Crystallographic Homogenization Research Articles

Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method

A semi-implicit elastic/crystalline viscoplastic finite element (FE) method based on a “crystallographic homogenization” approach is formulated for a multi-scale analysis. In the formulation, the asymptotic series expansion is introduced to define the displacement in the micro-continuum. This homogenization FE analysis is aimed at predicting the plastic deformation induced texture evolution of polycrystalline materials, the constituent microstructure of which is represented by an assembly of single crystal grains. The rate dependent crystal plasticity model is adopted for the description of microstructures. Their displacements are decomposed into two parts: the homogenized deformation defined in the macro-continuum and the perturbed one in the micro-continuum. This multi-scale formulation makes it possible to carry out an alternative transition from a representative micro-structure to the macro-continuum. This homogenization procedure satisfies both the compatibility and the equilibrium in the micro-structure. This developed code is applied to predict the texture evolution, and its performance is demonstrated by the numerical examples of texture evolution of FCC polycrystalline metals.

Process metallurgy analyses to design a high-bendability and high-springback property sheet by using two-scale finite element method

In this study, we develop bendability and springback prediction analysis code for an optimum crystal texture design to generate an ideal aluminum alloy sheet through the sheet rolling and heat treatment processes. To elucidate the relationships between the sheet metal formability and the crystal texture, we applied our two-scale finite element (FE) procedure based on the crystallographic homogenization method to analyze the bending and springback process. Our code employed two-scale FE model, such as the microscopic polycrystal structure and the macroscopic elastic plastic continuum by introducing the crystal orientation distribution, such as the texture characteristics. It means that our code can predict the plastic deformation of sheet metal in the macro-scale, and the crystal texture and hardening evolutions in the micro-scale. The macro-FE model consisted of the die, the punch and the sheet metal. The die and the punch were modelled as the rigid bodies in “V-bend” test problem. The crystal orientation distribution was employed in the microscopic polycrystal FE model, which was assigned by a three-dimensional representative volume element (RVE). This FE model was used as the initial textures for “V-bend” process analyses. The RVE model was featured as 3×3×3 equi-divided iso-parametric solid elements, totally 27 FEs with 216 crystal orientations. Bendability was evaluated by the coefficient of shear strain concentration index using two-scale FE results. On the other hand, the springback characteristics were evaluated by springback angle defined by the angular difference between before and after springback, which occurred in the punch and die removing process. We studied two relationships between (1) the bendability and (2) the springback. Furthermore, we designed the polycrystal texture through the asymmetric rolling and annealing heat treatment process to generate a high-bendability and high-springback property polycrystal material. Annealing heat treatment was modeled as the growth of Cube {0 0 1}〈1 0 0〉 orientation by using the Johnson–Mehl–Avrami׳s equation. In the process optimization, we adopted the asymmetric rolling ratio and the annealing heat treatment time for the design parameters, and the bendability factor and the springback angle as the objective functions. The response surface algorithm was used to optimize the design parameter for maximizing the bendability and minimizing the springback angle. As an optimized result, the asymmetric ratio 1.13 and the annealing heat treatment time 13.5 min were obtained.

Two-scale finite element analyses for bendability and springback evaluation based on crystallographic homogenization method

