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Reorganization of cellular space during the modeling of the microstructure evolution by frontal cellular automata
Computational Materials Science 60 (2012) 153–162
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Reorganization of cellular space during the modeling of the microstructure evolution by frontal cellular automata Dmytro S. Svyetlichnyy ⇑ AGH University of Science and Technology, Department of Metals Engineering and Industrial Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 13 December 2011 Received in revised form 12 March 2012 Accepted 14 March 2012 Available online 14 April 2012 Keywords: Cellular automata Microstructure Recrystallization Deformation Flat rolling
a b s t r a c t The prediction of the microstructure evolution is one of the most significant problems in materials science. The objective of this paper is development of principles of the cellular space reorganization and their implementation into the three-dimensional frontal cellular automata (FCA) for modeling of the microstructure evolution during the multi-stage deformation. Motivation for the reorganization and principles governing this process are presented in the paper. The FCA cells do not keep the form of undistorted cubes during the simulation, but they get deformed. That leads to the necessity of space reorganization. Different conditions on the boundaries of the cellular space are taken into consideration as they are closely connected with the developed algorithms of the reorganization. These two variants of the reorganization are described in detail. The first one can be used when cellular space can be reduced; and the second one, when the reduction of modeled volume is not acceptable. A short description of frontal cellular automata is presented in this paper. The model consists of two parts: deformation and microstructure evolution. The independence of the grain growing from the shape and the sizes of the cell is ensured by so-called ‘‘virtual front tracking’’ algorithm. The microstructural part of the model simulates two phenomena: nucleation and growth of new grains. In this paper only static recrystallization is considered. When the recrystallization is simulated, the nucleation and the grain boundary migration depend on the deformation parameters such as: temperature, strain, strain rate, dislocation density and crystallographic orientation. This phenomenon, along with the deformation, can be modeled over a wide range of multi-stage deformation processes. The process of flat rolling is chosen as the simplest example. The data needed for the FCA simulation is received from the simple numerical calculations. The results of the simulation of the microstructure evolution obtained during the last five passes of the flat rolling are also presented in the paper. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The prediction of the microstructure is one of the most important problems in materials science. There are different methods used for the modeling of the microstructure evolution. Cellular automata (CA) models [1], Monte Carlo Potts models [2], the finite element method (FEM) based models [3], the phase field [4,5], multi-phase-field [6] models, the front tracking method [7,8] and the vertex models [9] are among them. The application of the CA models, for the simulation of the different phenomena in the materials, has become incredibly important during the last years. CA are used for modeling of crystallization (solidification) [10–13], dynamic and static recrystallization [14–17], phase transformation [16,18,19], cracks propagation [20], severe plastic deformation [21–23], etc. The main asset of the CA is the ability for a close correlation of the real microstructure with its digital representation ⇑ Tel.: +48 126172580, 692983805; fax: +48 126172612. E-mail addresses: [email protected], [email protected] 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.029
on a micro-, meso- and macro-scale. One of the 3D CA modifications, known as the frontal CA (FCA) [17], which is capable to accelerate calculations, is used for the simulation of the microstructure evolution in the paper. The deformation is often neglected during modeling of the microstructure evolution by the cellular automata; whereas for the finite elements method (FEM) [3], models based on finite differences (FDM) or multi-phase-field [6] deformation are one of the main variables and results of the modeling. Lack of cell distortion during the modeling of microstructure evolution can be obtained only in the process without deformation. Large deformation leads to significant distortion of the elements, mesh in FEM or FDM, as well as cells in CA. A procedure known as ‘‘remeshing’’ is widely used in FEM codes to improve accuracy of the calculation when geometrical parameters of the elements become unfavorable. The similar procedure is to be implemented into the CA code. The objective of this paper is the development of the principles of the cellular space reorganization and their implementation into
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three-dimensional FCA for modeling of the microstructure evolution during the multi-stage deformation. Simulation of a separate phenomenon or a stage of technological processes can be carried out in the frame of simple CA models. Simulation of the same sequences of the technological stages can be carried out in a combination of CA with other numerical methods, such as the finite element method, the finite differences, the discrete element method, etc. Although the joint methods improve the accuracy of the simulations of the forming processes, the simplest modeling scheme without other advanced methods has been chosen since the main goal is not to obtain precise results, but present the possibility of the space reorganization.
2. General FCA model Accurate modeling of the microstructure evolution during the forming processes is a complex task which relies on the choice of appropriate methods. The choice depends on intended effect, accuracy, computation costs, etc. The cellular automata are recognized as one of the most optimal tools for the microstructure modeling, taking into account the IT costs and the results which can be achieved. They can be used for the microstructure evolution simulation, both as an all-sufficient tool as well as in connection with other methods. In the first case, the simulations are limited by the theoretical studies for the set of deformation conditions or for the process whose parameters can be easily calculated in an analytic way [24]. The use of CA combined with other methods makes it possible to obtain more accurate and more reliable results. CA are often jointed with the finite differences (FDM) or the finite element method (FEM). FEM is known as one of the best methods for the modeling of forming processes. Taking into account the role in the model, the schemes of the mutual use of CA and FEM can be divided into two groups. The first one contains a solution in which the simulations are carried out independently, without any feedback. The systems with the co-operation of both components can be included in the second group. One component is primary; another is secondary, and it is served to deliver the parameters and variables for the proper calculations by the primary method. CA can be secondary, only when it is of a simple structure and does not require high calculation costs, such as memory and calculation time. CA used for the microstructure evolution simulations, especially treedimensional ones, cannot be applied as the secondary method because of IT requirements. Thus, CA play the primary role in the theoretical research on the uneven deformation at the level of crystals as well as the properties of polycrystalline structures. The variant of independent simulations by FEM and CA, without any feedback, is chosen. The forming process is modeled by FEM on the macro-scale. The results of FEM simulation are used by CA for the microstructure modeling on the micro-scale. It is the so-called post-processing to obtain information about the microstructure. An arbitrary FEM code or the literature data can be used in such a strategy. It is enough to meet the requirements for the data needed for the post-processing calculations. The post-processing is described below. It is worth noting that main variables as time, temperature and strain rate tensor are required for the calculations. In this paper, the flat rolling is chosen to present the reorganization possibilities of the model based on FCA. The rolling process parameters was obtained by analytical method [24]. The parameters were the basis for further calculations by FCA, and they are presented below. The general model of the microstructure evolution during the forming processes is based on FCA and covers aspects such as the microstructure evolution, creation of the initial microstructure, deformation, recrystallization and phase transformation. All these aspects are presented below in the following sections.
2.1. Boundary conditions In order to understand the principles of space organization it is important to take into account boundary conditions. The boundary conditions determine how the cellular automata behave in cells that are located on the edge of cellular space. Historically, the boundary conditions for CA were first established for a moving particle. Therefore, the first classification bore traces of interaction of particles (or radiation) with a surface. There have been three types of the interaction introduced: reflection, transmission and absorption. The boundary conditions with the reflection imply perfectly elastic collision; the transmission means that the particle passes through the boundary and appears on the opposite side. When absorption is introduced, a particle leaves the space and ceases to exist. The space with reflection and transmission is closed and one with the absorption is open or half open, because particles often move only outwards. However, in CA for microstructure modeling a bit different classification of the boundary conditions has been adopted. The boundary conditions with the transmission are named as periodic conditions; and ones with the absorption are the so-called open conditions. In fact, other conditions cannot be found in the literature concerning the modeling of the microstructure by CA. There are two examples of microstructure presented in Fig. 1. They are obtained by two-dimensional cellular automata with the open (Fig. 1a) and periodic conditions (Fig. 1b). Since the space with periodic conditions can be considered as the surface of the torus (hipertorus) and the boundaries are the lines of virtual (not real) discontinuity, some grains are presented several times on the opposite sides of the space but they remain undivided. Contrary, all grains in the open space are presented only once. Two main differences between open and periodic conditions based on ‘‘boundary effect’’ can be noted. The first one can be seen in the final structure. Periodic conditions make the fine grains which are closer to the boundaries larger and the coarse grains smaller. The other difference is important for processes of grain growth (for example kinetics of recrystallization). Absence of ‘‘boundary effect’’ in periodic conditions, because of the equality of all cells in the space (nonexistence of the real boundaries), makes the process more realistic. That is why mainly periodic conditions are used in the modeling. Two additional boundary conditions are used in the model based on FCA. The first one is periodic conditions with displacement and the second one is full open conditions. The idea of the periodic conditions with displacement is that the neighboring cell is not directly on the opposite side, but with some horizontal, vertical, lateral displacement or arbitrary combination. For example, let us consider the space with nx, ny and nz cells on appropriate sides. The cell with coordinates {i = nx, j, k} lies on the boundary of the space and has the neighboring cell in direction x. The neighboring cell in the space with ordinary periodic conditions is of coordinates {1, j, k}, while the neighboring cell in the space with periodic conditions with displacement is of coordinates {1, j + dj, k + dk}, where dj and dk are appropriate displacements. The maximal number of displacements, which can be applied in the model, is six, two possible displacements for three directions. The displacements can be fixed or individual. However, in the algorithm presented, in the following section the only displacement along the axis x, equal to half of the space length (diz = nx/2), is used when neighboring cell in direction z is considered. The main weakness of the ‘‘open’’ boundary conditions is their half openness, when boundaries are transparent only in one direction, outward, and are opaque inward. The inner grains can grow outward the space, but any outer grain cannot grow inward the space. Such an ‘‘open’’ condition is named half open. An idea of full open condition is the following: when a growing grain reaches the
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Fig. 1. Microstructure obtained by two-dimensional cellular automata with the open (a) and periodic (b) boundary conditions.
