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[Ikumu Watanabe](https://orcid.org/0000-0002-7693-1675), Daigo Setoyama

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[Multiscale characterization of a polycrystalline aggregate subjected to severe plastic deformation with the finite element method](https://mdr.nims.go.jp/datasets/33e4a82c-e5e5-4c96-b4ff-f599aac0f0b8)

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Multiscale Characterization of a Polycrystalline Aggregate Subjected to Severe Plastic Deformation with the Finite Element MethodMultiscale Characterization of a Polycrystalline Aggregate Subjected  to Severe Plastic Deformation with the Finite Element MethodIkumu Watanabe1 and Daigo Setoyama21National Institute for Materials Science, Tsukuba 305–0047, Japan2TOYOTA Central R&D Labs., Inc., Nagakute 480–1118, JapanThe heterogeneous deformation of a polycrystalline aggregate under severe plastic deformation was reproduced with �nite element anal-ysis by using single-crystal plasticity to characterize the evolution process of the heterogeneity. Finite element analyses of a periodic polycrys-talline aggregate were carried out to simulate the deformation process corresponding to the multi-pass equal-channel angular extrusion process, which reached over 250% of the macroscopic logarithmic accumulated plastic strain. The numerical results were analyzed from multi-scale perspectives: the macroscopic response, evolution of the crystallographic orientation, and deformation state of the microstructure. This study addressed the importance of �nite element discretization to reproduce the heterogeneous deformation of a polycrystalline aggregate, including that of the inside grains, which is a key element for investigating the underlying �ne-graining mechanism.  [doi:10.2320/matertrans.MH201514](Received March 15, 2016; Accepted May 12, 2016; Published June 24, 2016)Keywords:　  �nite element analysis, crystal plasticity, severe plastic deformation, texture1.　  IntroductionFine-graining is currently a recognized strategy for im-proving the material properties in metals. It differs from the conventional materials design methodology by only adding alloy elements. A standard approach to fabricating �ne-grain metals is to impose a severe plastic deformation (SPD) to a metallic material. This is typically done by shape-preserving deformation processes, such as equal-channel angular extru-sion (ECAE)1,2), high-pressure torsion (HPT)3), and accumu-lative roll bonding (ARB)4,5). Hence, �ne-graining is a strate-gy for process-oriented material research and development.Computer-aided engineering (CAE) technologies, of which the �nite element (FE) method is representative, have been employed to simulate and optimize metal-forming processes in industry. These computational approaches have been ap-plied to SPD processes6–8). Recently, CAE technologies have also been applied to investigating the deformation mecha-nism at the micro-scale. FE analysis using a constitutive mod-el of a single crystal, which is commonly known as the crystal plasticity �nite element method (CPFEM), has drawn a great deal of attention as a cross-disciplinary research �eld be-tween mechanics and metallurgy9,10). Advanced crystal plas-ticity models have been developed to describe the strengthen-ing effect of �ne-graining on the basis of dislocation theo-ry11,12). In addition, polycrystal plasticity models have been coupled with homogenization theories and the constitutive model of a single crystal to estimate the evolution of a crys-tallographic texture corresponding to a metal-forming pro-cess. Homogenization theories are classi�ed according to the resolution of the microstructural representations. Mean-�eld theories13,14) and self-consistent theories15,16) are based on full-analytical and semi-analytical homogenization theories in which the crystal grains are modeled as a homogeneous body. Beyerlein et al.17–19) applied a self-consistent theory to investigate the evolution of the crystallographic texture under the ECAE process. They pointed out that a standard self-con-sistent theory has conceptual limitations with regard to the reproducibility of the crystallographic texture after multi-pass ECAE; a grain co-orientation model20) is required to ob-tain a better deformed texture within the framework of self-consistent theories. Within