Multiscale finite element simulations of piezoelectric materials based on two- and three-dimensional electron backscatter diffraction

Abstract

This article presents a computational procedure for evaluating effective homogenized material properties of polycrystalline piezoelectric materials and constructing two- and three-dimensional realistic microstructure models based on electron backscatter diffraction crystal orientation measurement. Microstructural features of a polycrystalline piezoelectric material, barium titanate, were investigated through electron backscatter diffraction measurements using an amorphous osmium coating to prevent charging. Realistic crystal orientations obtained from the electron backscatter diffraction measurements were introduced into multiscale finite element simulations based on homogenization theory to reveal the relationship between the macrostructure and the microstructure. First, a two-dimensional microstructural model was constructed, and the effect of the sampling area of the electron backscatter diffraction–measured crystal orientations was analyzed. We discuss the representative volume element determined from two points of view: the macroscopic homogenized material properties and the microscopic localized material behavior in response to external loads. Second, the surface of specimen was ground and polished at regular intervals and was measured by electron backscatter diffraction iteratively. Then a three-dimensional microstructural model was constructed by stacking in-plane crystal orientations in series along the out-of-plane direction, and the influence of the microstructural thickness, which indicates the stacking dimension in the out-of-plane direction, was investigated. We compare the macrostructural homogenized material properties between the two- and three-dimensional microstructures.

Keywords

piezoelectric material electron backscatter diffraction polycrystalline microstructure multiscale simulation finite element analysis representative volume element

Introduction

Ferroelectric materials, well known as perovskite oxides ABO₃, possess remarkable piezoelectric properties and have consequently been applied to actuators and sensors in microelectromechanical systems (MEMS). Ferroelectricity and piezoelectricity are caused by the spontaneous polarization that occurs in noncentrosymmetric crystal structures, such as tetragonal and rhombohedral structures. The material properties of these crystals, including the elastic, dielectric, and piezoelectric constants, show strong anisotropy. On the other hand, ferroelectric materials generally consist of many grains and domains at a microscopic scale, and each domain also shows strong anisotropy in mechanical and electrical behaviors according to the above-mentioned asymmetric crystal structure. Consequently, the macroscopic material properties of polycrystalline piezoelectric ceramics have a large dependence on the microscopic crystal morphology. Therefore, it is important to understand the microstructures and textures in these materials for the design and development of advanced piezoelectric materials.

The crystal orientation analytical system, composed of a scanning electron microscope (SEM) and electron backscatter diffraction (EBSD) (Dingley and Randel, 1992; Venables and Harland, 1973), is well established and is useful to characterize the crystal morphology in polycrystalline materials. An electron beam is irradiated onto a specimen in the SEM, and the crystal orientations of submicron-order grains and domains can then be analyzed by EBSD. Application of EBSD ranges from metals (Calcagnotto et al., 2010; Nakamachi et al., 2007; Wu et al., 2003, 2007; Zaafarani et al., 2006) to nonmetals (Koblischka-Veneva et al., 2002; Liu et al., 2008; Mainprice et al., 2004), progressing from two-dimensional tomographic analyses (Koblischka-Veneva et al., 2002; Liu et al., 2008; Mainprice et al., 2004; Wu et al., 2003, 2007) to three-dimensional analyses (Calcagnotto et al., 2010; Nakamachi et al., 2007; Zaafarani et al., 2006). In addition, some previous studies (Nakamachi et al., 2007; Wu et al., 2003) have reported finite element simulations incorporating the EBSD-measured microstructure. As realistic modeling proved important in understanding material properties and behaviors of polycrystals, these studies had a large impact on the field of material science and engineering. In the case of ferroelectric materials, EBSD measurements have been conducted for lead titanate (PbTiO₃) (Yang et al., 1994), barium titanate (BaTiO₃) (Koblischka-Veneva et al., 2002; Munoz-Saldana et al., 2001), lead zirconium titanate (PZT) (Farooq et al., 2008; Jeong et al., 2009; Lowe et al., 2008; Tai et al., 2002), lead lanthanum zirconium titanate (PLZT) (Faryna et al., 2006), and lead magnesium niobium titanate (PMNT) (Samardžija et al., 2007). EBSD measurements have been conducted on multiferroic materials besides ferroelectric materials (Koblischka et al., 2010). All these studies indicate that the microstructures clearly play a significant role in the overall material characteristics. However, there have been few investigations on analyzing the quantitative relationship between the microstructure and the macrostructure.