boundary, it can appear not only on opposite side, but also on arbitrary side, including the same side. Now it is not the same grain, but a new one. The side, which new outer grain grows into, and its coordinates (including the distance from the surface) are set at random. The combination of three types of the boundary conditions used in the model presented in the paper: half open, periodic and periodic with displacement. 2.2. Deformation and space reorganization The CA in three-dimensional version should be applied for the modeling of the microstructure evolution in the forming processes because the structure after the deformation demonstrates different images at the three perpendicular cross-sections. Deformation in CA is rarely concerned, but if even it is not neglected, any manipulations of the sizes or the shape of the cell or the cellular space do not take place. The real deformation is the problem that cannot be avoided in CA simulations, especially when the multi-stage deformation is modeled. The deformed structure must be used in the further modeling. The simplest solution is an introduction of the cell distortion into the model, when the sizes and the shape of the cells are to be changed according to the strain tensor. Consideration of the deformation in FCA is described in detail elsewhere [19]. Anisotropy of the space with deformed cells can be obtained by the use of the so called ‘‘virtual front tracking’’ algorithm. The algorithm is based on the idea presented by Rappaz and Gandin [10], Sanchez and Stefanescu [25] and Burbelko et al. [26]. Calculations in the present FCA model can begin with the cubical cells (but not necessarily). During the simulation with deformation they do not remain the same for long time. The deformation (for example during the flat rolling) effects not only on the shape and sizes of the cell (Fig. 2), but also on the grade of anisotropy of modeled space and representative volume. The ratio of the cell sides becomes too high, calculations are decelerated and quality of obtained structure is deteriorated. During the modeling with the use of the model based on finite element method (FEM) the procedure named ‘‘remeshing’’ is often applied in the same cases. A similar procedure should be used in the CA as well. In the model, while the strain is not large enough, FCA cells are deformed according to the real deformation. When the ratio reaches a preset value, the cellular space must be reorganized to obtain the shape of the cells (as well as the whole cellular space) far close to a cubical one. Such reorganization has been shortly presented for the first time at a conference [27], and it is described below in detail.
Fig. 2. The shape of the cell during the flat rolling without space reorganization.
There are two variants of the reorganization developed for the model. They can be chosen manually before simulation or automatically during the simulation. The first simplest variant is applied when deformation is accompanied with the microstructure refinement. The scheme of such reorganization is shown in Fig. 3. Here, the grey color represents the whole cellular space, while the white one – several cells. After the deformation or before the next deformation, ratios of side sizes (length/height, width/height) of the cell (or the representative volume) are tested. If either ratio becomes about 2.0, all space is halved. The first half of the cellular space is left for further simulations, whereas the second half is removed from the model. Each cell in the remaining part is halved as well. Both halves of the cell become the new cells and inherit all properties of the parent cell. This reorganization cuts the volume of the cellular space into half, but the number of cells in the space remains the same. The decrease of the volume limits the use of the approach when it does not lead to a significant reduction of the number of grains from one cut to another. As the periodic boundary condition is used for the more realistic modeling of the microstructure evolution, the halving must be taken into account. Actually, the halving in the space with the periodic boundary condition means two cuts. Then, we receive two opposite sides, which cannot be connected, because they have no coincident microstructure. As a result, open boundary conditions must be applied temporarily on these sides. They are not permanent conditions but transient. Those conditions evolve into partly-open, partly-periodic condition, and finally into the purely periodic conditions. Algorithm of such an evolution is the next (Fig. 3a). All grains on both sides of the space, which have been cut by halving, are perma-
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(a)
(b)
Fig. 3. Scheme of the halving of the cellular space (a) and boundary conditions before and after the halving (b).
nently considered as ones with the open boundary condition, and can grow only inside the space (upper right grain in Fig. 3a). They cannot appear on the opposite side. Yet every other grain which exists in the current structure (bottom left grain in Fig. 3a) or which appeared after the halving can grow not only inside the space, but also outward with its appearance on the opposite side of the space (bottom right). As a result, the boundary conditions depend on the properties of the grains, not on the space. The new grains receive the periodic boundary conditions, while the old grains stay with the open ones. The second variant is more complex and can be used when the volume of space cannot be reduced (Fig. 4). As in the first variant, after the side ratio reaches preset value (2–4), the space is halved, but then the halves are put together, one on the other. Such procedure supposes four cuts and at least two bondings. If the sides are known a priori to be cut and bonded, the boundary conditions can be organized before modeling. Otherwise, at first, the modeling is carried out with the periodic conditions up to the halving, and then, the modeling is repeated with other boundary conditions. For the cutting and bonding, the conditions must be as the ones presented in Fig. 4, where different lines determine coincident microstructure. A grain is demonstrated as an example. Thus, for the short sides, the simple periodic conditions are applied, while for the long sides periodic conditions with displacement are used. The displacement equals to a half of the space length (it is constant, measured in the number of cells). After cutting and bonding on the former long side, the simple periodic conditions are obtained. On the former short side periodic conditions with displacement are formed. The shape of the cells also changes. They are combined by the long sides and then they are halved as it is shown in Fig. 4. Cells which belong to the different grains are combined, taking into the consideration the neighboring cells. An example of modeled microstructure before and after the cutting and bonding is presented in Section 4. As the procedure can be repeated, the appropriate conditions must evolve either into the simple periodic conditions or into the periodic conditions with displacement, as it is described above for the first variant of the reorganization (the halving). Some modification of halving is used in calculations presented in the paper. Instead of purely periodic boundary conditions the
periodic boundary conditions with displacement are always applied. It allows for automatic choice of the reorganization method. 2.3. The cellular automaton Frontal cellular automata (FCA) [17,19] are used as the module of model presented in the paper. The use of the frontal cellular automata instead of the conventional ones makes it possible to reduce the computation time significantly, especially for the threedimensional models, as the large regions are excluded from the calculations in the current step and only the front of the changes is studied [17]. The multi-states cell automaton is presented schematically in Fig. 5 [19]. The set of the states Q = {q0, q1, q2, q3, q4} comprises the initial matrix state q0, the ‘‘frontal cell’’ q1, the ‘‘boundary cell’’ q2, the ‘‘cell inside the grain’’ q3 and the transient states q4. The set of conditions {I0–I4} is used by the transition rules. The transition rules define the next qi+1 state of the cell on the basis of the current qi states of the cell and the cells in its neighborhood and the transition condition Ik. Such rules are commonly presented either in a form of a table or by description. Shortly, the conditions are of the following meanings: I0 is the nucleation, I1 – the time delay, which allows to control the grain growth rate, I2 and I3 determine whether the cells are on the boundary or inside the grain (I3 ¼ I2 ) and I4 means that the growing grain reaches the current cell and involves it in the growing process. The automaton (Fig. 5) is characterized by a closed circuit of the states: q1 ðq2 ^ q3 Þq4 . The states q2 and q3 are not only the final states, but also the initial states for the next cycle of the modeling, and the states can be repeated several times. This property is useful for the modeling of some processes, for example, multi-pass flat rolling, where deformation in the passes alternates with the recrystallization in the pauses. Two kinds of neighborhood are used by the FCA. The Moore neighborhood is applied in the algorithm of the boundary motion during the grain growth simulation (state q1 and condition I4), while the unmovable grain boundaries are defined through von Neumann neighborhood (states q2, q3 and condition I2, I3). FCA works with a varied time step, which is defined by the minimal time for the boundary migration through the cell. The FCA can
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Fig. 4. Scheme of the cutting and bonding.
