tatic Recrystallization Phase-Field imulation Coupled with Crystal Plasticity Finite Element Method T. Takaki 1, A. Yamanaka, Y. Tomita 2 ummary We have proposed a simulation model for static recrystallization using the phase-field method coupled with the crystal plasticity finite element method. First, the plastic deformation behaviors of metallic materials are simulated by the crystal plasticity finite element method. econd, the nucleation and growth of recrystallized grains are examined by phasefield simulation, in which the crystallographic orientation of recrystallized grains, stored energy as a driving force for grain boundary migration, and information on nucleation obtained from the results of crystal plasticity finite element simulation are used. Introduction The mechanical properties of metallic materials are strongly affected by microstructures. It is, therefore, very important to understand the evolution of microstructures and texture during thermomechanical processing, particularly to develop a numerical model that can predict the effects of processing parameters on microstructures. Monte Carlo [1] and Cellular Automata methods [2] are successfully applied to the investigations of microstructures developed during annealing. In the past decade, phase-field methods have attracted considerable interest for the computation of complex interface morphologies [3]. The main advantage of the use of these methods is that the location of the interface is given implicitly by the phase field, which greatly simplifies the handling of migrating interfaces. Furthermore, we do not need to compute interfacial geometric quantities such as curvature, since they are already included in a phase-field model. The extension of this method to a three-dimensional problem is also simple. However, to conduct realistic simulation, the selected interface thickness should be sufficiently small. In this study, we propose a simulation model and a procedure for static recrystallization using the phase-field method. A phase-field model [4] that can predict the evolution of a polycrystalline microstructure by solidification and impingement is applied to the recrystallization. In this model, two ordered variables are used: one is the phase field φ which equals zero in the deformed matrix with a high dislocation density and unity in the recrystallized grain; the other is the crystallographic orientation θ of each recrystallized grain. From the numerical simulation point of view, an adaptive finite element method is employed to conduct phase-field simulation efficiently [5]. The stored energy, which is the driving force for the growth of the recrystallized-grain boundary, the crystallographic orientation of recrystallized grains, and information on nucleation are calculated by the crystal plasticity finite element method [6]. 1 Department of Marine Engineering, Kobe University, Higashinada, Kobe, 658-22, Japan 2 Graduate chool of cience and Technology, Kobe University, Nada, Kobe, 657-851, Japan
Crystal Plasticity Finite Element Method The deformation behaviors of polycrystal FCC metals are examined by finite element simulation based on the crystal plasticity theory accounting for statistically stored dislocations (Ds) and geometrically necessary dislocations (GNDs) [6]. In this chapter, we briefly summarize the crystal plasticity finite element method. The following constitutive equation for single crystals that takes the strain-rate-dependent large-strain theory into account is employed here. ij = D e ijkl d kl R (a) ij γ (a), (a) R (a) ij = D e ijkl P(a) kl +W (a) im σ mj σ im W (a) mj (1) where ij is the Jaumann rate of Kirchhoff stress, D e ijkl is the elastic modulus tensor, d ij is the deformation rate tensor, P (a) ij = 1 / ( ) 2 s (a) i m (a) j + m (a) i s (a) j, W (a) ij = 1 / ( ) 2 s (a) i m (a) j m (a) i s (a) j, σ ij is the Cauchy stress tensor, γ (a) is the shear strain rate, and the superscript (a) denotes the a-th slip system. The shear strain rate γ (a) on the a-th slip system is expressed by a power law. γ (a) (a) τ(a) = ȧ g (a) τ (a) g (a) 1 m 1 (2) where ȧ (a), τ (a), and g (a) are the reference shear strain rate, resolved shear stress, and critical resolved shear stress, respectively, and m is the strain rate sensitivity parameter. The critical resolved shear stress g (a) is assumed to