Abstract In this study, a relationship between the sheet metal formability, such as the bendability and springback property, and the crystal texture was investigated by using our two-scale finite element (FE) analysis code based on the crystallographic homogenization method (Nakamachi et al., 2010 [1] ). Our code employed a two-scale finite element model, such as the microscopic polycrystal structure and the macroscopic elastic plastic continuum, which can predict the anisotropic plastic deformation of sheet metal in the macro-scale, and the crystal texture and hardening evolutions in the micro-scale. The macro-FE model consisted of the die, punch and sheet metal. The die and punch were modeled as the rigid bodies for “V-bending” process analyses. The measured crystal orientation distribution was adopted as the initial texture in the microscopic polycrystal FE model, which corresponded to the three-dimensional representative volume element (RVE). RVE model was featured as 3×3×3 equi-divided solid finite elements, totally 27 FEs with 216 crystal orientations assigned at the integration points of micro-finite elements. The bendability was evaluated by the surface wrinkle growth index and the strain localization – the shear band formation – by using two-scale FE results. On the other hand, the springback property was defined by the angular difference between before and after the punch and die removing process – the post-process. At first, we investigated the bendability of single crystal sheets, which have typical preferred orientations of the copper alloy sheet, to elucidate the fundamental mechanism of shear band formation and wrinkle growth in the V-bending process. Next, we analyzed V-bending processes of four copper alloy polycrystal sheets to evaluate the bendability and springback property. The Cube-dominant texture sheet of the Corson series copper alloy and the {001}〈110〉-dominant texture sheet of the phosphor-bronze alloy show a high-bendability. On the other hand, the S-dominant texture sheet of the Corson series copper alloy and the Brass-dominant texture sheet of the phosphor-bronze alloy show a low-bendability. By contraries, the Cube-dominant and {001}〈110〉-dominant texture sheets have a low-springback property, and the S-dominant and Brass-dominant textures sheets have a high-springback property. Our two-scale FE results of four copper alloy sheets in the V-bending and the post-process were compared with experimental results, and finally the validity of our two-scale FE code was confirmed.

Simulation of yield surfaces for aluminium sheets with rolling and recrystallization textures

The influence of crystallographic texture and homogenization scheme on the yield surface for aluminium sheets has been studied numerically. A rate-dependent single crystal plasticity model was adopted to describe the behaviour of the grains, whereas the behaviour of the polycrystal was obtained either by the crystal plasticity finite element method (CP-FEM) or by the full-constraint Taylor approach. A representative volume element (RVE) of the material including 800 grains was considered. The grain orientations were defined according to typical rolling and recrystallization textures exhibited in rolled aluminium sheets. With CP-FEM, the RVE was discretized with finite elements, using in average 30 cubical solid elements per grain, whereas in the full-constraint Taylor approach, all grains are assumed to be subjected to the same deformation. Stress states located on the yield surface were determined by subjecting the RVE to proportional deformation in different directions, assuming yielding to occur at a given plastic work per unit volume. The yield surface was then described analytically by fitting the Yld2004-18p yield function to the obtained stress states at yield, and further employed to calculate stress ratios and Lankford coefficients in uniaxial tension. The resulting yield surfaces for aluminium with rolling and recrystallization textures are markedly different. The influence of homogenization scheme is moderate, but significant effects are found on the predicted Lankford coefficients.

426 ドメイン・スイッチングおよび構造相転移を考慮した強誘電体のマルチスケール非線形解析

The perovskite ferroelectric Pb(Zr_xTi_<1-x>)O_3 (PZT) at morphotropic phase boundary (MPB) consist of multiple crystal systems and it shows complicated hysteresis behavior and high piezoelectric responses depending on domain switching and structural phase transition. In the present work, a multiscale nonlinear finite element method based on crystallographic homogenization theory was developed to evaluate nonlinear behaviors at MPB. Some computational results of typical ferroelectric behaviors were presented for PZT dual phase models.

OS0612 ドメイン・スイッチングおよび構造相転移を考慮した強誘電体のマルチスケールシミュレーション

The perovskite ferroelectric Pb(Zr_xTi_<1-x>)O_3 (PZT) at morphotropic phase boundary (MPB) consist of multiple crystal systems and it shows complicated hysteresis behavior and high piezoelectric responses depending on domain switching and structural phase transition. In the present work, a multiscale nonlinear finite element method based on crystallographic homogenization theory was developed to evaluate nonlinear behaviors at MPB. We employed the incremental constitutive law taking into consideration with material property changes caused by domain switching and structural phase transition and the judgment formulation based on electromechanical energy. Some computational results of typical ferroelectric behaviors were presented for PZT.