q2 I0 I4 q0
I0 I4
q4 I4
Principles of modeling such phenomena as initial microstructure creation, recrystallization, and phase transformation by FCA are described in detail elsewhere [19]. They are realized in phenomenological module. As the accuracy of the microstructure evolution simulation is not the main goal of the paper, the only modeling of static recrystallization is shortly presented in the section. The model of recrystallization realized in FCA consists of two parts, which are the nucleation and the grain growth. Number of nuclei is calculated according to:
I2 I1
q1
I3
q3 Fig. 5. Cellular automaton (q – states, I – transition conditions).
be used for the simulation of static and dynamic recrystallization, phase transformation and solidification. Yet the example presented in the paper uses only static recrystallization. Application of the model to the hot flat rolling is described in one of the sections below. 2.4. The phenomenological and material modules Material module contains parameters for material and calculates current values of dislocation density and flow stress. The model of the dislocation density is described in detail elsewhere [28,29]. The equation considers both the hardening and the softening. It takes into account the strain rate e_ , the temperature T and the dislocation density q, according to the following equation, which is applied for all grains:
q_ ¼ a1 e_ n1 exp
Q1 Q2 a2 e_ n2 qm1 exp RT RT
ð1Þ
where Q1 and Q2 – the activation energies, a1, a2, n1, n2, m1 – the material constant. The flow stress is calculated according to equations:
rs ¼ r0 þ albðqav Þ1=2
ð2Þ
where r0 – the stress necessary to move the dislocation in the absence of the other dislocations, a – the constant, l – the shear modulus, b – the Burgers vector and qav – the average dislocation density.
NV ¼ aN enN exp
QN Vr RT
ð3Þ
where e – strain, T – temperature, R – gas constant, QN – activation energy, aN, nN, QN – the material constants, Vr – representative volume. The grain growth of recrystallized grain is defined by a boundary migration rate v, which depends on driving force of recrystallization p and grain boundary mobility m. The disorientation angle # is taken into account as well: v = mpf(#). The difference of the dislocation density of the old qold and new qnew grains is the driving force are calculated on the basis of stored dislocation energy: p = 0.5 lb2 (qold–qnew ). A curvature of the grain boundaries is neglected as another driving force because during the process of the recrystallization it is far smaller than the force from the difference of dislocation density. The grain boundary mobility m depends on the self-diffusion coefficient and therefore, it is defined by Arrhenius’ law:
m ¼ am exp
Qm RT
ð4Þ
where am, Qm – the material constants. 3. Flat rolling The frontal CA model is adapted by the technological module to the simulation of microstructure evolution during the hot flat rolling. The calculations are carried out for reverse mill with work rolls
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of 1000 mm diameter. The plate to be simulated is milled to 40.0 mm thickness and 10 m length. The rolling speed is 5 m/s. Cooling in air is assumed. Inter-pass time or time of pauses between the exit from the previous pass and the entrance to the next one is 2.3 s. Rolling pass schedule (Table 1 [24]) contains information about thicknesses, temperature and other process parameters. The schedule is one of the options of process carried out in one of the polish reverse rolling mills. It is a basis for modeling. The microstructure evolution is simulated for the point on the axis of symmetry of the plate. Steel, which on chemical composition is close to steel 45, is chosen as the modeled material. For each pass, technological module calculates strain, strain rate, length and time of rolling, inter-pass time and so on. Sizes of the cells before and after the pass are calculated in order to determine the time increment and a number of iteration during the rolling simulation. Temperature is assumed to be constant during the deformation, as well as strain rate. By the end of deformation, a number of new grains (nuclei) are calculated. Static recrystallization is modeled only, because even at 1100 °C and strain rate of 10 s1 peak strain is above 0.35. According to modeled condition (all temperatures are below 1100 °C, stain about 0.3 and strain rate about 90 s1), dynamic recrystallization does not take place. After deformation, the growth of recrystallized grains is simulated without the additional nucleation. Temperature changes are linearly approximated from pass to pass. A pause after the pass is a sum of inter-pass time and time of the rolling calculated on the basis of length of the plate and the rolling speed. 4. Simulation results Initial microstructure is created by the FCA with the following parameters (Fig. 6a). FCA space contains nx ny nz = 500 500 500 cells with the representative volume of ax ay az = 0.4 0.4 0.4 mm3. Initial microstructure include 200 grains, it gives
the average grain size of about 70 lm. Axis x is a rolling direction, y is a lateral direction, z is an axis along the thickness of the plate. Size y of the FCA space remains constant, x is elongated and z is reduced. Two variants of visualization can be chosen. The first one allows for colored representation, when forty five colors present the crystallographic orientation, but for paper another variant is preferred. Grey scale presents different states of microstructure. The old grains are charcoal grey, new recrystallized grains are light grey. Grains boundaries are black, migrated boundaries of the new growing grain are white. The simulation results are presented as isometric drawings with the microstructure seen on the cuboids’ faces (Figs. 6–11) and as graphs showing the changes of the average grain size and the flow stress (Figs. 12 and 13). The microstructure is presented in time, which is connected with the beginning or the ending of the deformation. Space reorganization is done automatically before the passes according to the following logical sequence:
ðng > ngmax Þ ^ ðax ¼ maxðax ; ay ; az ÞÞ ) HX ðng > ngmax Þ ^ ðay ¼ maxðax ; ay ; az ÞÞ ) HY ðng > ngmax Þ ^ ðaz ¼ maxðax ; ay ; az ÞÞ ) HZ
ð5Þ
ðax > 3az Þ ) BXZ ðax > 2az Þ ^ ðng > ngmin Þ ) HX ðay > 2az Þ ^ ðng > ngmin Þ ) HY where ax, ay and az – sizes of cellular space, ng – number of grains, ngmin and ngmax – minimal and maximal number of grains (ngmin = 500, ngmax = 4000), BXZ – cutting of the length of the cellular space (ax/2) and bonding in the thickness (2nz), HX, HY and HY – halving of the length (ax/2), the width (ay/2) and the thickness (az/ 2), respectively. A sequence of simulation and some results are presented in Table 2. Before the second pass, space is not reorganized because small number of grains (ng = 453 < ngmin = 500) does not allow
Table 1 Rolling schedule [24]. Pass number
Entrance thickness h0 (mm)
Exit thickness h1 (mm)
Temperature T (°C)
Reduction r (%)
Strain rate e_ (s1)
Pause after pass (s)
1 2 3 4 5
40.00 27.18 18.55 12.75 8.95
27.18 18.55 12.75 8.95 6.49
1098 1087 1061 1010 928
32 31.7 31.3 29.8 27.5
90.6 90.1 89.4 87.3 83.6
5.24 6.6 8.7 11.2 14.6
Fig. 6. Microstructure before (a) and after (b) the first pass. Representative volume Vr = 0.064 mm3.
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halving, and the sides ratio is not big enough for bonding (ax = 0.588 < 3az = 3 0.272 = 0.716). Then before the third, forth and fifth passes bonding or appropriate halving are fulfilled. After the deformation, small white points in the some figures can be observed (Figs. 6b, 7b, 9b, 11a). They represent nuclei. Because of high strain rate (80–90 s1) and temperature (1000– 1100 °C), static recrystallization is accompanied with simulated rolling. Complete recrystallization after the passes, except the last two, is observed as a consequence of relatively high temperature in
159
the first passes and long inter-pass time. After the last (fifth) pass and cooling, as a result of lower temperature, recrystallization is stopped (Fig. 11b). White boundaries and light grey grains mean that recrystallization is not finished. The procedure of cutting and bonding can be seen in Fig. 8. After cutting the nearest half, it goes to the top. The procedure (which is described in detail in Section 2.2) gives microstructure which is coincident in the place of joining. That is why, in Fig. 8 lines (or surfaces), where two parts are joined, cannot be observed. Halving
Fig. 7. Microstructure before (a) and after (b) the second pass. Representative volume Vr = 0.064 mm3.
Fig. 8. Microstructure after recrystallization after the second pass before (a) and after (b) cutting and bonding. Representative volume Vr = 0.064 mm3.
Fig. 9. Microstructure before (a) and after (b) the fourth pass. Representative volume Vr = 0.032 mm3.
