Issue 
Mechanics & Industry
Volume 24, 2023
History of matter: from its raw state to its end of life



Article Number  33  
Number of page(s)  9  
DOI  https://doi.org/10.1051/meca/2023030  
Published online  12 September 2023 
Regular Article
Statistical properties of effective elastic moduli of random cubic polycrystals
Nantes Université, Ecole Centrale Nantes, CNRS, GeM,
UMR 6183,
44000
Nantes,
France
^{*} email: shahram.khazaie@univnantes.fr
Received:
23
December
2022
Accepted:
4
August
2023
The homogenized elastic properties of polycrystals depend on the grain morphology and crystallographic orientations. For simplification purposes, the orientations of the grains are usually considered three independent Euler angles. However, experimental investigations reveal spatial correlations in these angles. The Karhunen–Loève expansion is used to generate random fields of Euler angles having exponential kernel functions with varying correlation lengths. The effective elastic moduli for numerically generated statistically equiaxed cubic polycrystals are estimated via the classical Eshelby–Kröner SelfConsistent homogenization model. The influence of the correlation lengths of the orientations’ random fields on the statistical properties of the effective elastic moduli has been investigated. Our results show that spatially correlated Euler angles could increase the variability of the homogenized elastic properties compared to the ones having uncorrelated Euler angles. Nevertheless, using independent random variables for Euler angles remains valid when correlation lengths are close to the average grain size.
Key words: Polycrystals / random fields / effective elastic moduli / Karhonen–Loève expansion
© N. Sheng et al., Published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Polycrystalline materials are widely used and play a key role in many engineering applications for which effective mechanical properties are often of great concern. The former are discrete crystallites or grains with different morphological and crystallographic characteristics. Numerous investigations have focused on how these characteristics ultimately influence the effective mechanical properties at the macroscopic scale.
In most previous numerical works, the crystallographic orientations of the grains in polycrystals are represented by introducing the randomness on a triplet of Euler angles (Θ_{1}, Θ, Θ_{2}), that are usually considered as statistically independent random variables [1–3]. This assumption simplifies the numerical generation process of the Euler angles in the polycrystalline samples. However, in real polycrystals, the crystallographic orientations may be spatially correlated due to the grain nucleation and growth mechanisms. For instance, grains with similar orientations tend to form lowenergy boundaries between them, facilitating cooperative growth along specific directions. This graintograin interaction leads to spatially correlated orientations within the polycrystal. Besides, experimental investigations also report the spatial correlation of the grain orientations [4–6]. In addition, in order to get the aimed performance, some special metalworking processes like extrusion and drawing are widely used when manufacturing polycrystalline materials. These processes may induce a strong crystallographic and morphological texture, referring to a particular spatial arrangement of grains with strongly correlated crystallographic orientations in the polycrystal. A common way to represent these spatial correlations is by constructing random fields of the Euler angles [6].
The impact of the parameters of the Euler angles’ random fields on the effective mechanical properties has not been investigated yet. In the present study, we consider an exponential kernel function, frequently used in the literature, and two different correlation lengths to cover a range of the size of inhomogeneities in the numerically simulated Euler angles’ random fields.
In this paper, the numerical simulation of the random fields of orientations, along with the polycrystal models, will be introduced in Sections 2.1 and 2.2. The classical Eshelby–Kröner SelfConsistent model explained in Section 2.3, is used in the present work to estimate the effective elastic moduli of the simulated polycrystals. The influence of the correlation length of the random fields of Euler angles on the statistical properties of the effective elastic moduli is presented and discussed in Section 3.
Basic information of synthetic polycrystal sample sets.
Fig. 1 A realization of a polycrystal having 10000 grains (left), the PDF of its grain volumes (middle), and the PDF of its grain sizes D_{eq} (histogram, righf) compared with the target one (black solid line, right). 
