Modeling of Recrystallization with Recovery by Frontal Cellular ...

Modeling of Recrystallization with Recovery by Frontal Cellular Automata Dmytro S. Svyetlichnyy, Jarosław Nowak, and Łukasz Łach AGH University of Science and Technology, Faculty of Metal Engineering and Industrial Computer Science, Kraków, Poland {svetlich,nowak}@metal.agh.edu.pl, [email protected]

Abstract. The objective of the paper is modeling of material softening after the deformation. The main problem of almost every model with digital material representation is consideration of recrystallization as the only mechanism of material softening. Static recovery is introduced into the model based on frontal cellular automata. An influence of static recovery on softening process is twofold. Static recovery effects on a decrease of dislocation density directly and on growing rate of recrystallized grain indirectly. Because of static recovery the recrystallization slows down and the time of recrystallization is extended. Simulation consists of two stages. During the deformation, distortion of the cells, evolution of dislocation density, nucleation and grain growth are considered, while after the deformation, the processes of softening are considered only. Comparison of simulation results with experimental data are presented as well. Keywords: cellular automata, microstructure, softening, recrystallization, recovery.

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-phasefield [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], rolling processes [19], [24] etc. 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 history of simulation of recrystallization has begun with publication Hesselbarth and Göbel [25]. Then Davies [1] has considered influence of nucleation and growth of nuclei on recrystallization. Ding and Guo [26] simulated joint problem of G.C. Sirakoulis and S. Bandini (Eds.): ACRI 2012, LNCS 7495, pp. 494–503, 2012. © Springer-Verlag Berlin Heidelberg 2012

Modeling of Recrystallization with Recovery by Frontal Cellular Automata

495

dynamic recrystallization and flow stress calculation on the base of modeling of dislocation evolution. Reviews of cellular automata in materials science were published by Raabe [27] in 2002, by Janssens [28] in 2010 and by Yang et al. [29] in 2011. Mukhopadhyay et al. [30] simulated static recrystallization accounting different types of nucleation with consideration the effect of dislocation density, temperature and grain boundaries misorientation angle on the mobility of a grain boundary. Flow stress is usually modeled in CA on the basis of dislocation theory. In the cellular automata, evolution of dislocation density is applied to every grain or every cell. During the deformation, dislocation density effects on nucleation and grain growth rate, while after the deformation, dislocation density is considered to be constant and recrystallization acts as the only softening process. It leads to constant grain growth rate. It is not a problem for models, which do not use digital material representation methods, because softening process is treated as a single, without division into separate phenomena. Then, parameters of the model (or equations) can respond to whole process, not separate phenomenon. Methods based on digital material representation including the cellular automata take into account evolution of dislocation density and grain growth during the recrystallization in two completely different ways. For evolution of dislocation density the dislocation theory is mainly used, while recrystallization is considered from geometric point of view. Then a problem with kinetics of recrystallization appears for almost every model with digital material representation, which considers recrystallization as the only mechanism of material softening. The objective of the paper is adequate modeling of material softening after the deformation, when kinetics of softening is close to experimental data.

2

The FCA Model

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 conventional ones makes it possible to reduce the computation time significantly, especially for the three-dimensional 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. 1 [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 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 = Ī2) and I4 means that the growing grain reaches the current cell and involves it in the growing process. Deformation in CA is rarely concerned. However, real deformation is the problem that cannot be avoided in CA simulations, especially when the multi-stage

496

D.S. Svyetlichnyy, J. Nowak, and Ł. Łach

q2 I0 ∨ I4 q0

I0 ∨ I4

q4 I4

I2 I1

q1

I3

q3 Fig. 1. Cellular automaton (q – states, I – transition conditions)

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].