We have employed multiscale finite element simulation based on homogenization theory (Gudes and Kikuchi, 1990; Kuramae et al., 2007; Nelli Silva et al., 1999; Uetsuji et al., 2004, 2008) and have attempted to estimate the relationship between the EBSD-measured realistic microstructure and the overall material properties for piezoelectric materials. In our previous study on BaTiO₃ (Uetsuji et al., 2006), crystal orientation maps with high resolution could not be obtained because of the charging of specimens. Therefore, we used a conductive layer of amorphous osmium to remove electric charge on the specimen during the EBSD measurements and were able to obtain high-resolution crystal orientation maps for PZT (Uetsuji et al., 2008) and BaTiO₃ (Uetsuji et al., 2011). The obtained EBSD crystal orientations were applied to multiscale finite element simulation. We investigated the effect of sampling conditions of EBSD crystal orientations on computational results. However, they were not enough to understand the representative volume element (RVE) of microstructural crystal orientations perfectly. In addition, EBSD-based microstructural modeling of piezoelectric materials has been confined to two-dimensional tomographic analysis. There is no expansion into three-dimensional microstructural modeling in our previous and other studies.

In this article, we reinvestigated the RVE of EBSD-measured high-resolution crystal morphology for BaTiO₃ (Uetsuji et al., 2011). The effect of the sampling area of EBSD-measured crystal orientations on material features was analyzed using a two-dimensional microstructural model through multiscale finite element simulations. We discuss RVEs determined from two points of view: macroscopic homogenized material properties and microscopic localized material behaviors in response to external loads. In addition, the surface layer of specimen was ground at regular intervals through mechanical and chemical polishing processes, and crystal orientations of each polished surface were measured by EBSD iteratively. Then a three-dimensional microstructural model was constructed by stacking in-plane crystal orientations in series along the out-of-plane direction. The influence of the microstructural thickness, which indicates the stacking dimension in the out-of-plane direction, was investigated. We present a comparison of macrostructural homogenized material properties for the two- and three-dimensional microstructures.

Overview of EBSD measurement and computational procedure

EBSD measurement

Figure 1 illustrates the overall scheme of multiscale finite element simulations based on the EBSD-measured microstructure. First, crystal orientations of a two-dimensional area in a BaTiO₃ polycrystalline piezoelectric material were analyzed through EBSD measurements. The measured BaTiO₃ had a perovskite-type tetragonal structure whose a- and c-axes lengths were 0.3994 and 0.4034 nm, respectively. The BaTiO₃ specimen surface subjected to observations was polished mechanically and chemically. The surface was coated with electrical conductive amorphous osmium layer to prevent charging by the electron beam. We employed an osmium coater (Meiwafosis Co., Ltd, Neoc-ST) and utilized EBSD (Oxford Instruments plc, Link ISIS C.7272) implemented in the SEM (JEOL Datum Ltd., JSM-5410). Details of the surface treatment and EBSD measurements were described previously (Uetsuji et al., 2011).

Figure 1.

Overall scheme of multiscale finite element simulations based on EBSD-measured microstructures.

Multiscale simulation

The EBSD-measured crystal orientations were incorporated into multiscale finite element simulation based on homogenization theory. A macrostructure is sufficiently large compared to the microstructure, and it can be assumed that the microstructures have periodicity on the microscopic scale. Then, the multiscale simulation based on homogenization theory enables us to estimate the macrostructural homogenized material properties as a volume average of the heterogeneous microstructures. It also enables us to evaluate the microscopic mechanical and electrical states in response to an external load applied to the macrostructure.

