. They seem to look different, however in terms of the boundaries distribution and the grains size they are close to each other. Next several cycles do not introduce significant changes in the microstructure evolution. However, the microstructure evolution is present during subsequent deformation process. Microstructures after 10, 15 and 20 cycles are presented in Fig. 8. It can be seen that the number of grains and the number of HABs slightly increase. Proposed model allows one also to track the changes in grain shapes with increasing strain – by comparing the simulated microstructures presented in Figs. 5, 7 and 8 respectively, it can be seen that the shape of the grains changes with increasing number of multiaxial deformation cycles, from equiaxed one to more elongated and vice versa. After the first deformation, boundaries appear primarily in the planes, which are parallel to the two easiest slip directions of the grain (Fig. 5b). Because the crystallographic orientation of the grains is different, the slip planes are of different

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165

Fig. 7. Simulated distribution (a), three-dimensional microstructure (b) and cross-section (c) of the cellular space with the boundaries disorientation angle after two cycles of MaxStrain deformation.

Fig. 8. Simulated microstructure after 10th (a), 15th (b) and 20th (c) cycles.

5. Discussion

Fig. 9. Distribution of the boundaries disorientation angle after 20th cycle.

slope. After the second deformation, dislocation cells structure is mainly formed, but grains are elongated. After the 7–8 cycles, the grains become equiaxial.

The simulation results of the microstructure refinement process (recrystallization in situ) during 15 cycles of the MaxStrain deformation confirm that it is possible to apply a very simple model for the proper representation of the evolution of grain boundaries during the SPD. The microstructures and grain boundaries distributions obtained with the computer simulation are quite similar to these observed in the experiments after 2, 5, 7, 10 and 15 cycles of MaxStrain deformation. Although all of the cycles can be modeled in one sequence, some differences can be observed both in the microstructures and grain boundary angles distributions. The differences can be explained by insignificant variation of the initial microstructure of the specimens and natural nonconformity of the MaxStrain method. But the noticeable discrepancies between simulated and real microstructures deformed at higher strains (between 15 and 20th cycles) is observed. Our first assumption is that this discrepansies were connected with a rapid increase of rotation rate after the 15th cycle. As it can be seen in the results presented in Fig. 9, rotation rate was increased in the model for the last 5 cycles from 1 up to 6 and 20° per cycle. Rotation rate varied in these cycles as well, but any combination did not give a result similar to experimental one. Either, the higher number of HABs and lower number of LABs or vice versa can be obtained in simulations. It allows one to draw a conclusion that initial

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microstructure or/and all sequences from the first cycles was different in this case. The grain boundary distribution can be separated into two components, presented by dashed lines in Fig. 9. The first ‘old’ component, random one, is connected with the initial or stable microstructure obtained in first cycles, and the second ‘new’ component represents an appearance of new boundaries during the last deformation cycles. In order to transform new LABs into HABs, relatively high deformation in needed. This is the reason why observed bimodal distribution of grain boundaries is typical for that case. When the number of new boundaries in the cycle decreases, the maximum frequency (Fig. 9) moves in the direction toward higher angles up to the moment when two extremes are merged into one – as it can be seen in Fig. 6. Then, the grain boundaries distribution after 20th cycle of deformation can be explained as follows: firstly, ‘old’ component of the distribution (Fig. 9) had been obtained, and then the second stage of refinement began. Comparing with the other distributions, two important points should be noted: higher number of HABs (the first ‘old’ component) and lower number of boundaries with very low angle, as well as more extensive distribution (higher dispersion) of the second ‘new’ component. It could be assumed, that both components are formed in the very early stages of deformation cycles, which was separated by deformations of subgrain clusters at a low rotation rate. Another statement that can be made assumes that the low third maxima observed for some experimental distributions in Fig. 2 are the results of the same process, which is described in more details for 20th cycle. Besides, the present study allows one to assume that the model should give more accurate results when parameters of both mechanisms, namely new boundaries appearing and the grain rotation, are identified commonly, not separately. It will be considered in the further studies. These two models will be probably modified as well. 6. Summary In the current work, the computer model based on threedimensional Frontal Cellular Automata for the simulation of the grain refinement during the multiaxial compression was developed. The development of grain boundaries and disorientation angles during the subsequent cycles of the MaxStrain deformation was simulated and compared with the experimental results for high SFE material i.e. microalloyed steel. The proposed model that consists of two parts (the formation of new low-angle grain boundaries and the rotation of the grains during the deformation) offers a very attractive numerical tool for the prediction of the microstructure evolution during SPD processing and may be used for theirs optimization. A relatively good agreement observed between the modeling and SEM/EBSD results demonstrates a high potential of the proposed modeling approach in the prediction and optimiza-

tion of the highly refined microstructures obtained in the Severe Plastic Deformation processes. Presented model is still being developed toward a further improvement of its accuracy and an extension of its use for other materials and processes. Acknowledgment Polish National Science Centre research project no. DEC-2012/ 05/B/ST8/00215 is acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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