3

Recrystallization

In presented FCA the words (I0 ∨ I4)I1(I2 ∨ I3) (Fig. 1) with the initial states q2 and q3 describe the recrystallization. The model of recrystallization realized in FCA consists of two parts; the nucleation and the growth of recrystallized grains. They work in cooperation with the dislocation model, which is described in the next section. The microstructure evolution depends on the nucleation rate and the grain boundary migration rate. The models of nucleation and growth of recrystallized grains are described in detail elsewhere [31-32]. In this paper they are presented shortly. The dislocation density is used for calculation of moment when recrystallization begins, as well as for calculation of the flow stress σ and the grain boundary migration rate v. The critical value of the dislocation density ρc for the nucleation initialization is of the following form (Roberts and B. Ahlblom [33]:

ρc =

8γ 3 20γε = τl 3blMτ 2

(1)

Number of nuclei is calculated according to: Q  NV = a N ε nN exp N Vr  RT 

(2)

where ε – strain, T – temperature, R – gas constant, QN – activation energy, aN, nN, QN – the material constants, Vr – representative volume.

Modeling of Recrystallization with Recovery by Frontal Cellular Automata

497

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 ϑ is taken into account as well: v = mpf(ϑ). The difference of the dislocation density of the old ρold and new ρnew grains is the driving force are calculated on the basis of stored dislocation energy: p = 0.5μb2( ρold - ρnew). 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: Q  m = a m exp m   RT 

(3)

where am, Qm – the material constants.

4

The Model of Dislocation Density

Model [31-32], [34] was developed on the basis of the Taylor dislocation theory [35]. It is used for determination of flow stress:

σ = σ 0 + αμ b ρ ,

(4)

where σ0 - stress necessary to move dislocation in the absence of other dislocations, α - constant, μ - shear modulus and b - Burgers vector, ρ – dislocation density. It is generally assumed that during the deformation the evolution of the dislocation density is dependent on two components: dρ = U (ε ) − Ω (ρ ) , dε

(5)

Here, U(ε) is regarded as the generation and the storage of the dislocation (hardening), while the term Ω(ρ) contributes an annihilation of the dislocation (dynamic recovery), ε - strain. When deformation ends, the left side of the equation (5) is meaningless, dynamic recovery stops, and dislocation density is considered to be constant. Then, constant dislocation density ( ρold - ρnew = const) and constant temperature (T = const) lead to the constant grain boundary mobility (m = const) and boundary migration rate (v = const). When constant boundary migration rate v is applied to simulation by two- or three dimensional cellular automata (2D or 3D CA), kinetics of softening, as well as recrystallization can be described by Avrami equation

(

χ = 1 − exp − at n

)

(6)

with Avrami exponent equal to n = 2 (2D CA) or n = 3 (3D CA) only (t – time, a – factor depended on recrystallization conditions). But real exponent is less then 2

498

D.S. Svyetlichnyy, J. Nowak, and Ł. Łach

(n < 2). Actually, arbitrary value of n (6) is accepted in analytical model of static recrystallization that can be obtained from experimental data. Equation (6) for static recrystallization can be rewritten in following form:   π Nv 3t 3  .   3

χ = 1 − exp −

(7)

Considering equation (7) it can be noted that only one element can be varied during the softening process, namely the grain boundary migration rate v. It is also known that apparent Avrami exponent n (6) with constant rate v can be differed from theoretical value of 3 if rate v is a function of time t. On the other hand at constant temperature, when boundary mobility remains constant, only stored energy, which depends on dislocation density ρ, can influence on rate v. So, dislocation density ρ of nonrecrystallized grains should be changed. And, in turn, dislocation density ρ can be decreased in process of static recovery only. It is the main causes why static recovery (R) can be introduced into the model of evolution of dislocation density (2):

ρ = U (ε )ε − Ω(ρ )ε − R(ρ )

(8)

In fact, static recovery affects the material softening in two ways, decreasing the dislocation density of non-recrystallized grains ρ and reducing the grain growth rate that is changing the recrystallization fraction χ. However not all dislocations can be removed by static recovery or the process for such dislocations is too long. As a result another type of dislocations is introduced to the model. This type of dislocation can be designated ρs – “structural” dislocations. Differential equation for ρs is of following form:

εs

dρ s+ = fρ − ρ s dε

(9)

where f – factor, which determines fraction of dislocation that cannot be removed by static recovery, εs – characteristic strain, sign “+” near the derivative means that dislocation density ρs can only increase. Thus, two types of dislocations can be taken into account and instead of ρ in equation (4) it is necessary to use the sum of these two dislocations types: ρ + ρs. During the deformation all three components of (8) effect on dislocation density ρ, in its absence (after deformation) only the last term. After the deformation “structural” dislocations ρs remain constant and the only recrystallization takes them away. Moreover, recrystallization removes dislocations ρ.