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Fig. 10. Microstructure after recrystallization after the forth pass before (a) and after (b) the halving in two axes. Representative volume Vr is reduced from 0.032 to 0.008 mm3.
Fig. 11. Microstructure after the fifth pass before (a) and after (b) recrystallization. Representative volume Vr = 0.008 mm3.
Fig. 12. Average grain size during the rolling.
Fig. 13. Flow stress during the rolling.
after the forth pass in two directions (in length and width) is presented in Fig. 10. A quarter is used in further simulation, the representative volume Vr is reduced from 0.032 to 0.008 mm3.
Changes of the average grain size are calculated during the whole process in the representative points. The average grain size dav is considered to be proportional to the ratio of the representa-
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D.S. Svyetlichnyy / Computational Materials Science 60 (2012) 153–162 Table 2 Sequence of the simulation. Pass number
Length width thickness, ax ay az (mm)
Representative volume Vr (mm3)
Number of grains before/after
Space reorgani-zation
Figures
1 2 3 4 5
0.4 0.4 0.4 0.588 0.4 0.272 0.431 0.4 0.371 0.314 0.4 0.255 0.224 0.2 0.179
0.064 0.064 0.064 0.032 0.008
200/453 453/576 576/994 674/1464 625/4850
– BXZ CX CX + CY –
Fig. 6 Figs. 7 and 8 – Figs. 9 and 10 Fig. 11
tive volume Vr to the entire area of the surface S of all the grains dav / V r =S:
dav ¼
Vr ðk1 n1 þ k2 n2 ÞSc
ð6Þ
where Vr – the representative volume of the cellular space, lm3; n1 and n2 – the number of the cells in the state q1 and q2; Sc – the average area of the cell’s faces, lm2, k1 and k2 – coefficients. There are several factors causing some difficulties, which influence the accuracy of the grain size. They are connected with the discretization of the cellular space. The real area S of the boundaries is less than the discrete one. The area of a cell’s faces Sc is not equal to the others and does not remain constant during the simulation (due to the deformation). The number of cells (not faces of the cells) is counted for the surface calculation. The frontal cells q1 (n1) and the boundary cells q2 (n2) are determined by different neighborhoods. What is more, the frontal cells are one-sided, but the immobile boundaries are two-sided i.e. firmed by the cells twice. By taking into account the above mentioned factors, appropriate coefficients (k1 and k2) are introduced. Coefficient k2 is determined in the way that the average grain size dav calculated for a structure without cells in state q1 according the equation:
dav ¼
rffiffiffiffiffiffiffiffiffiffiffi 3 6V av
p
sffiffiffiffiffiffiffiffiffi 3 6V r ¼ png
sented as a function of the accumulated strain (Fig. 13). A growth of the flow stress is observed from the pass to pass. It is caused by a decrease of temperature.
ð7Þ
gives the same results as Eq. (6). Here Vav is an average volume of the grain. The coefficient k1 is determined by the comparison of the structure with the cells in state q1 with the same structure, in which cells in the state q1 are replaced by the cells in state q2 or q3, taking into account different neighborhood and other factors. Accuracy of the average grain size calculations is evaluated at about 10%. Nevertheless, such calculations demonstrate the changes of the average grain size, not only qualitatively, but also quantitatively, describing the microstructure evolution. A change of grain size is demonstrated in Fig. 12. Static recrystallization is not completed after the fifth pass which is seen in Fig. 12, as well as in Fig. 11b. The average grain size reaches the minimal value because new recrystallized grains still do not fill the whole space and old grains do not disappear completely, but remain reduced significantly. During almost every pass, a decrease of grain size can be observed. The results can be explained by the method of calculation of the grain size (6). While volume is not changing during the deformation, the surface of the boundaries of the deforming grains is increasing. It leads to the decrease of the average grain size (6) during the pass. Additional abrupt decrement of the grain size, which can be seen in Fig. 12, is explained by space reorganization. Cubical cells after reorganization allows for more precise calculation of the surface of the boundaries then elongated or distorted cells. Generally, surface calculated after space reorganization increases, therefore the average grain size decreases abruptly. In the further it is anticipated if not eliminate this drawback, at least reduce the magnitude of this drop. The flow stress, which is calculated as described above (in Section 2.4, Eqs. (1) and (2)) is studied as well. The flow stress is pre-
5. Summary A proper simulation cannot be carried out without taking into consideration the real deformation. Introduction of the deformation into the model based on cellular automata causes several problems connected with distortion of the cells and the whole cellular space. It mainly increases anisotropy of model space. Anisotropy can be reduced by front-tracking algorithm, but the greater difference in resolution in different directions impedes precise calculations. Therefore, two algorithms for space reorganization have been developed and they are described in the paper. The first of them is ‘‘halving’’, and the second one is ‘‘cutting and bonding’’. Firstly, the boundary conditions have been enriched by the periodic condition with displacement. Then boundary conditions are assigned to the grains, not to the space (or the cells). It has opened a way to a more free operation with the space and to two algorithms for space reorganization. Both algorithms consider the cutting of the space. Cut grains lose their parts and boundaries of the space remain open for them. All other grains stay in the space with periodic conditions. For halving algorithm periodic conditions before and after the cutting can be with and without displacement; conditions without displacement lead only to the simpler algorithm. For the algorithm with bonding the only periodic conditions with displacement must be applied before the cutting. Despite some complication of the first algorithm, the periodic boundary conditions with displacement have been accepted for both algorithms. It has removed an indispensability of preliminary definition of the kind of reorganization and appropriate boundary conditions and it also makes it possible to reorganize automatically in view of current simulation state. To present the possibility of the reorganization algorithms, they have been implemented into the frontal CA model, which has been adapted to the simulation of the microstructure evolution during the multi-stage flat rolling process. Deformation and static recrystallization are considered for the five passes alternated by air cooling. Microstructure, the average grain size and flow stress are presented as the simulation results. The presented results favored the conclusion that the FCA model can be applied for the simulation of the microstructure evolution during the multi-stage deformation. Further research is directed to develop a material module in order to obtain more realistic results of microstructure evolution during multi-stages deformation processes. References [1] C.H.J. Davies, Scripta Mater. 36 (1997) 35–40. [2] E.A. Holm, G.N. Hassold, M.A. Miodownik, Acta Mater. 49 (2001) 2981–2991. [3] M. Bernacki, Y. Chastel, H. Digonnet, H. Resk, T. Coupe, R.E. Loge, Development of numerical tools for the multiscale modeling of recrystallization in metals, based on a digital material framework, Comp. Meth. Mater. Sci., 7 (2007) 142– 149.
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[4] D. Fan, L.Q. Chen, Acta Mater. 45 (1997) 611–622. [5] R. Kobayashi, J.A. Warren, W.C. Carter, Physica D 140 (2000) 141–150. [6] T. Takaki, Y. Hisakuni, T. Hirouchi, A. Yamanaka, Y. Tomita, Comput. Mater. Sci. 45 (2009) 881–888. [7] C.V. Thompson, H.J. Frost, F. Spaepen, Acta Metall. 35 (1987) 887–890. [8] H.J. Frost, C.V. Thompson, C.L. Howe, J. Whang, Scripta Metall. 22 (1988) 65–70. [9] D. Weygand, Y. Brechet, J. Lepinoux, Adv. Eng. Mater. 3 (2001) 67–71. [10] M. Rappaz, C.-A. Gandin, Acta Metall. Mater. 41 (1993) 345–360. [11] D. Raabe, Acta Mater. 52 (2004) 2653–2664. [12] A.A. Burbelko, W. Kapturkiewicz, D. Gurgul, Comp. Meth. Mater. Sci. 7 (2007) 182–188. [13] D.S. Svyetlichnyy, Modeling of macrostructure formation during the solidification by using frontal cellular automata, in: A. Salcido (Ed.), Cellular Automata – Innovative Modelling for Science and Engineering, InTech, Croatia, 2011, pp. 179–196. [14] P.J. Hurley, F.J. Humphreys, Acta Mater. 51 (2003) 3779–3793. [15] M. Qian, Z.X. Guo, Mater. Sci. Eng. A 365 (2004) 180–185. [16] M. Kumar, R. Sasikumar, P. Kesavan Nair, Acta Mater. 46 (1998) 6291–6303. [17] D.S. Svyetlichnyy, Comput. Mater. Sci. 50 (2010) 92–97. [18] P. Macioł, J. Gawa˛d, R. Kuziak, M. Pietrzyk, Int. J. Multiscale Comp. Eng. 8 (2010) 267–285.