2 Numerical model construction and computational framework
2.1 Numerical cubic polycrystal model
Using the opensource software NEPER [7], multiple synthetic polycrystals containing different numbers of grains are generated. These grains exhibit a statistically equiaxed morphology, with statistically equal dimensions in all three spatial directions (x, y, and z), without any prominent elongation or flattening on the grain shapes. Moreover, the values of the probability density function (PDF) of the aspect ratio should be peaked around 1 and should have a small coefficient of variation. Each polycrystal is a cube of side 1000 µm. The number of grains N_{g}, as well as the first and secondorder statistics of the grain size, e.g., the mean (${\overline{D}}_{\text{eq}}$), the standard deviation (${\sigma}_{{D}_{\text{eq}}}$), and the coefficient of variation (${\delta}_{{D}_{\text{eq}}}$) of the grain equivalent diameters D_{eq} of different sample sets, are given in Table 1. It should be noted that the equivalent diameter of each grain is defined as the diameter of a sphere having the same volume as the grain. Besides, the grain sizes follow a lognormal distribution, as reported in experimental investigations [8–10]. The values of ${\delta}_{{D}_{\text{eq}}}$ are set around 0.35 for all sample sets, considerably smaller than those observed in real polycrystalline samples, which can exceed 1 [8]. For each sample set, 200 realizations are generated (1200 samples in total). It should be mentioned that the size and the centroid location of the grains in the polycrystals vary between different realizations. Raster tessellation is used to generate microstructures in order to get closer to the morphological characteristics of the real grains [11,12]. Figure 1 (left) shows one of the realizations having 10000 grains, in which the grains are distinguished with different colors. The histogram of the volumes of the grains that also follows a lognormal distribution is depicted in the same Sigure (middle). Moreover, the histogram of D_{eq} for this realization closely matches the target PDF (black solid line), shown in Figure 1 (right).
Parameters of numerically simulated Euler angles.
2.2 Random fields of Euler angles
Based on the numerically generated polycrystal models, we subsequently aimed to simulate the crystallographic orientations of the grains. For comparison purposes, these Euler angles are considered either uncorrelated (${\text{\Theta}}_{1}^{uc}$, Θ^{uc}, ${\text{\Theta}}_{2}^{uc}$), or spatially correlated (Θ_{1}, Θ, Θ_{2}). For the case Uncorrelated in Table 2, the Euler angles are assumed to be statistically independent random variables, yielding uniform random rotations and a textureless random polycrystal. More specifically, ${\text{\Theta}}_{1}^{uc}$ and ${\text{\Theta}}_{2}^{uc}$ are uniformly distributed random variables between [0, 2π], while Θ^{uc} is a random variable following Gilbert’s sine distribution in [0, π] to achieve statistical isotropy in the polycrystal. These random Euler angles can be easily simulated as follows [13]:
$$\begin{array}{ccc}{\text{\Theta}}_{1}^{uc}=2\pi {X}_{1},& {\text{\Theta}}^{uc}=\mathrm{arccos}\left(1+2{X}_{2}\right),& {\text{\Theta}}_{2}^{uc}=2\pi {X}_{3},\end{array}$$(1)
in which X_{i} ∈ U(0,1) (i ∈ {1, 2, 3}) are three independent uniform random variables. The PDFs of numerically simulated uncorrelated Euler angles, based on 200 realizations of Poly6 model, are presented in Figure 2, and compared with the target distributions. The relative entropy D_{KL} (Kullback–Leibler divergence) [14,15] has been used as the measure to evaluate the distance between the PDFs of the numerically generated Euler angles P_{1} (x) and the target ones P_{2}(x). The Zormer is defined as:
$${\text{D}}_{\text{KL}}\left({P}_{1}\Vert {P}_{2}\right)={\displaystyle {\int}_{\infty}^{+\infty}\text{\hspace{0.17em}}{P}_{1}\left(x\right)\text{\hspace{0.17em}}\mathrm{log}\text{\hspace{0.17em}}\left(\frac{{P}_{1}\left(x\right)}{{P}_{2}\left(x\right)}\right)\text{\hspace{0.17em}d}x}.$$(2)
Considering the normalization through a nonlinear transformation ${\widehat{D}}_{\text{KL}}=1\mathrm{exp}\text{\hspace{0.17em}}\left({D}_{\text{KL}}\right)$, the plot of the normalized relative entropy ${\widehat{D}}_{\text{KL}}$, as a function of the number of realizations, is shown in Figure 2 (right) The small values of relative entropy after 2200 realizations indicate that the PDFs of simulated Euler angles are close to the target ones.