5

Experimental Studies

The stress relaxation method is applied to measure kinetics of material softening [36]. Here only one deformation is required. After the deformation specimen is kept in testing machine with the control of compressive force. The force changes reflect

Modeling of Recrystallization with Recovery by Frontal Cellular Automata

499

fraction of stress relaxation. It is a reliable and highly effective method which allows to receive kinetics for the given conditions in one test. The stress relaxation tests were carried out on a Gleeble3800 at IMŻ (Gliwice, Poland). The specimens were heated at a rate of 3 K/s up to the temperature of 1275, 1200 or 1100°C held 10 s. Then temperature was decreased to the deformation temperature (1000°C to 1200°C), and later compressed up to prescribed strain at constant strain rate. After the prior deformation, the strain was held constant at least 300 s and compressive force relaxed was recorded as a function of time. A carbon steel of grade 45 was used in the tests to prevent any interference of softening with precipitation of microalloy elements.

6

Simulation

Simulations reflect stress relaxation test. At first, parameters of the models are identified. Microscopic studies give parameters of nucleation model. Relaxation tests allow to obtain fraction f and parameters of static recovery and grain growth rate. Plastometric tests were a basis for calculation other parameters of dislocation model. After identification of model parameters, the initial microstructure with average grain size about dav = 100 μm (measured in studies 101.7 μm) was created according to algorithm described elsewhere [37]. The microstructure presented in Fig.2 is used in all further simulation. Modeled material has been subjected to deformation at the specified stain rate, up to a given strain at the prescribed temperature. After deformation, modeled material was held at the deformation temperature up to complete recrystallization. Two examples of microstructure after deformation and recrystallization are presented in Fig.3. The first variant shows microstructure obtained when material was subjected to deformation at T = 1100°C with strain rate ε = 1 s-1 up to strain ε = 0.18. Time of softening was about ts = 7.5 s. Average grain size is dav = 92 μm (measured 92.4 μm). The second variant of microstructure was obtained after deformation: T = 1100°C, ε = 0.1 s-1, ε = 0.18 and ts = 5.5 s. Average grain size is dav = 75 μm (measured 78.5 μm). Next results present kinetics of material softening after the deformation. In Fig.4 presents experimental data from stress relaxation test (symbols), simulation results obtained by new softening model, that takes into account recrystallization and static recovery (solid lines), by the previous model with the only recrystallization (solid lines with symbols) and approximation by equation (6), which is used in conventional analytical models (dashed lines). Simulations by the model with the only recrystallization give results that can be approximated by the equation (6) with Avrami exponent equal to 3. Therefore, only time of half recrystallization can be adjusted, not all curve. It is clearly seen on Fig. 4. Such results were the main reason for development of new model. Typically, approximation by equation (6), as observed in Fig. 4, gives good result, but when strain is low enough, recrystallization becomes very slow and effect of static

500

D.S. Svyetlichnyy, J. Nowak, and Ł. Łach

Fig. 2. Initial microstructure created by FCA, average grain size dav = 100 μm

a

b

Fig. 3. Final microstructure after deformation at temperature T = 1100°C and recrystallization: a – ε = 1 s-1, ε = 0.18, dav = 92 μm; b – ε = 0.1 s-1, ε = 0.18, dav = 75 μm

recovery can be separated from one of recrystallization. That can be seen in figure for the experimental curve with the smaller strain. Then the curve cannot be approximated by the equation (6) precisely. The new model based on cellular automata that takes into account recrystallization and static recovery allows for proper simulation both variants, when effect of those two phenomena can be separated or they acts simultaneously. Fig. 4 confirms it.