[19] D.S. Svyetlichnyy, ISIJ Int. 52 (2012) 559–568 (in publish). [20] S. Das, E.J. Palmiere, I.C. Howard, CAFE: a tool for modeling thermomechanical processes, in: Proc. Int. Conf. on ‘Thermomech. Processing: Mechanics, Microstructure & Control’, Sheffield, University of Sheffield, UK, 2002, pp. 296–301 (23–26 June). [21] D. Svyetlichnyy, J. Majta, K. Muszka, Steel Res. Int. 79 (2008) 452–458. [22] D.S. Svyetlichnyy, Mater. Sci. Forum 638–642 (2010) 2772–2777. [23] D. Svyetlichnyy, J. Majta, K. Muszka, Ł. Łach, AIP Conf. Proc. 1315 (2010) 1473– 1478. [24] D. Svyetlichnyy, Zastosowanie Technik Teorii Sterowania I Sztucznych Sieci Neuronowych W Modelowaniu On-Line Walcowania Wyrobów Płaskich, Politechnika Cze˛stochowska, Cze˛stochowa, 2004 (in Polish). [25] L.B. Sanchez, D.M. Stefanescu, Metall. Mater. Trans. A 35 (2004) 2471–2485. [26] A.A. Burbelko, E. Fras´, W. Kapturkiewicz, D. Gurgul, Mater. Sci. Forum 649 (2010) 217–222. [27] D.S. Svyetlichnyy, Modeling of microstructure evolution of steel during the rolling by using frontal cellular automata, in: Proc. 3rd Int. Conf. on ‘Thermomechanical Processing of Steels’, Padua, Italy, Associazione Italiana di Metallurgia, Milano, 10–12 September 2008. [28] D.S. Svyetlichnyy, ISIJ Int. 45 (2005) 1187–1193. [29] D.S. Svyetlichnyy, Mater. Sci. Technol. 25 (2009) 981–988.
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Reorganization of cellular space during the modeling of the microstructure evolution by frontal cellular automata Dmytro S. Svyetlichnyy ⇑ AGH University of Science and Technology, Department of Metals Engineering and Industrial Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 13 December 2011 Received in revised form 12 March 2012 Accepted 14 March 2012 Available online 14 April 2012 Keywords: Cellular automata Microstructure Recrystallization Deformation Flat rolling
a b s t r a c t The prediction of the microstructure evolution is one of the most significant problems in materials science. The objective of this paper is development of principles of the cellular space reorganization and their implementation into the three-dimensional frontal cellular automata (FCA) for modeling of the microstructure evolution during the multi-stage deformation. Motivation for the reorganization and principles governing this process are presented in the paper. The FCA cells do not keep the form of undistorted cubes during the simulation, but they get deformed. That leads to the necessity of space reorganization. Different conditions on the boundaries of the cellular space are taken into consideration as they are closely connected with the developed algorithms of the reorganization. These two variants of the reorganization are described in detail. The first one can be used when cellular space can be reduced; and the second one, when the reduction of modeled volume is not acceptable. A short description of frontal cellular automata is presented in this paper. The model consists of two parts: deformation and microstructure evolution. The independence of the grain growing from the shape and the sizes of the cell is ensured by so-called ‘‘virtual front tracking’’ algorithm. The microstructural part of the model simulates two phenomena: nucleation and growth of new grains. In this paper only static recrystallization is considered. When the recrystallization is simulated, the nucleation and the grain boundary migration depend on the deformation parameters such as: temperature, strain, strain rate, dislocation density and crystallographic orientation. This phenomenon, along with the deformation, can be modeled over a wide range of multi-stage deformation processes. The process of flat rolling is chosen as the simplest example. The data needed for the FCA simulation is received from the simple numerical calculations. The results of the simulation of the microstructure evolution obtained during the last five passes of the flat rolling are also presented in the paper. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The prediction of the microstructure is one of the most important problems in materials science. There are different methods used for the modeling of the microstructure evolution. Cellular automata (CA) models [1], Monte Carlo Potts models [2], the finite element method (FEM) based models [3], the phase field [4,5], multi-phase-field [6] models, the front tracking method [7,8] and the vertex models [9] are among them. The application of the CA models, for the simulation of the different phenomena in the materials, has become incredibly important during the last years. CA are used for modeling of crystallization (solidification) [10–13], dynamic and static recrystallization [14–17], phase transformation [16,18,19], cracks propagation [20], severe plastic deformation [21–23], etc. The main asset of the CA is the ability for a close correlation of the real microstructure with its digital representation ⇑ Tel.: +48 126172580, 692983805; fax: +48 126172612. E-mail addresses: [email protected], [email protected] 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.03.029
on a micro-, meso- and macro-scale. One of the 3D CA modifications, known as the frontal CA (FCA) [17], which is capable to accelerate calculations, is used for the simulation of the microstructure evolution in the paper. The deformation is often neglected during modeling of the microstructure evolution by the cellular automata; whereas for the finite elements method (FEM) [3], models based on finite differences (FDM) or multi-phase-field [6] deformation are one of the main variables and results of the modeling. Lack of cell distortion during the modeling of microstructure evolution can be obtained only in the process without deformation. Large deformation leads to significant distortion of the elements, mesh in FEM or FDM, as well as cells in CA. A procedure known as ‘‘remeshing’’ is widely used in FEM codes to improve accuracy of the calculation when geometrical parameters of the elements become unfavorable. The similar procedure is to be implemented into the CA code. The objective of this paper is the development of the principles of the cellular space reorganization and their implementation into
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three-dimensional FCA for modeling of the microstructure evolution during the multi-stage deformation. Simulation of a separate phenomenon or a stage of technological processes can be carried out in the frame of simple CA models. Simulation of the same sequences of the technological stages can be carried out in a combination of CA with other numerical methods, such as the finite element method, the finite differences, the discrete element method, etc. Although the joint methods improve the accuracy of the simulations of the forming processes, the simplest modeling scheme without other advanced methods has been chosen since the main goal is not to obtain precise results, but present the possibility of the space reorganization.
2. General FCA model Accurate modeling of the microstructure evolution during the forming processes is a complex task which relies on the choice of appropriate methods. The choice depends on intended effect, accuracy, computation costs, etc. The cellular automata are recognized as one of the most optimal tools for the microstructure modeling, taking into account the IT costs and the results which can be achieved. They can be used for the microstructure evolution simulation, both as an all-sufficient tool as well as in connection with other methods. In the first case, the simulations are limited by the theoretical studies for the set of deformation conditions or for the process whose parameters can be easily calculated in an analytic way [24]. The use of CA combined with other methods makes it possible to obtain more accurate and more reliable results. CA are often jointed with the finite differences (FDM) or the finite element method (FEM). FEM is known as one of the best methods for the modeling of forming processes. Taking into account the role in the model, the schemes of the mutual use of CA and FEM can be divided into two groups. The first one contains a solution in which the simulations are carried out independently, without any feedback. The systems with the co-operation of both components can be included in the second group. One component is primary; another is secondary, and it is served to deliver the parameters and variables for the proper calculations by the primary method. CA can be secondary, only when it is of a simple structure and does not require high calculation costs, such as memory and calculation time. CA used for the microstructure evolution simulations, especially treedimensional ones, cannot be applied as the secondary method because of IT requirements. Thus, CA play the primary role in the theoretical research on the uneven deformation at the level of crystals as well as the properties of polycrystalline structures. The variant of independent simulations by FEM and CA, without any feedback, is chosen. The forming process is modeled by FEM on the macro-scale. The results of FEM simulation are used by CA for the microstructure modeling on the micro-scale. It is the so-called post-processing to obtain information about the microstructure. An arbitrary FEM code or the literature data can be used in such a strategy. It is enough to meet the requirements for the data needed for the post-processing calculations. The post-processing is described below. It is worth noting that main variables as time, temperature and strain rate tensor are required for the calculations. In this paper, the flat rolling is chosen to present the reorganization possibilities of the model based on FCA. The rolling process parameters was obtained by analytical method [24]. The parameters were the basis for further calculations by FCA, and they are presented below. The general model of the microstructure evolution during the forming processes is based on FCA and covers aspects such as the microstructure evolution, creation of the initial microstructure, deformation, recrystallization and phase transformation. All these aspects are presented below in the following sections.