Crystallographic homogenization finite element method and its application on simulation of evolution of plastic deformation induced texture

A crystallographic homogenization method is proposed and implemented to predict the evolution of plastic deformation induced texture and plastic anisotropy (earring) in the stamping of polycrystalline sheet metals. The microscopic inhomogeneity of crystal aggregate has been taken into account with the microstructure made up of a representative aggregate of single crystal grains. Multi-scale analysis is performed by coupling the microscopic crystal plasticity with the macroscopic continuum response through the present homogenization procedure. The macroscopic stress is defined as the volume average of the corresponding microscopic crystal aggregations, which simultaneously satisfies the equation of motion in both micro- and macro-states. The proposed numerical implementation is based on a finite element discretization of the macro-continuum, which is locally coupled at each Gaussian point with a finite element discretization of the attached micro-structure. The solution strategy for the macro-continuum and the pointwise-attached micro-structure is implemented by the simultaneous employment of dynamic explicit FE formulation. The rate-dependent crystal plasticity model is used for the constitutive description of the constituent single crystal grains. It has been confirmed that Taylor's constant strain homogenization assumption yields an undue concentration of the preferred crystal orientation compared with the present homogenization in the prediction of texture evolution, with the latter having relaxed the constraints on the crystal grains. Two kinds of numerical examples are presented to demonstrate the capability of the developed code: 1) The texture evolution of three representative deformation modes, and 2) Plastic anisotropy (earring) prediction in the hemispherical cup deep drawing process of aluminum alloy A5052 with initial texture. By comparison of simulation results with those obtained employing direct crystal plasticity calculation adopting Taylor assumption, conclusions are drawn that the proposed dynamic explicit crystallographic homogenization FEM is able to more accurately predict the plastic deformation induced texture evolution and plastic anisotropy in the deep drawing process.

316 結晶均質化法マルチスケール有限要素解析による板材のミクロ-マクロ変形特性評価(GS-4 加工(2))

316 結晶均質化法マルチスケール有限要素解析による板材のミクロ-マクロ変形特性評価(GS-4 加工(2))

Multi-scale parallel finite element analyses of LDH sheet formability tests based on crystallographic homogenization method

A multi-scale parallel finite element (FE) procedure based on the crystallographic homogenization method was applied to the LDH sheet formability test analysis. For the multi-scale structure, two scales are considered. One is a microscopic polycrystal structure and the other is a macroscopic elastic plastic continuum. The analysis code can predict the formability of sheet metal in macro-scale, simultaneously the crystal texture and hardening evolutions in the micro-scale (Nakamachi E et al. Int J Plasticity 2007;23:450–8). Since huge computation time is required for the nonlinear dynamic multi-scale FE analysis, parallel computing technique based on domain partitioning of FE model for macro-continuum is introduced into the multi-scale code using the message passing interface (MPI) library and PC cluster (Kuramae H et al. In: Proceedings of the eighth international conference on computational plasticity, Part 1, 2005. p. 622–5). The explicit time stepping solution scheme in the nonlinear multi-scale FE dynamic problem is well-suited for parallel computing on distributed memory environment such as PC cluster because solving simultaneous equation is not required. We measured crystal morphologies of four automotive sheet metals, aluminum alloy sheet metals A6022-T43 and A5182-O, an asymmetrically rolled aluminum alloy sheet metal A6022-ASR, and mild steel HC220YD, by using the scanning electron microscope (SEM) with electron back scattered diffraction (EBSD) analyses, and defined a three-dimensional representative volume element (RVE) of micro polycrystal structure, which satisfy the periodicity condition of crystal orientation distribution. We evaluate not only macroscopic formability of the automotive sheet metals by the multi-scale LDH test analysis, but also microcrystalline texture evolution during plastic deformation. Furthermore, a relationship between the macroscopic formability and the microcrystal texture evolution was discussed through looking at multi-scale FE results. It is concluded that the mild steel HC220YD was the highest formability than the aluminum alloy sheet metals because of remaining and generating the γ-fiber texture, such as {1 1 1}〈1 1 0〉–{1 1 1}〈1 1 2〉 orientations, during plastic deformation.