The spatially correlated crystallographic orientations of the grains can be represented by constructing correlated nonGaussian random fields of Euler angles We first generated a vector containing three Gaussian random fields Z(p) = [Z_{1} (p), Z_{2} (p), Z_{3} (p)], where p is the spatial centroid position of each grain, having the given target autocovariance functions (ACFs) via the Karhunen– Loève expansion [16,17]. The target ACFs of Θ_{1}, Θ and Θ_{2} are supposed to be isotropic so that they only depend on the distance between the centroid positions of the grains. In this paper, we consider an exponential ACF based on [18]:
$${C}_{XX}\text{\hspace{0.17em}}\left({p}_{m},{p}_{n}\right)=\mathrm{exp}\text{\hspace{0.17em}}\left(2\frac{\Vert {p}_{m}{p}_{n}\Vert}{{l}_{X}}\right),$$(3)
in which p_{m} and p_{n} (m,n ∈{1,…, N_{g}}) are the centroid positions of any two grains in the polycrystal and l_{X} ( X ∈ (Θ_{1}, Θ, Θ_{2})) is the correlation length of the random field. In our numerical study, the case CS, with a relatively short correiation length ${l}_{X}^{s}$ and the case CL, with a relatively long correlation length ${l}_{X}^{l}$ are considered as shown in Table 2. The correlation lengths ${l}_{X}^{s}$ and ${l}_{X}^{l}$ were defined as a multiple of the mathematical expectation of the grain equivalent diameter ${\overline{D}}_{\text{eq}}$ in each realization as:
$$\begin{array}{l}{l}_{X}^{s}=\left({l}_{{\text{\Theta}}_{1}}^{s},{l}_{\text{\Theta}}^{s},{l}_{{\text{\Theta}}_{2}}^{s}\right)=\left({\overline{D}}_{\text{eq}},{\overline{D}}_{\text{eq}},{\overline{D}}_{\text{eq}}\right),\hfill \\ {l}_{X}^{l}=\left({l}_{{\text{\Theta}}_{1}}^{l},{l}_{\text{\Theta}}^{l},{l}_{{\text{\Theta}}_{2}}^{l}\right)=10\left({\overline{D}}_{\text{eq}},{\overline{D}}_{\text{eq}},{\overline{D}}_{\text{eq}}\right).\hfill \end{array}$$(4)
It should be noted that the Uncorrelated case may be imagined as the scenario where the ratio between the correlation length and average grain diameter is 1/2. The KLE is a series expansion method based on the spectral decomposition of the ACF of the random field. In practice, the KLE can be approximated by truncating at a finite number of terms in order to numerically simulate the Gaussian random fields Z_{į} (p):
$$\begin{array}{cc}{Z}_{i}\text{\hspace{0.17em}}\left(p\right)\approx {\displaystyle \sum _{k=1}^{{N}_{g}}\text{\hspace{0.17em}}\sqrt{{\lambda}_{k}^{i}}{\gamma}_{k}^{i}{\psi}_{k}^{i}\left(p\right),}& i\in \left\{1,2,3\right\}\end{array},$$(5)
where {${\lambda}_{k}^{i}$, ${\psi}_{k}^{i}$} are the eigenvalues with their corresponding eigenfunctions of the target ACF, and ${\gamma}_{k}^{i}$ is a vector of independent standard Gaussian random variables. The memoryless transforms of the standard Gaussian processes are used to simulate nonGaussian processes [19,20]. Based on the known marginal cumulative distribution functions (CDFs), the simulation of the target nonGaussian random fields could be performed by the socalled nonGaussian translation processes [21]:
$$\begin{array}{cc}{\tilde{Z}}_{i}\text{\hspace{0.17em}}\left(p\right)={F}_{{\tilde{Z}}_{i}}^{1}\text{\hspace{0.17em}}\left(\Phi \left({Z}_{i}\text{\hspace{0.17em}}\left(p\right)\right)\right),& i\in \left\{1,2,3\right\},\end{array}$$(6)
in which ${F}_{{\tilde{Z}}_{i}}\left(x\right)={f}_{\infty}^{x}\text{\hspace{0.17em}}{p}_{{\tilde{Z}}_{i}}\left(y\right)\text{\hspace{0.17em}dy}$ is the CDF of ${\tilde{Z}}_{i}$ with target PDF ${p}_{{\tilde{Z}}_{i}}$, and Φ(·) is the target CDF of the standard Gaussian distribution. We subsequently used this method to simulate the correlated Euler angles with different parameters for each grain in each realization of different polycrystal models listed in Table 1. The PDFs of the nonGaussian random fields of the Euler angles $\left({\text{\Theta}}_{1},\text{\Theta},{\text{\Theta}}_{2}\right)=\left({\tilde{Z}}_{1},{\tilde{Z}}_{2},{\tilde{Z}}_{3}\right)$ with the parameters CS are shown in Figure 3, for 200 realizations of Poly6. According to this figure, the PDFs of the numerically simulated correlated Euler angles appropriately match the target PDFs. Furthermore, it is expected that augmenting the number of realizations will lead to a more accurate convergence with the target PDFs.