In this section, the basic equations for multiscale simulation are described briefly. Details of the formulations and the computational approach have been published previously (Kuramae et al., 2007; Uetsuji et al., 2004, 2008). As shown in Figure 1, x is employed for the coordinates of the macrostructure Ω, and y = x /λ for the coordinates defined in the microstructure Y, where λ is the ratio of the macroscopic representative length to the microscopic one. An overall structure is considered, which is a combined domain between the macrostructure and the microstructure. Here it is assumed that the displacement and electric potential in the overall structure can be expressed as an asymptotic expansion with respect to the scale ratio. Then the microstructural and macrostructural correlative equations are obtained from the virtual work equations for the overall structure. The microstructural equations are given by

\int_{Y} (^{micro} C_{ijkl}^{E} \frac{\partial χ_{kmn}^{uu} (x, y)}{\partial y_{l}} +^{micro} e_{kij} \frac{\partial χ_{mn}^{ϕ u} (x, y)}{\partial y_{k}}) \frac{\partial δ u_{i}^{1} (x, y)}{\partial y_{j}} dY = - \int_{Y}^{micro} C_{ijmn}^{E} \frac{\partial δ u_{i}^{1} (x, y)}{\partial y_{j}} dY

(1)

\int_{Y} (^{micro} e_{ikl} \frac{\partial χ_{kmn}^{uu} (x, y)}{\partial y_{l}} -^{micro} \in_{ik}^{S} \frac{\partial χ_{mn}^{ϕ u} (x, y)}{\partial y_{k}}) \frac{\partial δ ϕ^{1} (x, y)}{\partial y_{i}} dY = - \int_{Y}^{micro} e_{imn} \frac{\partial δ ϕ^{1} (x, y)}{\partial y_{i}} dY

(2)

\int_{Y} (^{micro} C_{ijkl}^{E} \frac{\partial χ_{km}^{u ϕ} (x, y)}{\partial y_{l}} +^{micro} e_{kij} \frac{\partial χ_{m}^{ϕ ϕ} (x, y)}{\partial y_{k}}) \frac{\partial δ u_{i}^{1} (x, y)}{\partial y_{j}} dY = - \int_{Y}^{micro} e_{mij} \frac{\partial δ u_{i}^{1} (x, y)}{\partial y_{j}} dY

(3)

\int_{Y} (^{micro} e_{ikl} \frac{\partial χ_{km}^{u ϕ} (x, y)}{\partial y_{l}} -^{micro} \in_{ik}^{S} \frac{\partial χ_{m}^{ϕ ϕ} (x, y)}{\partial y_{k}}) \frac{\partial δ ϕ^{1} (x, y)}{\partial y_{i}} dY = \int_{Y}^{micro} \in_{im}^{S} \frac{\partial δ ϕ^{1} (x, y)}{\partial y_{i}} dY

(4)

where $^{micro} C_{ijkl}^{E}$ , $^{micro} \in_{ik}^{S}$ , and $^{micro} e_{kij}$ denote the tensors of elastic compliance constants, dielectric constants, and piezoelectric stress constants of domains in the microstructure, respectively. $u_{i}^{1} (x, y)$ indicates the microscopic perturbation of displacement and $ϕ^{1} (x, y)$ is the microscopic perturbation of electric potential. Additionally, $χ_{kmn}^{uu} (x, y)$ is the characteristic displacement function describing the microscopic displacement perturbation per unit macroscopic strain, and $χ_{m}^{ϕ ϕ} (x, y)$ is the characteristic electrical potential function describing the microscopic electric potential perturbation per unit macroscopic electric field. Additionally, $χ_{km}^{u ϕ} (x, y)$ and $χ_{mn}^{ϕ u} (x, y)$ are the characteristic coupled functions describing the microscopic displacement perturbation per unit macroscopic electric field and the microscopic electric potential perturbation per unit macroscopic strain, respectively. These characteristic functions are obtained as the solutions of microstructural equations under a periodic boundary condition. On the other hand, the macrostructural equations are simplified as follows

\int_{Ω} (^{macro} C_{ijkl}^{E} \frac{\partial u_{k}^{0} (x)}{\partial x_{l}} +^{macro} e_{kij} \frac{\partial ϕ^{0} (x)}{\partial x_{k}}) \frac{\partial δ u_{i}^{0} (x)}{\partial x_{j}} d Ω = \int_{Γ_{t}} t_{i} δ u_{i}^{0} (x) d Γ