Modeling of Recrystallization with Recovery by Frontal Cellular Automata

501

softening fraction x

1

0.5

1100°C, 1 s-1, ε=0.18 1100°C, 0.1 s-1, ε=0.09 new model Avrami equation old CA 0 0.01

0.1

1

10

time, s

Fig. 4. Softening fraction

7

Summary

In the paper a new model based on frontal cellular automata is presented. FCA contains a module for simulation of evolution of dislocation density. The modification of dislocation model consists in accounting for static recovery. It allows to obtain curves that can be described by Avrami equation (6) with almost arbitrary Avrami exponent. It also makes possible to model processes where effect of recrystallization and static recovery can be divided into two events. Effectiveness of new model is confirmed by comparison simulation results with experimental data. Acknowledgements. Support of the Polish Ministry of Education and Science is greatly appreciated (Grant Nos. N N508 620140 and 2011/01/N/ST8/03658).

References 1. Davies, C.H.J.: Growth of nuclei in a cellular automaton simulation of recrystallization. Scr. Mater. 36, 35–40 (1997) 2. Holm, E.A., Hassold, G.N., Miodownik, M.A.: The influence of anisotropic boundary properties on the evolution of misorientation distribution during grain growth. Acta Mat. 49, 2981–2991 (2001) 3. Bernacki, M., Chastel, Y., Digonnet, H., Resk, H., Coupe, T., Loge, R.E.: Development of numerical tools for the multiscale modeling of recrystallization in metals, based on a digital material framework. Comp. Meth. Mat. Sci. 7, 142–149 (2007) 4. Fan, D., Chen, L.Q.: Computer simulation of grain growth using a continuum field model. Acta Mat. 45, 611–622 (1997)

502

D.S. Svyetlichnyy, J. Nowak, and Ł. Łach

5. Kobayashi, R., Warren, J.A., Carter, W.C.: A continuum model of grain boundaries. Physica D 140, 141–150 (2000) 6. Takaki, T., Hisakuni, Y., Hirouchi, T., Yamanaka, A., Tomita, Y.: Multi-phase-field simula-tions for dynamic recrystallization. Comput. Mater. Sci. 45, 881–888 (2009) 7. Thompson, C.V., Frost, H.J., Spaepen, F.: The relative rates of secondary and normal grain growth. Acta Metall. 35, 887–890 (1987) 8. Frost, H.J., Thompson, C.V., Howe, C.L., Whang, J.: A two-dimensional computer simulation of capillarity-driven grain growth: preliminary results. Scripta Metall. 22, 65–70 (1988) 9. Weygand, D., Brechet, Y., Lepinoux, J.: A Vertex Simulation of Grain Growth in 2D and 3D. Adv. Engng. Mater. 3, 67–71 (2001) 10. Rappaz, M., Gandin, C.-A.: Probabilistic modeling of microstructure formation in solidification processes. Acta Metal. Mater. 41, 345–360 (1993) 11. Raabe, D.: Mesoscale simulation of spherulite growth during polymer crystallization by use of a cellular automaton. Acta Mater. 52, 2653–2664 (2004) 12. Burbelko, A.A., Kapturkiewicz, W., Gurgul, D.: Problem of the artificial anisotropy in solidification modeling by cellular automata method. Comp. Meth. Mat. Sci. 7, 182–188 (2007) 13. Svyetlichnyy, D.S.: Modeling of macrostructure formation during the solidification by using frontal cellular automata. In: Salcido, A. (ed.) Cellular Automata - Innovative Modelling for Science and Engineering, pp. 179–196. InTech, Croatia (2011) 14. Hurley, P.J., Humphreys, F.J.: Modelling the recrystallization of single-phase aluminium. Acta Mater 51, 3779–3793 (2003) 15. Qian, M., Guo, Z.X.: Cellular automata simulation of microstructural evolution during dynamic recrystallization of an HY-100 steel. Mater. Sci. Eng. A 365, 180–185 (2004) 16. Kumar, M., Sasikumar, R., Kesavan Nair, P.: Competition between nucleation and early growth of ferrite from austenite – studies using cellular automation simulations. Acta Mater. 46, 6291–6303 (1998) 17. Svyetlichnyy, D.S.: Modeling of the microstructure: from classical cellular automata approach to the frontal one. Comput. Mater. Sci. 50, 92–97 (2010) 18. Macioł, P., Gawąd, J., Kuziak, R., Pietrzyk, M.: Internal variable and cellular automatafinite element models of heat treatment. Int. J. Multiscale Comp. Eng. 8, 267–285 (2010) 19. Svyetlichnyy, D.S.: Simulation of microstructure evolution during shape rolling with the use of frontal cellular automata. ISIJ Int. 52, 559–568 (2012) 20. Das, S., Palmiere, E.J., Howard, I.C.: CAFE: a tool for modeling thermomechanical processes. In: Proc. Int. Conf. on Thermomechamical Processing: Mechanics, Microstructure & Control, Sheffield, UK, pp. 296–301. University of Sheffield, Sheffield (2003) 21. Svyetlichnyy, D., Majta, J., Muszka, K.: Modeling of microstructure evolution of BCC metals subjected to severe plastic deformation. Steel Res. Int. 79, 452–458 (2008) 22. Svyetlichnyy, D.S.: Modeling of microstructure evolution in process with severe plastic deformation by cellular automata. Mater. Sci. Forum, 638–642, 2772–2777 (2010) 23. Svyetlichnyy, D., Majta, J., Muszka, K., Łach, Ł.: Modeling of microstructure evolution of BCC metals subjected to severe plastic deformation. In: AIP Conf. Proc., vol. 1315, pp. 1473–1478 (2010) 24. Svyetlichnyy, D.S.: Reorganization of cellular space during the modeling of the microstructure evolution by frontal cellular automata. Comput. Mater. Sci. (in press, 2012) 25. Hesselbarth, H.G., Göbel, I.R.: Simulation of recrystallization by cellular automata. Acta Metal. Mater. 39, 2135–2143 (1991)

