An integrated fast Fourier transform-based phase-field and crystal plasticity approach to model recrystallization of three dimensional polycrystals

Abstract

A fast Fourier transform (FFT) based computational approach integrating phase-field method (PFM) and crystal plasticity (CP) is proposed to model recrystallization of plastically deformed polycrystals in three dimensions (3-D). CP at the grain level is employed as the constitutive description to predict the inhomogeneous distribution of strain and stress fields after plastic deformation of a polycrystalline aggregate while the kinetics of recrystallization is obtained employing a PFM in the plastically deformed grain structure. The elasto-viscoplastic equilibrium is guaranteed during each step of temporal phase-field evolution. Static recrystallization involving plasticity during grain growth is employed as an example to demonstrate the proposed computational framework. The simulated recrystallization kinetics is compared using the classical Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory. This study also gives us a new computational pathway to explore the plasticity-driven evolution of 3D microstructures.

Introduction

Microstructure plays a crucial role in determining the properties of polycrystalline materials, which therefore stimulated enormous efforts to tailor the microstructure of polycrystals by a combination of thermal and mechanical processes. One widely used process is static recrystallization (SRX) by annealing of plastically deformed grain structures  [1], [2]. The kinetics of recrystallization, i.e. the volume fraction of recrystallized grains as a function of time, is often described by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model  [3], [4] based on the assumptions that the nucleation rate is constant or the number of nucleation sites is fixed, constant growth velocity, and spherical grain shapes until impingement. JMAK theory assumes a homogeneous deformed state with a constant driving force and does not provide the microstructural details during recrystallization. To overcome these shortcomings, numerous attempts have been made to model the recrystallization process using meso-scale computational methods such as Monte Carlo Potts model  [5], [6], [7] cellular automata model  [8], [9] and isogeometric method  [10], [11] to take into account the evolution of grain structures during recrystallization.

On the other hand, phase-field method (PFM) has been widely applied to model various meso-scale phenomena, e.g. solidification  [12], [13], solid-state transformation  [14], recrystallization  [15], [16], [17], [18] and grain growth  [19], [20], [21]. It can easily handle time-dependent growth geometries and describe complex microstructure morphologies, which make it particularly suitable for modeling microstructure evolution where morphological complexities are common. However, most of existing PFMs incorporate the strain energy contribution to microstructure evolution in the elastic regime. Both experimental and computational results have shown that the stresses in the context of polycrystals or microstructures can significantly exceed the elastic limit. Therefore, a PFM including not only the driving forces originating from the elastic fields, but also the driving forces resulting from the plastic activities is necessary for modeling microstructure evolution.

Plastic deformation can be introduced in two different ways in the context of PFM. Since plasticity in crystals is primarily due to the generation and motion of dislocations, one approach is to explicitly introduce mobile dislocations  [22], [23], [24] using continuous fields for each slip system. In this approach, it is necessary to resolve the dislocation core size using several numerical grid spacing. In this case, the description of a realistic dislocation core size, which is important for the short-range interaction between dislocations, would require a very refine grid size  [23]. Therefore, the spatial length scale in this approach is limited, and large scale simulations are computationally expensive. In addition, plastic deformation mechanisms other than dislocation glide (e.g. climb and/or cross-slip at high temperatures, or twining in materials with law stacking-fault energy) are not included in this approach.

Another approach is to directly include a plastic strain field defined at the meso-scale in PFM  [25]. For example, Boussinot et al.  [26] employs a decrease in the lattice misfit to account for the plastic activity. Zhou et al.  [27] relates the plastic strain to the inter-dislocation distance, i.e. the dislocation density. In particular, the crystal plasticity (CP) theory has been rigorously formulated  [28] and extensively used to obtain the micromechanical response of plastically deforming polycrystalline aggregates.

A number of efforts have made to couple PFM and plasticity theory at the meso-scale. The first attempt to couple PFM with an isotropic plasticity model was proposed by Guo et al.  [29], who investigated the stress fields around defects such as holes and cracks. Later, Ubachs et al.  [30] proposed a general formalism to incorporate phase-field and isotropic viscoplasticity with non-linear hardening for investigating tin–lead solder joints undergoing thermal cycling. Subsequently, similar approaches have been introduced to study crystal growth  [31], martensites  [32], superalloys [27], [33] and diffusion controlled growth kinetics  [34], [35].

There have also been attempts to employ the PFM integrating plasticity to simulate the recrystallization, either in the context of a dislocation density based plastic model  [15], or in a CP framework, including both hardening and viscosity  [8], [16]. For example, Takaki et al.’s model  [15] assumed a homogeneous dislocation density field in each grain. However, ignoring the intragranular heterogeneity may lead to a poor representation of recrystallization kinetics and microstructure evolution. On the other hand, a successful PFM/CP coupling depends, to a large extent, on the availability of efficient, and yet reliable, CP implementations. In this sense, while the finite element method (FEM) has been extensively used to deal with problems involving CP (for an excellent review on CP-FEM, see  [28]), the large number of degrees of freedom required by such CP-FEM calculations limits the size of the aggregates that can be investigated by this method.

Conceived as an alternative to CP-FEM, a formulation inspired by image-processing techniques and based on the spectral FFT algorithm has been recently proposed to predict the micromechanical behavior of plastically deforming heterogeneous polycrystals  [36], [37], [38], [39], [40]. Owing to being free from any large matrix inversion, this spectral FFT formulation is very computationally efficient. It is numerically demonstrated that the computational time of the CP-FEM solver is about 25–40 times more than that of the CP-FFT counterpart when achieving the same level of fidelity  [41]. Such cheap computation makes the FFT solver an excellent candidate to incorporate fine-scale microstructural information in plastic deformation simulations.

In this paper, we propose to couple our previous FFT-based PFM  [19], [20] and the CP-FFT model  [36], [37], [38], [39], [40], by taking advantage of the high efficiency in the FFT solver, to model the recrystallization of plastically deformed polycrystalline materials in 3-D. In this approach, the plastic strain field is first calculated with the CP-FFT approach for its subsequent use by the FFT-based PFM for the determination of the driving forces for recrystallization. The use of the FFT algorithm in both PFM and CP not only guarantees their seamless integration from a numerical perspective, but also helps us to significantly enhance the computational efficiency. The elasto-viscoplastic equilibrium is solved during each step of temporal phase-field evolution. The proposed computational framework is applicable to other plasticity-driven phase-field evolution processes.

The plan of the paper is as follows: in Section  2 we summarize the essential aspects of the FFT-based CP model and present how elastic equilibrium for each step of phase-field evolution is solved. Next, Section  3 describes the formulation of the PFM for static recrystallization of plastically deformed polycrystals, paying special attention to the determination of the plastic driving forces. Examples of phase-field simulation of recrystallization process are described in Section  4, where a simply case with only one deformed crystal is designed to validate the recrystallization kinetics of the proposed model, followed by a realistic case with multiple deformed grains. Finally, we draw our conclusions in Section  5.