(5)

\int_{Ω} (^{macro} e_{ikl} \frac{\partial u_{k}^{0} (x)}{\partial x_{l}} -^{macro} \in_{ik}^{S} \frac{\partial ϕ^{0} (x)}{\partial x_{k}}) \frac{\partial δ ϕ^{0} (x)}{\partial x_{i}} d Ω = \int_{Γ_{ρ}} ρ δ ϕ^{0} (x) d Γ

(6)

where $u_{i}^{0} (x)$ and $ϕ^{0} (x)$ are displacement and electric potential of the macrostructure, respectively; t_i is the traction applied on the macrostructure surface Γ _t ; ρ is the extrinsic charge density on the macrostructure structure surface Γ $ρ;^{macro} C_{ijkl}^{E}$ , $^{macro} \in_{ik}^{S}$ , and $^{macro} e_{kij}$ are macrostructural homogenized material properties, and they are provided by the following equations

^{macro} C_{ijmn}^{E} = \frac{1}{| Y |} \int_{Y} (^{micro} C_{ijmn}^{E} +^{micro} C_{ijkl}^{E} \frac{\partial χ_{kmn}^{uu} (x, y)}{\partial y_{l}} +^{micro} e_{kij} \frac{\partial χ_{mn}^{ϕ u} (x, y)}{\partial y_{k}}) d Y

(7)

\begin{matrix} ^{macro} e_{mij} = \frac{1}{| Y |} \int_{Y} (^{micro} e_{mij} +^{micro} e_{kij} \frac{\partial χ_{m}^{ϕ ϕ} (x, y)}{\partial y_{k}} +^{micro} C_{ijkl}^{E} \frac{\partial χ_{km}^{u ϕ} (x, y)}{\partial y_{l}}) d Y \\ = \frac{1}{| Y |} \int_{Y} (^{micro} e_{mij} +^{micro} e_{mkl} \frac{\partial χ_{kij}^{uu} (x, y)}{\partial y_{l}} -^{micro} \in_{mk}^{S} \frac{\partial χ_{ij}^{ϕ u} (x, y)}{\partial y_{k}}) d Y \end{matrix}

(8)

^{macro} \in_{im}^{S} = \frac{1}{| Y |} \int_{Y} (^{micro} \in_{im}^{S} +^{micro} \in_{ik}^{S} \frac{\partial χ_{m}^{ϕ ϕ} (x, y)}{\partial y_{k}} -^{micro} e_{ikl} \frac{\partial χ_{km}^{u ϕ} (x, y)}{\partial y_{l}}) d Y

(9)

An isoparametric linear solid element was employed for finite element modeling of microstructures and macrostructures. A two-scale coupling finite element analytical code was developed to solve the above macrostructural and microstructural equations, and it was used here to evaluate the macrostructural homogenized properties and the microstructural localized states in response to external loads.

Two-dimensional microstructure

EBSD-measured microstructure

An EBSD crystal orientation map was obtained for the poled BaTiO₃ specimen. The scanning interval of the EBSD was set to 0.64 µm. The mean diameter of a grain was estimated to be approximately 6.71 µm. The microstructural texture was investigated by changing the sampling area of the crystal orientations from 19.7 × 19.7 µm² to 90.8 × 90.8 µm². When the sampling area was converted to the number of grains based on the mean diameter of the grains, it was changed from 11 to 233 grains. Figure 2 shows a comparison of the crystal orientation map, the (001) pole figure, and the inverse pole figure among various sampling areas. The grains in the figure are shaded based on their crystallographic orientation, which are given in the color-coded stereographic triangle at the upper left. From Figure 2, the frequency of the <001>, <111>, and <101> directions is higher than that of the <110> and <100> directions.

Figure 2.

Comparison of EBSD-measured crystal orientation map, (001) pole figure, and inverse pole figure among various sampling areas.