Proposition of a First-Principles Aided Triple-Scale Analysis for Biocompatible Piezoelectric Thin Films

A process crystallography algorithm based on the first-principles aided triple-scale analysis is newly developed to design biocompatible piezoelectric thin films fabricated on a substrate. The pseudo-potential method within the density functional theory was used to predict the crystal morphology of thin film, such as preferred orientations and their fractions as well as the structural stability of possible conformations of crystal clusters on substrates. A crystal morphology at the micro scale was selected and macro homogenized properties of piezoelectric thin film were obtained through a double-scale finite element analysis based on the crystallographic homogenization theory. Further, our analysis was applied to the existent biocompatible piezoelectric BaTiO3 thin films, fabricated on SrTiO3(110), SrTiO3(001) and MgO(100) substrates. Numerical results of the preferred orientations of the micro structure and the homogenized dielectric constants of the macro structure showed good agreements with experimental results. Additionally, the proposed process crystallography algorithm was applied to the new biocompatible piezoelectric MgSiO3 thin film generation, which has been found by the first-principles calculation in the previous study. As a result, the computational result indicates that the Cr(110) substrate is most suitable for stable crystal growth of [101] oriented MgSiO3 and shows high piezoelectric stress constants, such as e33= 5.39 C/m2 and e31= -3.64 C/m2.

Proposition of a First-Principles Aided Triple-Scale Analysis for Biocompatible Piezoelectric Thin Films

A computational scheme of first-principles aided triple-scale analysis based on a process crystallography was proposed to design biocompatible piezoelectric thin films fabricated on substrate. To predict the crystal morphology of thin film such as preferred orientations and their ratio, the structural stability of all conformations of crystal cluster on substrate was evaluated by employing pseudo-potential method in density functional theory. Then, the predicted crystal morphology was introduced into micro structure and macro homogenized properties of piezoelectric thin film were estimated through dual-scale finite element analysis based on crystallographic homogenization theory. From the application to the existing biocompatible piezoelectric BaTiO3thin films fabricated on SrTiO3 (110), SrTiO3 (001) and MgO (100), computational results of the preferred orientations of microstructure and the homogenized dielectric constants of macrostructure had good correlation with experimental ones. Additionally, the proposed computation was applied to a new biocompatible piezoelectric MgSiO3thin film, which has been found by first-principles calculations in our previous studies. As a result, the computation indicated that Cr (110) substrate is most suitable for stable crystal growth of MgSiO3orientated to [101] direction and it outputs the high piezoelectric stress constants e33=5.39 C/m2 ande31=-3.64 C/m2.

A Proposal for Optimization of Crystal Orientations in Piezoelectric Ceramics by Multiscale Finite Element Analysis through Crystallographic Homogenization Method

Polycrystalline piezoelectric ceramics have a large possibility to exhibit higher performance in a macroscopic scale by design of crystal morphology in a microscopic scale. In this paper, a multi-scale finite element method by using crystallographic homogenization theory has been applied to a typical piezoelectric ceramics, barium titanate (BaTiO3) in order to estimate macroscopic properties considering microscopic crystal morphology. Then, microscopic crystal orientation distribution has been optimized by steepest decent method to maximize macroscopic piezoelectric strain constant d333. Computational results indicated that BaTiO3 polycrystal consisting of three layers with different crystal orientations has the highest piezoelectric performance d333=258 pC/N, that is beyond single crystal because of mechanical effect in microscopic inhomogeneous structure.

Identification of Material Properties of PZT Single Crystals through Crystallographic Homogenization Method

Single crystals of lead zirconium titanate (PZT) are difficult to fabricate. Thus, not all material properties of PZT have been fully characterized. In this paper, the mechanical and electrical properties of a PZT single crystal, which can be assumed to be identical to those of a crystal grain in a polycrystal, have been computed from those of a polycrystalline PZT ceramic by the steepest decent method and multiscale finite element modeling based on crystallographic homogenization method. Crystallographic homogenization enables us to predict macroscopic properties of ceramics taking into account the inhomogeneous microstructure of an aggregate of crystal grains. The crystal morphology of the PZT ceramic was measured by the SEM·EBSD technique, and the result was used in the microscopic finite element model. Then, the mechanical and electrical properties of the crystal grain were derived by the steepest decent method so that its macroscopic properties would correspond to the measured properties of the PZT ceramic. The proposed computational method was applied to barium titanate (BT) and validated by comparison of the computed material properties with known properties of the BT single crystal. Finally, the computed material properties, such as the elastic compliance, and the dielectric and piezoelectric constants, were presented for the PZT single crystal.