2.1. Boundary conditions In order to understand the principles of space organization it is important to take into account boundary conditions. The boundary conditions determine how the cellular automata behave in cells that are located on the edge of cellular space. Historically, the boundary conditions for CA were first established for a moving particle. Therefore, the first classification bore traces of interaction of particles (or radiation) with a surface. There have been three types of the interaction introduced: reflection, transmission and absorption. The boundary conditions with the reflection imply perfectly elastic collision; the transmission means that the particle passes through the boundary and appears on the opposite side. When absorption is introduced, a particle leaves the space and ceases to exist. The space with reflection and transmission is closed and one with the absorption is open or half open, because particles often move only outwards. However, in CA for microstructure modeling a bit different classification of the boundary conditions has been adopted. The boundary conditions with the transmission are named as periodic conditions; and ones with the absorption are the so-called open conditions. In fact, other conditions cannot be found in the literature concerning the modeling of the microstructure by CA. There are two examples of microstructure presented in Fig. 1. They are obtained by two-dimensional cellular automata with the open (Fig. 1a) and periodic conditions (Fig. 1b). Since the space with periodic conditions can be considered as the surface of the torus (hipertorus) and the boundaries are the lines of virtual (not real) discontinuity, some grains are presented several times on the opposite sides of the space but they remain undivided. Contrary, all grains in the open space are presented only once. Two main differences between open and periodic conditions based on ‘‘boundary effect’’ can be noted. The first one can be seen in the final structure. Periodic conditions make the fine grains which are closer to the boundaries larger and the coarse grains smaller. The other difference is important for processes of grain growth (for example kinetics of recrystallization). Absence of ‘‘boundary effect’’ in periodic conditions, because of the equality of all cells in the space (nonexistence of the real boundaries), makes the process more realistic. That is why mainly periodic conditions are used in the modeling. Two additional boundary conditions are used in the model based on FCA. The first one is periodic conditions with displacement and the second one is full open conditions. The idea of the periodic conditions with displacement is that the neighboring cell is not directly on the opposite side, but with some horizontal, vertical, lateral displacement or arbitrary combination. For example, let us consider the space with nx, ny and nz cells on appropriate sides. The cell with coordinates {i = nx, j, k} lies on the boundary of the space and has the neighboring cell in direction x. The neighboring cell in the space with ordinary periodic conditions is of coordinates {1, j, k}, while the neighboring cell in the space with periodic conditions with displacement is of coordinates {1, j + dj, k + dk}, where dj and dk are appropriate displacements. The maximal number of displacements, which can be applied in the model, is six, two possible displacements for three directions. The displacements can be fixed or individual. However, in the algorithm presented, in the following section the only displacement along the axis x, equal to half of the space length (diz = nx/2), is used when neighboring cell in direction z is considered. The main weakness of the ‘‘open’’ boundary conditions is their half openness, when boundaries are transparent only in one direction, outward, and are opaque inward. The inner grains can grow outward the space, but any outer grain cannot grow inward the space. Such an ‘‘open’’ condition is named half open. An idea of full open condition is the following: when a growing grain reaches the
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Fig. 1. Microstructure obtained by two-dimensional cellular automata with the open (a) and periodic (b) boundary conditions.
boundary, it can appear not only on opposite side, but also on arbitrary side, including the same side. Now it is not the same grain, but a new one. The side, which new outer grain grows into, and its coordinates (including the distance from the surface) are set at random. The combination of three types of the boundary conditions used in the model presented in the paper: half open, periodic and periodic with displacement. 2.2. Deformation and space reorganization The CA in three-dimensional version should be applied for the modeling of the microstructure evolution in the forming processes because the structure after the deformation demonstrates different images at the three perpendicular cross-sections. Deformation in CA is rarely concerned, but if even it is not neglected, any manipulations of the sizes or the shape of the cell or the cellular space do not take place. The real deformation is the problem that cannot be avoided in CA simulations, especially when the multi-stage deformation is modeled. The deformed structure must be used in the further modeling. The simplest solution is an introduction of the cell distortion into the model, when the sizes and the shape of the cells are to be changed according to the strain tensor. Consideration of the deformation in FCA is described in detail elsewhere [19]. Anisotropy of the space with deformed cells can be obtained by the use of the so called ‘‘virtual front tracking’’ algorithm. The algorithm is based on the idea presented by Rappaz and Gandin [10], Sanchez and Stefanescu [25] and Burbelko et al. [26]. Calculations in the present FCA model can begin with the cubical cells (but not necessarily). During the simulation with deformation they do not remain the same for long time. The deformation (for example during the flat rolling) effects not only on the shape and sizes of the cell (Fig. 2), but also on the grade of anisotropy of modeled space and representative volume. The ratio of the cell sides becomes too high, calculations are decelerated and quality of obtained structure is deteriorated. During the modeling with the use of the model based on finite element method (FEM) the procedure named ‘‘remeshing’’ is often applied in the same cases. A similar procedure should be used in the CA as well. In the model, while the strain is not large enough, FCA cells are deformed according to the real deformation. When the ratio reaches a preset value, the cellular space must be reorganized to obtain the shape of the cells (as well as the whole cellular space) far close to a cubical one. Such reorganization has been shortly presented for the first time at a conference [27], and it is described below in detail.
Fig. 2. The shape of the cell during the flat rolling without space reorganization.
There are two variants of the reorganization developed for the model. They can be chosen manually before simulation or automatically during the simulation. The first simplest variant is applied when deformation is accompanied with the microstructure refinement. The scheme of such reorganization is shown in Fig. 3. Here, the grey color represents the whole cellular space, while the white one – several cells. After the deformation or before the next deformation, ratios of side sizes (length/height, width/height) of the cell (or the representative volume) are tested. If either ratio becomes about 2.0, all space is halved. The first half of the cellular space is left for further simulations, whereas the second half is removed from the model. Each cell in the remaining part is halved as well. Both halves of the cell become the new cells and inherit all properties of the parent cell. This reorganization cuts the volume of the cellular space into half, but the number of cells in the space remains the same. The decrease of the volume limits the use of the approach when it does not lead to a significant reduction of the number of grains from one cut to another. As the periodic boundary condition is used for the more realistic modeling of the microstructure evolution, the halving must be taken into account. Actually, the halving in the space with the periodic boundary condition means two cuts. Then, we receive two opposite sides, which cannot be connected, because they have no coincident microstructure. As a result, open boundary conditions must be applied temporarily on these sides. They are not permanent conditions but transient. Those conditions evolve into partly-open, partly-periodic condition, and finally into the purely periodic conditions. Algorithm of such an evolution is the next (Fig. 3a). All grains on both sides of the space, which have been cut by halving, are perma-
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(a)
(b)
Fig. 3. Scheme of the halving of the cellular space (a) and boundary conditions before and after the halving (b).