Modeling of Recrystallization with Recovery by Frontal Cellular Automata

503

26. Ding, R., Guo, Z.X.: Coupled quantitative simulation of microstructure evolution and plastic flow during dynamic recrystallization. Acta Mater. 49, 3163–3175 (2001) 27. Raabe, D.: Cellular automata in materials science with particular reference to recrystallization simulation. Annu. Rev. Mater. Res. 32, 53–76 (2002) 28. Janssens, K.G.F.: An introductory review of cellular automata modeling of moving grain boundaries in polycrystalline materials. Math. Comput. Simul. 80, 1361–1381 (2010) 29. Yang, H., Wu, C., Li, H.W., Fan, X.G.: Review on cellular automata simulations of microstructure evolution during metal forming process: Grain coarsening, recrystallization and phase transformation. Sci. China Tech. Sci. 54, 2107–2118 (2011) 30. Mukhopadhyay, P., Loeck, M., Gottstein, G.: A cellular operator model for the simulation of static recrystallization. Acta Mater. 55, 551–564 (2007) 31. Svyetlichnyy, D.S.: The coupled model of a microstructure evolution and a flow stress based on the dislocation theory. ISIJ Int. 45, 1187–1193 (2005) 32. Svyetlichnyy, D.S.: Modification of coupled model of microstructure evolution and flow stress: experimental validation. Mater. Sci. Technol. 25, 981–988 (2009) 33. Roberts, W., Ahlblom, B.: A nucleation criterion for dynamic recrystallization hot working. Acta Metall. 26, 801–813 (1978) 34. Svyetlichnyy, D.S., Nowak, J., Łach, Ł., Pidvysotskyy, V.: Numerical simulation of Flow Stress by Internal Variables Model. Steel Res. (in press, 2012) 35. Taylor, G.I.: The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. A. 145, 362–415 (1934) 36. Karjalainen, L.P.: Stress relaxation method for investigation of softening kinetics in hot deformed steels. Mater. Sci. Technol. 11, 557–565 (1995) 37. Svyetlichnyy, D.S., Łach, Ł.: Digital material representation of given parameters. Steel Res. (in press, 2012)