computational discretization approaches based on the homogenization theory, of which the FE method is representative, the deformation state of the mi-crostructure is considered in addition to the macroscopic re-sponse and evolution of the crystallographic texture21,22). Here, the heterogeneous deformation of the inside grains is explicitly treated. The interaction between neighboring grains leading to grain co-orientation behavior can be considered within such computational discretization approaches to de-scribe the morphology of the polycrystalline aggregate, in-cluding the inside crystal grains. However, �ne discretization of the inside grains is de�nitely required in order to discuss the deformation mechanism caused by heterogeneous defor-mations in the microstructure, such as grain sub-division and �ne-graining. Although such computational approaches have the potential to characterize the heterogeneous deformation process of a microstructure subjected to SPD, application studies on SPD problems in this �eld remain an issue owing to the high computational costs.This study used computational simulations of a polycrys-talline aggregate subjected to SPD to investigate the hetero-geneous deformation of the microstructure, including inside grains, and its evolution from multiscale perspectives: the macroscopic response, evolution of the crystallographic ori-entation, and deformation state of the microstructure. First, numerical simulations corresponding to multi-pass ECAE were carried out to reproduce the evolution of the deformed microstructure; the macroscopic deformation state of sin-gle-pass ECAE was repeatedly imposed upon a polycrystal-line aggregate to obtain a deformation microstructure having over 250% of the macroscopic logarithmic accumulated plas-tic strain. The difference between the B and C routes of ECAE was addressed. Finally, the reproducibility of the heteroge-neous deformation of inside grains by using a FE model dis-cretized with a �ne mesh was considered.Materials Transactions, Vol. 57, No. 9 (2016) pp. 1404 to 1410 Special Issue on Advanced Materials Science in Bulk Nanostructured Metals III ©2016 The Japan Institute of Metals and Materialshttp://dx.doi.org/10.2320/matertrans.MH201514http://dx.doi.org/10.2320/matertrans.MH2015142.　  Computational Method Using Finite Element Analy-sis of a Polycrystalline AggregateIn this study, a computational framework based on FE analysis of a periodic microstructure and a constitutive model of a single crystal were employed to solve the deformation problem of a polycrystalline aggregate.2.1　  Boundary value problem for a periodic microstruc-tureBridging the microstructure and bulk property is a classical research �eld in continuum mechanics, where the bulk prop-erty is estimated from averaging the state of the microstruc-ture. In such homogenization approaches, the microstructure is modeled as a representative volume element (RVE), or a representative part of the objective microstructure. In compu-tational approaches22,23), the RVE is usually assumed to be a periodic microstructure, and the boundary value problem of the RVE is formulated as ΩYP : ∇Yη(1)dΩY = 0 ∀η(1) ∈ Wperiodic, (1)where Y is the coordination system on a micro-scale, P is the �rst Piola–Kirchhoff stress, η(1) is the variation in the period-ic displacement u(1), dΩY denotes the differential volume of the overall RVE ΩY, and Wperiodic is the Sobolev space of the periodic function. The displacement �eld (w) on a micro-scale is de�ned as w =  H̄  Y +  u(1), where H̄   is the macroscopic dis-placement gradient. Then, the displacement gradient (H) on a micro-scale is given as H = ∇Yw = H̄ + ∇Yu(1), (2)where ∇Y is the gradient operator. The macroscopic stress is de�ned as the volume average of the corresponding micro-scopic variable in the overall RVE. P̄ :=1ΩY ΩYPdΩY . (3)Following the de�nition of periodicity and eq. (2), the dif-ference in displacements at points A and B that satis�es the periodicity is written as follows: wA − wB = H̄(YA − YB). (4)Equation (4) relates the macroscopic displacement gradient H̄   to the displacement �eld w on a micro-scale. By using eqs. (3) and (4), FE analysis of the periodic microstructure is carried out by controlling the macroscopic displacement gra-dient H̄   or macroscopic stress P̄  .2.2　  Constitutive model of a single crystalThe heterogeneity of a