nently considered as ones with the open boundary condition, and can grow only inside the space (upper right grain in Fig. 3a). They cannot appear on the opposite side. Yet every other grain which exists in the current structure (bottom left grain in Fig. 3a) or which appeared after the halving can grow not only inside the space, but also outward with its appearance on the opposite side of the space (bottom right). As a result, the boundary conditions depend on the properties of the grains, not on the space. The new grains receive the periodic boundary conditions, while the old grains stay with the open ones. The second variant is more complex and can be used when the volume of space cannot be reduced (Fig. 4). As in the first variant, after the side ratio reaches preset value (2–4), the space is halved, but then the halves are put together, one on the other. Such procedure supposes four cuts and at least two bondings. If the sides are known a priori to be cut and bonded, the boundary conditions can be organized before modeling. Otherwise, at first, the modeling is carried out with the periodic conditions up to the halving, and then, the modeling is repeated with other boundary conditions. For the cutting and bonding, the conditions must be as the ones presented in Fig. 4, where different lines determine coincident microstructure. A grain is demonstrated as an example. Thus, for the short sides, the simple periodic conditions are applied, while for the long sides periodic conditions with displacement are used. The displacement equals to a half of the space length (it is constant, measured in the number of cells). After cutting and bonding on the former long side, the simple periodic conditions are obtained. On the former short side periodic conditions with displacement are formed. The shape of the cells also changes. They are combined by the long sides and then they are halved as it is shown in Fig. 4. Cells which belong to the different grains are combined, taking into the consideration the neighboring cells. An example of modeled microstructure before and after the cutting and bonding is presented in Section 4. As the procedure can be repeated, the appropriate conditions must evolve either into the simple periodic conditions or into the periodic conditions with displacement, as it is described above for the first variant of the reorganization (the halving). Some modification of halving is used in calculations presented in the paper. Instead of purely periodic boundary conditions the
periodic boundary conditions with displacement are always applied. It allows for automatic choice of the reorganization method. 2.3. The cellular automaton Frontal cellular automata (FCA) [17,19] are used as the module of model presented in the paper. The use of the frontal cellular automata instead of the conventional ones makes it possible to reduce the computation time significantly, especially for the threedimensional models, as the large regions are excluded from the calculations in the current step and only the front of the changes is studied [17]. The multi-states cell automaton is presented schematically in Fig. 5 [19]. The set of the states Q = {q0, q1, q2, q3, q4} comprises the initial matrix state q0, the ‘‘frontal cell’’ q1, the ‘‘boundary cell’’ q2, the ‘‘cell inside the grain’’ q3 and the transient states q4. The set of conditions {I0–I4} is used by the transition rules. The transition rules define the next qi+1 state of the cell on the basis of the current qi states of the cell and the cells in its neighborhood and the transition condition Ik. Such rules are commonly presented either in a form of a table or by description. Shortly, the conditions are of the following meanings: I0 is the nucleation, I1 – the time delay, which allows to control the grain growth rate, I2 and I3 determine whether the cells are on the boundary or inside the grain (I3 ¼ I2 ) and I4 means that the growing grain reaches the current cell and involves it in the growing process. The automaton (Fig. 5) is characterized by a closed circuit of the states: q1 ðq2 ^ q3 Þq4 . The states q2 and q3 are not only the final states, but also the initial states for the next cycle of the modeling, and the states can be repeated several times. This property is useful for the modeling of some processes, for example, multi-pass flat rolling, where deformation in the passes alternates with the recrystallization in the pauses. Two kinds of neighborhood are used by the FCA. The Moore neighborhood is applied in the algorithm of the boundary motion during the grain growth simulation (state q1 and condition I4), while the unmovable grain boundaries are defined through von Neumann neighborhood (states q2, q3 and condition I2, I3). FCA works with a varied time step, which is defined by the minimal time for the boundary migration through the cell. The FCA can
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Fig. 4. Scheme of the cutting and bonding.
q2 I0 I4 q0
I0 I4
q4 I4
Principles of modeling such phenomena as initial microstructure creation, recrystallization, and phase transformation by FCA are described in detail elsewhere [19]. They are realized in phenomenological module. As the accuracy of the microstructure evolution simulation is not the main goal of the paper, the only modeling of static recrystallization is shortly presented in the section. The model of recrystallization realized in FCA consists of two parts, which are the nucleation and the grain growth. Number of nuclei is calculated according to:
I2 I1
q1
I3
q3 Fig. 5. Cellular automaton (q – states, I – transition conditions).
be used for the simulation of static and dynamic recrystallization, phase transformation and solidification. Yet the example presented in the paper uses only static recrystallization. Application of the model to the hot flat rolling is described in one of the sections below. 2.4. The phenomenological and material modules Material module contains parameters for material and calculates current values of dislocation density and flow stress. The model of the dislocation density is described in detail elsewhere [28,29]. The equation considers both the hardening and the softening. It takes into account the strain rate e_ , the temperature T and the dislocation density q, according to the following equation, which is applied for all grains:
q_ ¼ a1 e_ n1 exp
Q1 Q2 a2 e_ n2 qm1 exp RT RT
ð1Þ
where Q1 and Q2 – the activation energies, a1, a2, n1, n2, m1 – the material constant. The flow stress is calculated according to equations:
rs ¼ r0 þ albðqav Þ1=2
ð2Þ
where r0 – the stress necessary to move the dislocation in the absence of the other dislocations, a – the constant, l – the shear modulus, b – the Burgers vector and qav – the average dislocation density.
NV ¼ aN enN exp
QN Vr RT
ð3Þ
where e – strain, T – temperature, R – gas constant, QN – activation energy, aN, nN, QN – the material constants, Vr – representative volume. The grain growth of recrystallized grain is defined by a boundary migration rate v, which depends on driving force of recrystallization p and grain boundary mobility m. The disorientation angle # is taken into account as well: v = mpf(#). The difference of the dislocation density of the old qold and new qnew grains is the driving force are calculated on the basis of stored dislocation energy: p = 0.5 lb2 (qold–qnew ). A curvature of the grain boundaries is neglected as another driving force because during the process of the recrystallization it is far smaller than the force from the difference of dislocation density. The grain boundary mobility m depends on the self-diffusion coefficient and therefore, it is defined by Arrhenius’ law:
m ¼ am exp
Qm RT
ð4Þ
where am, Qm – the material constants. 3. Flat rolling The frontal CA model is adapted by the technological module to the simulation of microstructure evolution during the hot flat rolling. The calculations are carried out for reverse mill with work rolls
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of 1000 mm diameter. The plate to be simulated is milled to 40.0 mm thickness and 10 m length. The rolling speed is 5 m/s. Cooling in air is assumed. Inter-pass time or time of pauses between the exit from the previous pass and the entrance to the next one is 2.3 s. Rolling pass schedule (Table 1 [24]) contains information about thicknesses, temperature and other process parameters. The schedule is one of the options of process carried out in one of the polish reverse rolling mills. It is a basis for modeling. The microstructure evolution is simulated for the point on the axis of symmetry of the plate. Steel, which on chemical composition is close to steel 45, is chosen as the modeled material. For each pass, technological module calculates strain, strain rate, length and time of rolling, inter-pass time and so on. Sizes of the cells before and after the pass are calculated in order to determine the time increment and a number of iteration during the rolling simulation. Temperature is assumed to be constant during the deformation, as well as strain rate. By the end of deformation, a number of new grains (nuclei) are calculated. Static recrystallization is modeled only, because even at 1100 °C and strain rate of 10 s1 peak strain is above 0.35. According to modeled condition (all temperatures are below 1100 °C, stain about 0.3 and strain rate about 90 s1), dynamic recrystallization does not take place. After deformation, the growth of recrystallized grains is simulated without the additional nucleation. Temperature changes are linearly approximated from pass to pass. A pause after the pass is a sum of inter-pass time and time of the rolling calculated on the basis of length of the plate and the rolling speed. 4. Simulation results Initial microstructure is created by the FCA with the following parameters (Fig. 6a). FCA space contains nx ny nz = 500 500 500 cells with the representative volume of ax ay az = 0.4 0.4 0.4 mm3. Initial microstructure include 200 grains, it gives
the average grain size of about 70 lm. Axis x is a rolling direction, y is a lateral direction, z is an axis along the thickness of the plate. Size y of the FCA space remains constant, x is elongated and z is reduced. Two variants of visualization can be chosen. The first one allows for colored representation, when forty five colors present the crystallographic orientation, but for paper another variant is preferred. Grey scale presents different states of microstructure. The old grains are charcoal grey, new recrystallized grains are light grey. Grains boundaries are black, migrated boundaries of the new growing grain are white. The simulation results are presented as isometric drawings with the microstructure seen on the cuboids’ faces (Figs. 6–11) and as graphs showing the changes of the average grain size and the flow stress (Figs. 12 and 13). The microstructure is presented in time, which is connected with the beginning or the ending of the deformation. Space reorganization is done automatically before the passes according to the following logical sequence:
ðng > ngmax Þ ^ ðax ¼ maxðax ; ay ; az ÞÞ ) HX ðng > ngmax Þ ^ ðay ¼ maxðax ; ay ; az ÞÞ ) HY ðng > ngmax Þ ^ ðaz ¼ maxðax ; ay ; az ÞÞ ) HZ
ð5Þ
ðax > 3az Þ ) BXZ ðax > 2az Þ ^ ðng > ngmin Þ ) HX ðay > 2az Þ ^ ðng > ngmin Þ ) HY where ax, ay and az – sizes of cellular space, ng – number of grains, ngmin and ngmax – minimal and maximal number of grains (ngmin = 500, ngmax = 4000), BXZ – cutting of the length of the cellular space (ax/2) and bonding in the thickness (2nz), HX, HY and HY – halving of the length (ax/2), the width (ay/2) and the thickness (az/ 2), respectively. A sequence of simulation and some results are presented in Table 2. Before the second pass, space is not reorganized because small number of grains (ng = 453 < ngmin = 500) does not allow
Table 1 Rolling schedule [24]. Pass number
Entrance thickness h0 (mm)
Exit thickness h1 (mm)
Temperature T (°C)
Reduction r (%)
Strain rate e_ (s1)
Pause after pass (s)
1 2 3 4 5
40.00 27.18 18.55 12.75 8.95
27.18 18.55 12.75 8.95 6.49
1098 1087 1061 1010 928
32 31.7 31.3 29.8 27.5
90.6 90.1 89.4 87.3 83.6
5.24 6.6 8.7 11.2 14.6
Fig. 6. Microstructure before (a) and after (b) the first pass. Representative volume Vr = 0.064 mm3.