be a Bailey-Hirsch type function [7]. g (a) = g (a) + (b) ω ab aµ b ρ (b) a (3) where g (a) and ρ (a) a are the critical resolved shear stress at the initial state and the dislocation density that accumulates on the a-th slip system, respectively. a is a numerical factor on the order of.1 and µ is the elastic shear modulus. The interactions between slip systems are controlled by the interaction matrix ω ab. where ρ (a) The dislocation density on the a-th slip system is assumed to be ρ (a) a = ρ (a) + ρ (a) G, and ρ (a) G are the densities of the statistically stored and the geometrically necessary dislocations, respectively. The evolution equation of ρ (a) ρ (a) = 1 b is described by ( ) 1 L (a) 2y cρ (a) a γ (a) (4)
( ) where L (a) is the mean free path of dislocations and is chosen to be L (a) = L (a) ρ (a) n /ρ(a) a, in which L (a), ρ(a), and n are the initial mean free path of dislocations, the initial dislocation density, and a material parameter, respectively. y c is the characteristic length, and b is the magnitude of the Burgers vector. The density of geometrically necessary dislocations ρ (a) G is expressed as the sum of two density components of the edge and screw dislocations indicated by ρ (a) G,edge = 1 b γ (a) ξ, ρ(a) G,screw = 1 b γ (a) ζ (5) where ξ and ζ denote the directions parallel and perpendicular to the slip direction on the slip plane. The stored energy of plastic deformation E store is calculated using E store = βρ a µ b 2, where ρ a is assumed the average of the dislocation densities ρ (a) a on all slip systems, and β is a constant. The stored energies computed by the crystal plasticity finite element method are transferred into the phase-field method and are used as a driving force of grain boundary migration and information on nucleation in the phase-field simulation. Phase-Field Method The nucleation and growth of recrystallized grains are simulated by the phase-field method. The model proposed here is an extension of the solidification and impingement phase-field model [4] to the recrystallization model. The phase-field model presented here starts from the free-energy functional F = [ ] f (φ)+ α2 2 φ 2 + g(φ)s θ + h(φ) ε2 2 θ 2 dv (6) where f (φ) is the free energy density of a highly deformed matrix or a recrystallized grain. α and ε are the gradient energy coefficients that determine the magnitude of the penalty induced by the presence of grain boundaries. g(φ) and h(φ) are selected as monotonically increasing functions, because the effects of crystalline orientation are eliminated for the recrystallized grains growing in the deformed matrix. Here, we use the simplest form g(φ) = h(φ) =φ 2. f (φ) has the shape of a double well at φ. f (φ)=(1 p(φ)) f m (ρ a )+p(φ) f r (ρ a )+Wq(φ) (7) Here, we use q(φ)=φ 2 (1 φ) 2 which is a double-well potential with minima at φ = and 1, scaled by the well height W and p(φ)=φ 3 ( 1 15φ + 6φ 2), which satisfies p()=, p(1)=1 and p ()=p (1)= because p (φ)=3q(φ). f m (ρ a ) and f r (ρ a ) are the stored energies in the deformed matrix and the recrystallized grain, respectively. Although the
migration of the recrystallized grain boundary is driven by the differences in stored energy in both phases, we set f r (ρ a )= because the dislocation density in the recrystallized grains is a few orders of magnitude smaller than that in the deformed matrix. Therefore, f m (ρ a )=E store and f r (ρ a )=. Assuming that the system evolves in time so that its total free energy decreases monotonically, the evolution equation for the phase field φ: [ t = M φ (φ, θ,t ) α 2 2 η f (φ) g(φ) s θ h(φ) while the orientation field θ should evolve with: θ t = M θ (φ, θ,t ) 1 [ φ 2 g(φ)ε 2 2 θ + h(φ)s θ ] θ ε 2 ] 2 θ 2 (8) (9) where M φ (φ, θ,t )=M φ and M θ (φ, θ,t )=(1 φ) 2 M θ are the mobilities for φ and θ, respectively. The phase field parameters are given by α = 3δσ/b, W = 6σb/δ, and M φ = M 2W/6α [5]. Here, σ is the interface energy, δ is the interface width, M is the mobility of the grain boundary, and b is a constant that can be determined from the definition of the interface region. We chose the orientation parameters M θ = 3M φ, ε = α/5, and s = α 2W/π. Equations (8) and (9) are discretized by the adaptive finite element method that uses fine meshes only around the grain boundary to efficiently perform the simulation [5]. Numerical