Macroscopic homogenized material properties

The macroscopic homogenized material properties originating from the EBSD microstructural texture were investigated through the multiscale finite element simulation. Figure 3 shows the finite element modeling of the EBSD-measured realistic microstructure. Although the EBSD-measured crystal orientations are distributed within a two-dimensional region, each point includes the three-dimensional orientation in the microstructural coordinate system. Consequently, three-dimensional finite element modeling is necessary to use EBSD-measured crystal orientations. A plate-shaped regular cubic mesh, whose number of elements corresponded exactly to one of the measurement points, was employed for the microstructural finite element model. The crystal orientation of each measurement point was mapped independently into one element. As the scanning interval of EBSD measurement was set to less than tenth of mean diameter of a grain, 90° domains in the grains were detected and introduced into finite element model. On the other hand, because EBSD measurement cannot differentiate 0° and 180° domains, it was assumed that 0° and 180° domains were fully poled and they were a single domain orienting to the global poling direction.

Figure 3.

Finite element modeling of two dimensionally distributed EBSD crystal orientations. The EBSD crystal orientation of each measurement point was mapped independently into one element. The sampling area was changed within the range from 19.7 × 19.7 µm² to 90.8 × 90.8 µm², and the number of mesh division was changed from 31 × 31 × 1 to 143 × 143 × 1.

In order to investigate the influence of the microstructural relative dimension, the sampling area of the EBSD crystal orientations was changed within the range from 19.7 × 19.7 µm² to 90.8 × 90.8 µm² in the same way as in Figure 2. According to the sampling area, the number of measurement points and the mesh division was changed from 31 × 31 × 1 to 143 × 143 × 1. The material properties of a BaTO₃ single crystal (Jaffe, 1971) were used for each grain in the microstructure.

Figure 4 shows the computational results for the relationship between the macrostructural homogenized material properties and the sampling area. This figure summarizes (a) nine components of elastic compliance constants $^{macro} s_{ij}^{E}$ , (b) three components of dielectric constants $^{macro} \in_{ij}^{T}$ , and (c) five components of piezoelectric strain constants $^{macro} d_{ij}$ . While the lower horizontal axis indicates the sampling area of the EBSD-measured crystal orientations, the upper one indicates the number of grains estimated from the mean diameter of the grains. In addition, Figure 5 shows the difference in material properties obtained from each sampling area compared with the ones obtained from the maximum sampling area of 90.8 × 90.8 µm². If the sampling area of the microstructure became larger and the number of grains increased, the macrostructural homogenized material properties converged at a constant value and their error decreased. These computations indicated that the sampling area must be larger than 3700 µm² (105 grains) to make the relative error of the homogenized material properties 5% or less. The experimental values of the piezoelectric strain constants d₃₃ and d₃₁, as described in a brochure provided by Murata Manufacturing Co., Ltd., are 133 and −44.0 pC/N, respectively. In the case of the computational values obtained from the maximum sampling area, d₃₃ is 106 pC/N, and d₃₁ and d₃₂ are −43.3 and −45.2 pC/N, respectively. The computational values of the macrostructural homogenized material properties have reliable correspondence to the experimental values.

Figure 4.

Relationship between the macrostructural homogenized material properties and the sampling area of EBSD-measured crystal orientations. The macrostructural homogenized property of vertical axis means the volume average of microstructure, which was obtained through multiscale finite element simulation based on homogenization theory: (a) elastic compliance constants, (b) relative dielectric constants, and (c) piezoelectric strain constants.

Figure 5.

Variation of the errors of macrostructural homogenized material properties of each sampling area compared with those of the maximum sampling area of EBSD-measured crystal orientations: (a) elastic compliance constants, (b) relative dielectric constants, and (c) piezoelectric strain constants.

Focusing our attention on the isotropy in the y₁–y₂ plane, we investigated the difference between the respective components of a pair of the macrostructural homogenized material properties. Figure 6(a) shows the relative error between the two respective components of a pair in case of the elastic compliance constants. All the pairs had no strong dependence on the sampling area, and their errors were sufficiently small to be regarded as a transversely isotropic body. Figure 6(b) indicates that the relative errors of the dielectric constants $^{macro} \in_{11}^{T}$ and $^{macro} \in_{22}^{T}$ , and shear piezoelectric strain constants $^{macro} d_{15}$ and $^{macro} d_{24}$ decreased almost to zero according to the sampling area. The transverse piezoelectric strain constants $^{macro} d_{31}$ and $^{macro} d_{32}$ had a relatively large error at every sampling area because their absolute values are smaller and the error is overestimated compared with those for the other constants. Additionally, although elastic compliance and dielectric constants depend only on the direction, piezoelectric strain constants change their signs according to the sense of spontaneous polarization. Thus, the respective components of a pair of piezoelectricity are more difficult to have a good agreement than other material constants.