Optimum Design of Micro Crystal Morphology in Piezoelectric Ceramics by Multiscale Finite Element Analysis

In order to improve piezoelectric response of polycrystals in a macro scale, micro distribution of crystal orientations was optimized by using multiscale finite element method based on crystallographic homogenization theory. As a result, two characteristic optimum solutions were obtained for micro structure. One is a polycrystalline structure laminated with three specified grains for macro piezoelectric strain constant d333, the other is a polycrystalline structure consisting of two regularly-neighboring grains with specified orientations for d331. As the maximum micro strain occurs by electric and mechanical effects, the optimized micro structures exhibit the highest macro piezoelectricity that is beyond single crystal. Their micro structures are available for piezoelectric ceramics such as BaTiO3 and Pb (Zr, Ti) O3, that have the maximum piezoelectric strain constant under off-axis electric field.

2601 板成形問題のための結晶均質化並列マルチスケール有限要素解析(OS26.並列・マルチスケール計算手法の開発と応用(1))

2601 板成形問題のための結晶均質化並列マルチスケール有限要素解析(OS26.並列・マルチスケール計算手法の開発と応用(1))

Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models

In this study, two multi-scale analyses codes are newly developed by combining a homogenization algorithm and an elastic/crystalline viscoplastic finite element (FE) method (Nakamachi, E., 1988. A finite element simulation of the sheet metal forming process. Int. J. Numer. Meth. Eng. 25, 283–292; Nakamachi, E., Dong, X., 1996. Elastic/crystalline viscoplastic finite element analysis of dynamic deformation of sheet metal. Int. J. Computer-Aided Eng. Software 13, 308–326; Nakamachi, E., Dong, X., 1997. Study of texture effect on sheet failure in a limit dome height test by using elastic/crystalline viscoplastic finite element analysis. J. Appl. Mech. Trans. ASME(E) 64, 519–524; Nakamachi, E., 1998. Elastic/crystalline viscoplastic finite element modeling based on hardening–softening evaluation equation. In: Proc. of the 6th NUMIFORM, pp. 315–321; Nakamachi, E., Hiraiwa, K., Morimoto, H., Harimoto, M., 2000a. Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification. Int. J. Plasticity 16, 1419–1441; Nakamachi, E., Xie, C.L., Harimoto, M., 2000b. Drawability assessment of BCC steel sheet by using elastic/crystalline viscoplastic finite element analyses. Int. J. Mech. Sci. 43, 631–652); (1) a “semi-implicit” finite element (FE) code and (2) a “dynamic explicit” FE code. These were applied to predict the plastic strain induced yield loci and the formability of sheet metal in the macro scale, and simultaneously the crystal texture and hardening evolutions in the micro scale. The isotropic and kinematical hardening laws are employed in the crystalline plasticity constitutive equation. For the multi-scale structure, two-scales are considered. One is a microscopic polycrystal structure and the other a macroscopic elastic plastic continuum. We measure crystal morphologies by using the SEM-EBSD apparatus with a unit of about 3.8 μm voxel, and define a three dimensional (3D) representative volume element (RVE) for the micro polycrystal structure, which satisfy the periodicity condition of crystal orientation distribution. A “micro” finite element modeling technique is newly established to minimize the total number of finite elements in the micro scale. Next, the “semi-implicit” crystallographic homogenization FE code, which employs the SEM-EBSD measured RVE, is applied to the 99.9% pure-iron uni-axial tensile problem to predict the texture evolution and the subsequent yield loci in the various strain paths. These “semi implicit” results reveal that the plastic strain induced anisotropy in the micro and macro levels can be predicted by our FE analyses. The kinematical hardening law leads a distinct plastic strain induced anisotropy. Our “dynamic-explicit” FE code is applied to simulate the limit dome height (LDH) test problem of the mild steel DQSK, the high strength steel HSLA and the aluminum alloy AL6022 sheet metals, which were adopted as the NUMISHEET2005 Benchmark sheet metals (Smith, L.M., Pourboghrat, F., Yoon, J.-W., Stoughton, T.B., 2005. NUMISHEET2005. In: Proc. of 6th Int. Conf. Numerical Simulation of 3D Sheet Metal Forming Processes, PART A and B(Benchmark), pp. 409–451) to estimate formability. The “dynamic explicit” results reveal that the initial crystal orientation distribution has a large affects to a plastic strain induced texture and anisotropic hardening evolutions and sheet formability.