polycrystalline aggregate is de�ned by the anisotropy of a crystal grain and its orientations. The anisotropic mechanical behavior of a crystal grain is charac-terized by an elastoplastic constitutive model of a single crys-tal.2.2.1　  Constitutive model of a single crystalIn metallic materials, the elastic deformation is generally small enough that the elastic response can be regarded as lin-ear. Anisotropic linear elasticity can be employed to describe the elastic deformation. Based on the multiplicative decom-position of the elastoplastic deformation gradient F =  FeFp, the constitutive equation at an intermediate con�guration (i.e., the con�guration pulled back from the current con�gu-ration by the elastic deformation gradient) is de�ned with the fourth-order elastic constant Ĉe  as follows: Ŝ = Ĉe :12FeTFe − 1 , (5)where Ŝ  is the second Piola–Kirchhoff stress and 1 is the sec-ond-order identity tensor at the intermediate con�guration.For the plasticity of a single crystal, the anisotropic plastic deformation is characterized by slip systems dependent on the crystal structure. The yield function of the α-th slip sys-tem is de�ned in a strain-rate-independent format as follows: φ(α) := |τ(α)| − q(α) ≤ 0, (6)where τ  (α) is the stress norm and q(α) is the relevant yield stress, including the plastic hardening. The stress norm, which is generally called the resolved shear stress, is de�ned at the intermediate con�guration as τ(α) := FeTFeŜ : s(α)0 ⊗ m(α)0  (7)where s(α)0  and m(α)0  represent the slip direction vector and nor-mal vector, respectively, of the slip surface of the α-th slip system at the intermediate con�guration. nslip is the number of the slip systems. For the plasticity of a single crystal, the crystal lattice does not rotate with the slip deformation. Then, the vectors of the slip system at the current con�guration can be denoted as s(α) =  Fes(α)0   and m(α) =  Fe−Tm(α)0  . Here the fol-lowing evolution equation of the plastic deformation gradient Fp is employed: ḞpFp−1 =nslipα=1γ(α)sign τ(α) s0 ⊗ m0, (8)where γ(α) is a plastic �ow of the active slip system α. The slip history variable is de�ned as the time integration of the �ow, that is, ξ(α) =t0γ(α)dt ∀α ∈ [1, nslip]. (9)In this study, the yield stress, or critical resolved shear stress, was de�ned phenomenologically as a nonlinear func-tion of the slip history variables ξ(1), · · · , ξ(nslip) : q(α) := τ(α)0 + δτ(α)1 − exp−h(α)0δτ(α)nslipβ=1Ωαβξ(β) , (10)where τ(α)0 , δτ(α), h(α)0 , and Ωαβ are material constants.In this study, exponential mapping using the evolution law of the elastic/plastic deformation gradient and a generalized inverse matrix on an implicit stress-update algorithm were employed for ef�cient and robust FE simulations23,24).2.2.2　  Model settingsIn this study, two FE meshes were prepared for the follow-ing simulations, as shown in Fig. 1. Both were assumed to satisfy the geometric periodicity condition and were com-posed of 54 and 16 crystal grains with the same volume and geometry of a truncated octahedron. These were discretized as eight-node hexahedral �nite elements. Models (a) and (b) 1405Multiscale Characterization of a Polycrystalline Aggregate Subjected to Severe Plastic Deformation with the Finite Element Methodwere discretized into 4320 elements (80 elements per grain) and 37,807 elements (2160 elements per grain), respectively. The crystallographic orientations of each grain were provided in a random fashion. The objective material was assumed to be pure copper composed of face-centered cubic (FCC) crys-tals. The initial pole �gures of {111} are also shown in Fig. 2, which contains the number of points of crystal grains and the slip planes. The FE model (b’), which had the same structure as model (b) and was discretized into 1280 elements (80 ele-ments per grain), was prepared for comparison.For the elastic constitutive model, the elastic constants at room temperature were taken from a database25) as follows: Ĉe1111 = 170,000.MPa, Ĉe1122 = 120,000.MPa,Ĉe1212 = 75,000.MPa (11)For the plasticity, the material constants of eq. (10) were de-termined on the basis of experimental data on wire-drawing26) as follows: τ(α)0 = 30.MPa, δτ(α) = 180.MPa, h(α)0 = 70.MPa (12)Here, the interaction matrix Ωαβ between slip systems is de-�ned as two parts: self-hardening (α =  β) and latent hardening (α ≠  β). The range of the