Figure 6.

Variation of the relative errors between the two respective components of a pair in case of macrostructural homogenized material properties according to the sampling area of EBSD-measured orientations: (a) elastic compliance constants and (b) dielectric and piezoelectric strain constants.

Microscopic localized material behaviors

A cubic mesh was employed for macrostructural finite element modeling, and a uniform electric field of 100 kV/m was applied along the third axis of the macrostructure. The mechanical and electrical states in the microstructure were investigated in order to reveal the effect of the sampling area on microscopic localized response. Five points, indicated in Figure 7, were picked as representative intragranular points, and their mechanical and electric states in response to a macrostructural external load were monitored by changing the sampling area from 19.7 × 19.7 µm² to 90.8 × 90.8 µm², in the same way as described in the previous section.

Figure 7.

Monitoring points of microstructural localized states in response to a macrostructural external load. They are picked up as representative intragranular points and exist within the minimum sampling area in the center.

Although the macrostructural external load is simple, every grain is under multiaxial electromechanical coupling state inside the microstructure composed of many differently oriented grains. Consequently, energy was employed as a total indicator of the complicated states. Especially, we focused attention on two dominant energies. One is the electric energy that is directly connected to the external electric load, and the other is the piezoelectric energy that is caused by the mismatch in mechanical deformation among adjacent grains. Figure 8 shows the variation in the (a) electrical energy and (b) piezoelectric energy at five points in the microstructure according to the sampling area. In addition, Figure 9 shows the error in the energy obtained from each sampling area compared with ones obtained from the maximum sampling region of 90.8 × 90.8 µm². As the sampling area became larger, the error in the electrical and piezoelectric energies decreased. The error became under 5% when the sampling area was larger than 5500 µm² (155 grains). These computations suggest that it is necessary to set a larger RVE for evaluation of microscopic localized responses compared to the macroscopic homogenized material properties.

Figure 8.

Variation in electrical and piezoelectric energies at the microstructural monitoring points according to the sampling area: (a) electrical energy and (b) piezoelectric energy.

Figure 9.

Processions of the errors of electrical and piezoelectric energies at the microstructural monitoring points obtained from each sampling area compared with those obtained from the maximum sampling area of the EBSD-measured crystal orientations: (a) electrical energy and (b) piezoelectric energy.

Three-dimensional microstructure

A three-dimensional microstructure was constructed by repeated surface treatment of the specimen and EBSD measurement. Surface treatment consisted of 20-min mechanical polishing using 3-µm diamond particles with a polishing sheet, 20-min chemical polishing at pH 3.5 using colloidal particles, and 10 s of coating with osmium. Approximately 1.66 µm thickness was estimated to be removed by such a treatment. Then the scanning interval of EBSD was set to 0.64 µm, and a crystal orientation map of 81.3 × 63.5 µm², which was the maximum region that could be measured at once, was obtained. That region includes 146 grains, and it is enough large to become a RVE of two-dimensional microstructure. After the specimen and SEM sample holder were taken out from SEM, the specimen was polished in a fixed state with SEM sample holder. Then they were set back into SEM and EBSD measurement was performed again without changing the in-plane position of SEM stage. Although it was difficult to keep exactly the same in-plane position, two-dimensional EBSD crystal orientation maps at different positions in the thickness direction were obtained through repetition of such a polishing and measuring process. Two-dimensional EBSD crystal orientation maps obtained by repeating this process 13 times were introduced in series into a three-dimensional finite element model as shown in Figure 10. In the same way as the two-dimensional microstructural model, in this case, a linear isoparametric solid element was employed and a 100 × 128 × 13-divided regular cubic mesh was produced. The crystal orientation of one EBSD measurement point was input into one element of the microstructural model. Figure 11 demonstrates a three-dimensional microstructure based on EBSD measurements. Although the three-dimensional microstructure has some misalignment of in-plane position among adjacent layers, it could take the inhomogeneity of crystal orientations in the thickness direction into account.