Procedures and Results As a first step, the stored energy, which is the driving force for recrystallized grain growth, is calculated by crystal plasticity finite element simulation. Figure 1 shows the computational model and boundary condition for the polycrystal FCC metals subjected to compression in a plane strain state. A polycrystal model consisting of 77 grains with random orientations is used. The finite element mesh, in which each quadrilateral consists of four crossed-triangular elements, is regular with 64 elements along each side of the square region. The polycrystal model is compressed up to u/l =.3 at a strain rate u/l = 1 3 1/s. Figure 2 illustrates the deformation pattern and the stored energy distributions at u/l =.3 for the simplest two-slip system model. Next, the nucleation and grain growth are simulated by the phase-field method. The phase-field simulation is conducted in the domain enclosed by the dashed line in Fig.2. The size of the computational domain is 12.4 64 µm. An adaptive finite element method based on the concept of the quadtree data structure is employed in discretizing Eqs.(8) and (9) [5]. ix levels of refinement, namely, level to level 5, are used, and the element size at level is set to dx = 2 5 dx = 6.4 µm, at which dx =.2µm is the minimum element size at level 5. The finite elements for crystal plasticity after deformation are not regular as
u π L = 183 µm (64 elements) x1 6 [J/m 3 ] 2 L = 183 µm (64 elements) Figure 1: Computational model and boundary condition for crystal plasticity simulation Figure 2: Deformation pattern and stored energy distribution at u/l =.3 compression shown in Fig.2, and the element size is different from that for the phase field. Therefore, the stored energy and crystal orientation calculated for the crystal plasticity triangle element are mapped to the nodes in the phase field mesh by Winslow smoothing. Figure 3 (a) shows the recrystallization nuclei together with the stored energy, and initial adaptive mesh. Here, we assume twenty recrystallization nuclei at the initial state. Figures 3 (b) to (e) indicate the time evolution of recrystallized grain growth, and its orientation and adaptive mesh. The growth rate of the recrystallized grains is large at the early stage, because the nuclei are placed in the region in which the magnitude of stored energy is high. After some time, the impingement of the grains can be observed and a new grain boundary between recrystallized grains is produced. Finally, almost all region are filled with recrystallized grains. The following are assumed in the present simulation: σ = 1 J/m 2, δ =4dx, M = 1.42 1 11 m 4 /Js, b = 2.2, and dt =1µs. Conclusion A static recrystallization simulation model and procedure have been proposed using the phase-field method coupled with the crystal plasticity finite element method. Reference 1. Radhakrishnan, B., arma, G. B. and Zacharia, T. (1998): Modeling the kinetics and microstructural evolution during static recrystallization-monte Carlo simulation of recrystallization, Acta Materialia, Vol. 46, pp. 4415-4433. 2. Marx, V., Reher, F. R. and Gottstein, G. (1999): imulation of primary recrystallization using a modified three-dimensional cellular automaton, Acta Materialia, Vol. 47, pp. 1219-123. 3. Kobayashi, R. (1993): Modeling and numerical simulations of dendritic crystal growth, Physica D, Vol. 63, pp. 41-423. 4. Warren, J. A., Kobayashi, R., Lobkovsky, A. E., and Carter, W. C. (23): Extending phase field models of solidification to polycrystalline materials, Acta Materi-
(a) (b) (c) (d) Figure 3: (a) Recrystallization nuclei together with stored energy and adaptive mesh under initial conditions, (b)-(d) time slices of recrystallized-grain growth, orientation, and adaptive mesh in 5, 15 and 3 steps, respectively. alia, Vol. 51, pp. 635-658. 5. Takaki, T., Fukuoka, T., Tomita, T. (25): Phase-Field imulation During Directional olidification of A Binary Alloy Using Adaptive Finite Element Method, J. Crystal Growth, (in print). 6. Higa, Y., awada, Y. and Tomita, Y. (23): Gomputational imulation of Characteristic Length Dependent Deformation Behavior of Polycrystalline Metals, Trans. Japan ociety Mech. Eng., Vol. 69, pp. 523-529. 7. Ohashi, T. (1994): Numerical modelling of plastic multislip in metal crystals of FCC type, Philos. Mag. A, Vol. 7, pp. 793-83.