ratio of the self-hardening and latent hardening coef�cients is known to be [1.0,  1.4] for FCC crys-tals27). Then, the following values were assumed for Ωαβ : Ωαα = 1.0, Ωαβ = 1.1 (if α β) (13)Note that the above material constants were determined to reproduce the experimental stress–strain curve of the objec-tive materials phenomenologically because the constitutive model and material constants do not consider the macroscop-ic response of a polycrystalline aggregate and all micro- and nanoscopic mechanisms.Figure 3 shows the stress–strain responses of the single crystal when uniaxial tensile stresses are applied in three dif-ferent directions. The equivalent strain is de�ned as ε∗ :=23dev12ln FFT : dev12ln FFT . (14)Based on the above constitutive model and material con-stants, FE analyses were carried out on models (a) and (b’) to evaluate the macroscopic initial anisotropy. Macroscopic uni-axial tensile stresses were applied in the three orthogonal di-rections. Figure 4(a) shows the macroscopic equivalent stress–strain curves. Although the same material constants were employed in both simulations, the macroscopic harden-ing behaviors were different. In general, these macroscopic responses converged with an isotropic response as more crys-tal grains were considered.FE analyses of macroscopic uniaxial tensile stress for Y1-direction were also carried out on Models (b) and (b’), which had the same morphology, to investigate the effect of the discretization. As shown in Fig. 4(b), the �ne-mesh model (b) provided a slightly softer response than the coarse mesh model (b’), which is a well-known feature of a standard FE Fig. 1　Finite element models of the periodic polycrystalline aggregate. (a) 54 grains, 4320 elements, (b) 16 grains, 34,560 elements.Fig. 2　Initial pole �gures of {111} of �nite element models (Fig. 1). (a) 54 grains, (b) 16 grains.Fig. 3　Anisotropic stress–strain responses of a copper single crystal.1406 I. Watanabe and D. Setoyamaformulation28). The results converged with a true solution to the mathematical problem when the �ner mesh was used for the discretization.The number of elements, FE formulation, degrees of free-dom, and performance of the interpolation function are essen-tial to describing the mechanical behavior. In our previous studies29,30), FE model (a) was con�rmed to basically be suf-�cient to evaluate the macroscopic averaged response. How-ever, the macroscopic response was only one of the analysis objectives. Depending on the objective, convergence should be con�rmed from not only the macroscopic viewpoint but also the microscopic viewpoint, such as regarding the defor-mation state and stress distribution of the microstructure. Considering the computational cost, it is not easy to develop a fully converged FE model in practice. In the current state, the discretization of FE model (b) was our best effort in the following simulations.3.　  Finite Element Simulations of Polycrystalline Aggre-gate Subjected to Severe Plastic DeformationSPD corresponding to multi-pass ECAE was imposed on the FE model of a periodic polycrystalline aggregate (Fig. 1(a)) to examine the evolution processes of the micro-structure.3.1　  Numerical simulationsThe macroscopic displacement gradient corresponding to an orthogonal ECAE was repeatedly imposed on the FE mod-el for the simulations. The deformation state of an orthogonal ECAE can be described as a simple shear deformation. Then, the macroscopic displacement gradient can be described as H̄ =0 0 00 0 0γ 0 0 , (15)where γ is the imposed simple shear strain. Based on the re-sults of the FE simulation of the corresponding ECAE sys-tem, γ was estimated to be 2.1, which is about 1.05 of the equivalent strain. A single pass of the ECAE is de�ned as the macroscopic displacement gradient (15) being imposed on the periodic polycrystalline aggregate shown in Fig. 1(a) and the macroscopic stress being released during the unloading process. Based on the single-pass simulation, the multi-pass simulation involves rotating and repeatedly imposing the macroscopic displacement gradient of (15). The billet was ro-tated 90◦  and 180◦  around the extruding direction (i.e., Y3-di-rection) during each pass on routes B and C, respectively.3.2　  Results and discussionFigure 5(a) shows the resulting macroscopic equivalent stress–strain curves. The macroscopic equivalent strain