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halving, and the sides ratio is not big enough for bonding (ax = 0.588 < 3az = 3 0.272 = 0.716). Then before the third, forth and fifth passes bonding or appropriate halving are fulfilled. After the deformation, small white points in the some figures can be observed (Figs. 6b, 7b, 9b, 11a). They represent nuclei. Because of high strain rate (80–90 s1) and temperature (1000– 1100 °C), static recrystallization is accompanied with simulated rolling. Complete recrystallization after the passes, except the last two, is observed as a consequence of relatively high temperature in
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the first passes and long inter-pass time. After the last (fifth) pass and cooling, as a result of lower temperature, recrystallization is stopped (Fig. 11b). White boundaries and light grey grains mean that recrystallization is not finished. The procedure of cutting and bonding can be seen in Fig. 8. After cutting the nearest half, it goes to the top. The procedure (which is described in detail in Section 2.2) gives microstructure which is coincident in the place of joining. That is why, in Fig. 8 lines (or surfaces), where two parts are joined, cannot be observed. Halving
Fig. 7. Microstructure before (a) and after (b) the second pass. Representative volume Vr = 0.064 mm3.
Fig. 8. Microstructure after recrystallization after the second pass before (a) and after (b) cutting and bonding. Representative volume Vr = 0.064 mm3.
Fig. 9. Microstructure before (a) and after (b) the fourth pass. Representative volume Vr = 0.032 mm3.
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Fig. 10. Microstructure after recrystallization after the forth pass before (a) and after (b) the halving in two axes. Representative volume Vr is reduced from 0.032 to 0.008 mm3.
Fig. 11. Microstructure after the fifth pass before (a) and after (b) recrystallization. Representative volume Vr = 0.008 mm3.
Fig. 12. Average grain size during the rolling.
Fig. 13. Flow stress during the rolling.
after the forth pass in two directions (in length and width) is presented in Fig. 10. A quarter is used in further simulation, the representative volume Vr is reduced from 0.032 to 0.008 mm3.
Changes of the average grain size are calculated during the whole process in the representative points. The average grain size dav is considered to be proportional to the ratio of the representa-
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D.S. Svyetlichnyy / Computational Materials Science 60 (2012) 153–162 Table 2 Sequence of the simulation. Pass number
Length width thickness, ax ay az (mm)
Representative volume Vr (mm3)
Number of grains before/after
Space reorgani-zation
Figures
1 2 3 4 5
0.4 0.4 0.4 0.588 0.4 0.272 0.431 0.4 0.371 0.314 0.4 0.255 0.224 0.2 0.179
0.064 0.064 0.064 0.032 0.008
200/453 453/576 576/994 674/1464 625/4850
– BXZ CX CX + CY –
Fig. 6 Figs. 7 and 8 – Figs. 9 and 10 Fig. 11
tive volume Vr to the entire area of the surface S of all the grains dav / V r =S:
dav ¼
Vr ðk1 n1 þ k2 n2 ÞSc
ð6Þ
where Vr – the representative volume of the cellular space, lm3; n1 and n2 – the number of the cells in the state q1 and q2; Sc – the average area of the cell’s faces, lm2, k1 and k2 – coefficients. There are several factors causing some difficulties, which influence the accuracy of the grain size. They are connected with the discretization of the cellular space. The real area S of the boundaries is less than the discrete one. The area of a cell’s faces Sc is not equal to the others and does not remain constant during the simulation (due to the deformation). The number of cells (not faces of the cells) is counted for the surface calculation. The frontal cells q1 (n1) and the boundary cells q2 (n2) are determined by different neighborhoods. What is more, the frontal cells are one-sided, but the immobile boundaries are two-sided i.e. firmed by the cells twice. By taking into account the above mentioned factors, appropriate coefficients (k1 and k2) are introduced. Coefficient k2 is determined in the way that the average grain size dav calculated for a structure without cells in state q1 according the equation:
dav ¼
rffiffiffiffiffiffiffiffiffiffiffi 3 6V av
p
sffiffiffiffiffiffiffiffiffi 3 6V r ¼ png
sented as a function of the accumulated strain (Fig. 13). A growth of the flow stress is observed from the pass to pass. It is caused by a decrease of temperature.
ð7Þ
gives the same results as Eq. (6). Here Vav is an average volume of the grain. The coefficient k1 is determined by the comparison of the structure with the cells in state q1 with the same structure, in which cells in the state q1 are replaced by the cells in state q2 or q3, taking into account different neighborhood and other factors. Accuracy of the average grain size calculations is evaluated at about 10%. Nevertheless, such calculations demonstrate the changes of the average grain size, not only qualitatively, but also quantitatively, describing the microstructure evolution. A change of grain size is demonstrated in Fig. 12. Static recrystallization is not completed after the fifth pass which is seen in Fig. 12, as well as in Fig. 11b. The average grain size reaches the minimal value because new recrystallized grains still do not fill the whole space and old grains do not disappear completely, but remain reduced significantly. During almost every pass, a decrease of grain size can be observed. The results can be explained by the method of calculation of the grain size (6). While volume is not changing during the deformation, the surface of the boundaries of the deforming grains is increasing. It leads to the decrease of the average grain size (6) during the pass. Additional abrupt decrement of the grain size, which can be seen in Fig. 12, is explained by space reorganization. Cubical cells after reorganization allows for more precise calculation of the surface of the boundaries then elongated or distorted cells. Generally, surface calculated after space reorganization increases, therefore the average grain size decreases abruptly. In the further it is anticipated if not eliminate this drawback, at least reduce the magnitude of this drop. The flow stress, which is calculated as described above (in Section 2.4, Eqs. (1) and (2)) is studied as well. The flow stress is pre-
5. Summary A proper simulation cannot be carried out without taking into consideration the real deformation. Introduction of the deformation into the model based on cellular automata causes several problems connected with distortion of the cells and the whole cellular space. It mainly increases anisotropy of model space. Anisotropy can be reduced by front-tracking algorithm, but the greater difference in resolution in different directions impedes precise calculations. Therefore, two algorithms for space reorganization have been developed and they are described in the paper. The first of them is ‘‘halving’’, and the second one is ‘‘cutting and bonding’’. Firstly, the boundary conditions have been enriched by the periodic condition with displacement. Then boundary conditions are assigned to the grains, not to the space (or the cells). It has opened a way to a more free operation with the space and to two algorithms for space reorganization. Both algorithms consider the cutting of the space. Cut grains lose their parts and boundaries of the space remain open for them. All other grains stay in the space with periodic conditions. For halving algorithm periodic conditions before and after the cutting can be with and without displacement; conditions without displacement lead only to the simpler algorithm. For the algorithm with bonding the only periodic conditions with displacement must be applied before the cutting. Despite some complication of the first algorithm, the periodic boundary conditions with displacement have been accepted for both algorithms. It has removed an indispensability of preliminary definition of the kind of reorganization and appropriate boundary conditions and it also makes it possible to reorganize automatically in view of current simulation state. To present the possibility of the reorganization algorithms, they have been implemented into the frontal CA model, which has been adapted to the simulation of the microstructure evolution during the multi-stage flat rolling process. Deformation and static recrystallization are considered for the five passes alternated by air cooling. Microstructure, the average grain size and flow stress are presented as the simulation results. The presented results favored the conclusion that the FCA model can be applied for the simulation of the microstructure evolution during the multi-stage deformation. Further research is directed to develop a material module in order to obtain more realistic results of microstructure evolution during multi-stages deformation processes. References [1] C.H.J. Davies, Scripta Mater. 36 (1997) 35–40. [2] E.A. Holm, G.N. Hassold, M.A. Miodownik, Acta Mater. 49 (2001) 2981–2991. [3] M. Bernacki, Y. Chastel, H. Digonnet, H. Resk, T. Coupe, R.E. Loge, Development of numerical tools for the multiscale modeling of recrystallization in metals, based on a digital material framework, Comp. Meth. Mater. Sci., 7 (2007) 142– 149.
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