A Crystallographic Homogenization Elastic/Crystal Viscoplastic Finite Element Analysis of Pure Iron Polycrystal Material by Employing the Kinematic Hardening Rule

In this study, multi scale analyses, by using the crystallographic homogenization based semi-implicit finite element code, have been carried out to indicate a plastic strain induced texture evolution and macro-continuum yield loci evolution. This crystallographic homogenization finite element code has employed the double scale hierarchical structure, which consists of a polycrystal microstructure and a macro-continuum. We focus to discuss “How to define a proper microstructure for the double scale finite element analysis, by employing the real polycrystal aggregation, which has been measured by SEM-EBSD apparatus”. For scaling up from the polycrystal microstructure to the macro-continuum, we looked at the periodicity of crystal orientation distribution, which is one of morphological factors to feature the polycrystal structure itself, which is named as “texture”. Through a statistical stydy of these measured polycrystal morphologies, finally we obtaine a realistic polycrystal unit cell of the microstructure, which can be adopted for multi scale finite element analyses. The availability of this code to predict the plastic deformation induced anisotropy and kinematical hardening is confirmed.

Evaluation of Ferroelectric Properties of Piezoelectric Ceramics Based on Crystallographic Homogenization Method and Crystal Orientation Analysis by SEM·EBSD Technique

Macroscopic ferroelectric properties of piezoelectric polycrystals are strongly affected by microscopic inhomogeneous crystal structure. In our previous study, a multi-scale finite element method based on crystallographic homogenization method has been developed to estimate macroscopic properties considering microscopic crystal morphology. In this paper, the crystal orientation distribution of polycrystalline barium titanate has been measured by SEM·EBSD technique, and the measured crystal orientation distribution has been introduced to the microscopic finite element model. As the prediction of macroscopic properties depends on the sampling conditions of the measured crystal orientations, the effects of number of sampling points and sampling area have been investigated. As a result, the effective sampling conditions have been clarified to estimate macroscopic ferroelectric properties.

A Proposal for Optimization of Crystal Orientations in Piezoelectric Ceramics by Multiscale Finite Element Analysis Based on a Crystallographic Homogenization Method

Polycrystalline piezoelectric materials have a large possibility to exhibit higher performance ina macroscopic scale by design of crystal morphology in a microscopic scale. In this paper, a multiscale finite element method by using crystallographic homogenization method has been applied to a typical piezoelectric ceramics, barium titanate (BaTiO3) in order to estimate macroscopic properties considering microscopic crystal morphology. Then, microscopic crystal orientation distribution has been optimized by steepest decent method to maximize macroscopic piezoelectric strain constant d333. Computational results indicated that BaTiO3 polycrystal consisting of three layers with different crystal orientations has the highest piezoelectric performance d333=258 pC/N, that is beyond single crystal because of mechanical effect in microscopic inhomogeneous structure.

1730 結晶均質化法による強誘電性ヒステリシス挙動のマルチスケール有限要素解析(S16-1 圧電・強誘電材料,S16 先進材料の強度・機能評価とメゾメカニックス)

1730 結晶均質化法による強誘電性ヒステリシス挙動のマルチスケール有限要素解析(S16-1 圧電・強誘電材料,S16 先進材料の強度・機能評価とメゾメカニックス)