be-came almost zero after four passes on route B and two and four passes on route C. To evaluate the amount of imposed strain, the macroscopic accumulated plastic strain was de-�ned as the volume average of the logarithmic accumulated plastic strain: εp∗ :=1ΩY ΩYln[1 + ξ]dΩY , (16)where the accumulated plastic strain ξ is de�ned as ξ :=nslipα=1ξ(α), (17)Fig. 4　Macroscopic equivalent stress–strain curves of polycrystalline ag-gregates for the macroscopic uniaxial tensile stress. (a) Effect of number of grains, (b) Effect of discretization.Fig. 5　Macroscopic stress–strain curves of multi-pass ECAE. (a) macro-scopic equivalent stress–strain curves, (b) macroscopic equivalent stress- plastic strain curves.1407Multiscale Characterization of a Polycrystalline Aggregate Subjected to Severe Plastic Deformation with the Finite Element Methodthat is, the accumulated plastic strain ξ represents the amount of the inelastic strain at the crystal scale. Based on the above de�ned variable, the macroscopic responses in Fig. 5(a) were redrawn as shown in Fig. 5(b) to re�ect the relationship be-tween the macroscopic equivalent stress and macroscopic ac-cumulated plastic strain, which reached over 250% of the logarithmic strain. Figure 5(b) indicates that route B was more effective at imposing the plastic strain than route C. Fig-ure 6 and Fig. 7 depict the deformation states and distribu-tions of the equivalent stress and accumulated plastic strain of the microstructures after four-pass ECAE. Even though the macroscopic stress was almost zero in both cases, a relatively high stress remained in the microstructure. The deformation states were completely different between routes B and C. In the case of route B, the microstructure was deformed in a complicated manner, and the accumulated plastic strain was distributed homogeneously by the imposition of different macroscopic strain modes at each pass. In the case of route C, in the other hand, the shape of the microstructure was pre-served even after four-pass ECAE, and the accumulated plas-tic strain was distributed locally because of the monotonic macroscopic strain modes. Note that further deformation analysis of multi-pass ECAE would possibly lose computa-tional accuracy, especially for route B, because the deformed FE model of route B contains distorted elements, as shown in Fig. 6 and Fig. 7.Figure 8 graphs the evolution of crystallographic orienta-tions as a density plot. This was used to analyze the heteroge-neous deformation states, where the horizontal axis is the extrusion direction. In experimental work18), the texture of the simple shear strain was observed after every pass of ECAE. Figure 8 clearly shows that the crystallographic tex-ture of the simple shear strain appeared with the �rst and sec-ond passes on route B and �rst and third passes on route C. However the texture was weak on the other passes, which in-volved a reversed strain. Li et al.18) reported the same tenden-cy with the standard self-consistent model; the grain co-rota-tion behavior contributed to obtaining the texture during the ECAE pass with the reversed strain. This implies that the heterogeneous deformation of the polycrystalline aggregate was inadequately represented–that is, the discretization of the Fig. 6　Deformation state and equivalent stress distribution of microstructure after four-pass ECAE.Fig. 7　Deformation state and accumulated plastic strain distribution of microstructure after four-pass ECAE.Fig. 8　Evolution of pole �gure of {111} in multi-pass ECAE.1408 I. Watanabe and D. SetoyamaFE model was insuf�cient to characterize the heterogeneous state of the inside grains.4.　  Characterization of Heterogeneous Deformation of Inside GrainsThe deformation analysis of a polycrystalline aggregate was carried out by using a FE model discretized with a �ne mesh to focus on the heterogeneous deformation state of the inside grains.4.1　  Numerical simulationsSimilar to the work presented in the previous section, the macroscopic deformation corresponding to two-pass ECAE on route C was imposed on a FE model, as shown in Fig. 1(b). For comparison, the same calculation was carried out with the coarse-mesh model (b’).4.2　  Results and discussionThe macroscopic stress–strain responses are shown in Fig. 9. The macroscopic equivalent stress and strain were al-most the same with the �ne and coarse