Figure 10.

Finite element modeling of three dimensionally distributed EBSD-measured crystal orientations. The EBSD crystal orientation of each measurement point was mapped independently into one element.

Figure 11.

Three-dimensional microstructural model based on EBSD measurement.

Macrostructural homogenized material properties were investigated by changing the thickness of the microstructure model. As the number of crystal orientation maps and the number of mesh divisions in the thickness direction were changed from 1 to 13, the microstructural thickness was varied from 1.66 to 21.6 µm. Figure 12 shows the computational results for the relationship between the macrostructural homogenized material properties and the thickness of the microstructure. In addition, Figure 13 shows the variation in material properties obtained from each thickness compared with those obtained from the maximum thickness of 21.6 µm. When the microstructural thickness increased, the macrostructural homogenized material properties converged at a constant value and their error gradually decreased. Although some components whose values were relatively small were overestimated, most of the components had an error under 5% when the thickness was over 5.0 µm. There was no dependence on the thickness when the thickness was greater than 15.0 µm.

Figure 12.

Relationship between the macrostructural homogenized material properties and the thickness of the EBSD-measured microstructure: (a) elastic compliance constants, (b) relative dielectric constants, and (c) piezoelectric strain constants.

Figure 13.

Processions of the errors of macrostructural homogenized material properties obtained from each thickness are compared with those obtained from the maximum thickness of the EBSD-measured microstructure: (a) elastic compliance constants, (b) relative dielectric constants, and (c) piezoelectric strain constants.

Figure 14 compares all components of the macrostructural homogenized material properties between the two- and three-dimensional microstructural models. While the two-dimensional model indicates computational results obtained from the maximum sampling area of 90.8 × 90.8 µm² described in the previous section, the three-dimensional model indicates the computational results obtained from the maximum sampling volume of 81.3 × 63.5 × 21.6 µm³. Figure 14 indicates that the two-dimensional model overvalues in-plane components and undervalues out-of-plane components compared to the three-dimensional model. The error between the both models was approximately 11% at a maximum.

Figure 14.

Comparison of the macrostructural homogenized material properties between the two- and three-dimensional microstructural models: (a) elastic compliance constants, (b) relative dielectric constants, and (c) piezoelectric strain constants.

Conclusion

Multiscale finite element simulations based on the EBSD-measured microstructure were presented for a polycrystalline piezoelectric material. First, we constructed a two-dimensional microstructure and investigated the influence of the sampling area of the EBSD-measured crystal orientations on macrostructural homogenized material properties and the microstructural localized states in response to an external load. For the computational results, the evaluation of microstructural localized states requires about 1.5 times larger sampling area compared to the macrostructural homogenized material properties. The rough standard of RVE is approximately 150 grains for multiscale simulations using a realistic microstructural model. Second, we constructed a three-dimensional microstructure based on EBSD-measured crystal orientations and investigated the influence of the microstructural thickness. The dependence on the microstructural thickness disappeared almost entirely at thickness over 15 µm. Thus, the two-dimensional microstructural model overvalues in-plane components and undervalues out-of-plane components compared to the three-dimensional microstructural model.

Footnotes

Acknowledgements

The authors thank Professor Eiji Nakamachi of Doshisha University for helpful discussions and comments.

Y. Uetsuji was financially supported by a Grant-in-Aid for Young Scientists (B) (No. 22760087) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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Multiscale finite element simulations of piezoelectric materials based on two- and three-dimensional electron backscatter diffraction–measured microstructures

Abstract

Keywords

Introduction

Overview of EBSD measurement and computational procedure

EBSD measurement

Multiscale simulation

Two-dimensional microstructure

EBSD-measured microstructure

Macroscopic homogenized material properties

Microscopic localized material behaviors

Three-dimensional microstructure

Conclusion

Footnotes

Acknowledgements

References