meshes. However, there was a remarkable difference in the macroscopic accu-mulated plastic strain. The deformation state and distribution of accumulated plastic strain are drawn in Fig. 10. In the case of coarse mesh (b’), the deformation state was homogeneous in comparison with that of the �ne mesh (b). Therefore, the difference in the macroscopic accumulated plastic strain was caused by the discretization of the microstructure.Figure 11 shows the states of the crystallographic orienta-tion {111}. Figure 12 indicates the intensity of pole �gure Fig. 11 along the horizontal line, which is normalized with the average values of overall pole �gures. The �ne mesh mod-el provided the smoother distribution of the density in these pole �gures. That is, the coarse mesh model has still a short-age of the ability to express the heterogeneous deformation state. In this context, the crystallographic texture of simple shear was observed in the case of the �ne mesh, even after the second pass of route C. Figure 13 shows the distributions of the nominal vector component of the {111} slip plane as the corresponding data from the microscopic perspective. These Fig. 10　Deformation state and accumulated plastic strain distribution of the microstructure after two-pass ECAE on route B.Fig. 11　Evolution of pole �gure of {111} with multi-pass ECAE.Fig. 12　Comparison of intensity of pole �gure along horizontal line of Fig. 11.Fig. 9　Macroscopic stress–strain curves of the �ne-mesh model.1409Multiscale Characterization of a Polycrystalline Aggregate Subjected to Severe Plastic Deformation with the Finite Element Methodvalues were continuously distributed through the original grain boundary, which represents the interaction effect be-tween the crystal grain and grain co-rotation behavior. These results are consistent with those of the previous study18), which addressed the importance of the grain co-rotation. Thus, we conclude that �ne discretization is conceptually re-quired to represent the heterogeneous deformation at the mi-cro-scale. However, such computations are expensive. For reference, the simulation of the �ne-mesh model consumed a CPU time of about 2.0 ×  106 s.5.　  ConclusionsWe carried out FE analyses to reproduce the heterogeneous deformation after multi-pass ECAE where a large amount of macroscopic logarithmic accumulated plastic strain was im-posed on a polycrystalline aggregate. The numerical results were analyzed from multi-scale viewpoints, where the under-lying mechanism of the macroscopic response was explained on the basis of the evolution of the crystallographic texture and the deformation state of the microstructure. Based on the results, computational discretization methods have the poten-tial to deal with the deformation process by explicitly consid-ering the interaction effect between crystal grains. However, massive computational efforts are required to carry out prac-tical simulations using an FE mesh that is suf�ciently �ne to reproduce the heterogeneity of the deformed microstructure.This study demonstrated the dif�culty with setting up FE models of a microstructure. Obviously, it would be more dif-�cult to validate the numerical simulations by using advanced constitutive models11,12). A method to validate simulations and models should be established in this �eld. The simulation in this study can be used as a benchmark for FE mesh valida-tion.AcknowledgementThis research was supported by Grants-in-Aid for Scientif-ic Research on Innovative Areas “Bulk Nanostructured Met-als” (No. 23102513 and No. 25102711) and Young Scientists (No. 15K18205).REFERENCES 1)   V.M. Segal: Mater. Sci. Eng. A 271 (1999) 322–333. 2)   M. Furukawa, Z. Horita, M. Nemoto and T.G. Langdon: J. Mater. Sci. 36 (2001) 2835–2843. 3)   A. Zhilyaev and T. Langdon: Prog. Mater. Sci. 53 (2008) 893–979. 4)   Y. 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Owen, Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons Ltd., Hoboken, NJ, 2008. 29)   K. Terada, I. Watanabe and M. Akiyama: Int. J. Multiscale Computa-tional Engineering 4 (2006) 445–460. 30)   I. Watanabe, K. Terada, E.A. de Souza Neto and D. Peric: J. Mech. Phys. Solids 56 (2008) 1105–1125.Fig. 13　Distribution of the nominal vector component of the {111} slip plane. (a) Y1-component, (b) Y2-component.1410 I. Watanabe and D. 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