Statistical convergence has been investigated on the numerically generated Euler angles by calculating the relative errors of the relevant statistical quantities over 200 realizations for each model shown in Figure 4. Here, the relative errors were defined as the normalized gradient of the cumulative mean and the cumulative coefficient of variation. The convergence criterion was set as the relative error being less than 2%. Figure 4 illustrates the statistical convergence of the generated Euler angles with the parameters CL for Poly1 up to 200 realizations. The results highlight a good convergence of the first and secondorder statistics of the random fields using 200 realizations. For all other cases, similar observations have been made.
Fig. 2 PDFs of numerically simulated arid target uncorrelated Euler angles with statistical isotropy in 200 realizations of Poly6 model (left), and heir measured normalized relative entropy, as a function of the number of realizations (right). 
Fig. 3 The nonGaussian random fields of Euler angles (Θ_{1},Θ,Θ_{2}) having CS parameters compared with the target PDFs in 200 realizations for the model Poly6. 
2.3 Computational framework for effective elastic moduli
The classical Eshelby–Kröner SelfConsistent (SC) model will be subsequently used to estimate the effective elasticity tensor of polycrystals. Eshelby [22] has shown that according to the equivalent inclusion method, under the assumption of linear elasticity and small deformation, in an unbounded homogeneous isotropic medium submitted to an applied uniform macroscopic strain ϵ_{kl} in the far field, the strain in the embedded ellipsoid is also uniform with ${\u03f5}_{ij}^{\text{inc}}={A}_{ijkl}{\u03f5}_{kl}$, where A_{įjkl} is the socalled localization tensor, and ${\u03f5}_{ij}^{\text{inc}}$ and ϵ_{kl} are the uniform strains inside the ellipsoid and in the far field, respectively. Based on Eshelby’s theory, Kroner [23] introduced the SC method to estimate the elastic moduli of polycrystals from single crystal elastic constants of cubic symmetry. Lubarda [24] proposed the method to estimate the effective elastic moduli for the isotropic polycrystals with cubic crystals, which is an extension of the SC method. He showed that if a single cubic crystal with a spherical shape is surrounded by an effective isotropic polycrystalline aggregate, the concentration tensor has the form:
$${A}_{ijkl}={I}_{ijkl}+a\text{\hspace{0.17em}}\left({\delta}_{ij}{\delta}_{kl}+2{I}_{ijkl}5{T}_{ijkl}\right),$$(7)
in which ${I}_{ijkl}=\frac{1}{2}\text{\hspace{0.17em}}\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right)$ is the fourthorder identity tensor, δ_{ij} (i,j ∈ {1,2,3}) is the Kronecker delta, being s 1 when i = j, and 0 otherwise. The explicit formula of a writes:
$$\begin{array}{l}a=\frac{\left({c}_{11}+2{c}_{12}+6b\right)\left({c}_{11}{c}_{12}2b\right)}{3\left[8{b}^{2}+9{c}_{11}b+\left({c}_{11}{c}_{12}\right)\left({c}_{11}+2{c}_{12}\right)\right]},\hfill \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}8{b}^{3}+\left(5{c}_{11}+4{c}_{12}\right)\text{\hspace{0.17em}}{b}^{2}{c}_{44}{\left(7{c}_{11}4{c}_{12}\right)}^{b}\hfill \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{c}_{44}\left({c}_{11}{c}_{12}\right)\left({c}_{11}+2{c}_{12}\right)=0,\hfill \end{array}$$(8)
where c_{IJ} (I J ∈ {11,12, 44}) are the components of elastic constants in the crystallite’s local reference, and b is the positive real root of the given cubic polynomial [23].
In equation (7), T_{ijkl} is the rotation tensor and can be expressed in the form ${T}_{ijkl}={\displaystyle \sum _{m\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{3}{t}_{im}{t}_{jm}{t}_{km}{t}_{lm}}\text{\hspace{0.17em}}\left(\left\{i,j,k,l\right\}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\left\{1,2,3\right\}\right)$. The corresponding rotation matrix t is defined as [13]:
$$\text{t\hspace{0.17em}=\hspace{0.17em}}\left[\begin{array}{ccc}\mathrm{cos}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)+\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)& \mathrm{cos}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)& \mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\\ \mathrm{cos}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)+\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)& \mathrm{cos}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)& \mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{1}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\\ \mathrm{sin}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left(\text{\Theta}\right)& \mathrm{cos}\text{\hspace{0.17em}}\left({\text{\Theta}}_{2}\right)\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\left(\text{\Theta}\right)& \mathrm{cos}\text{\hspace{0.17em}}\left(\text{\Theta}\right)\end{array}\right].$$(9)
Based on the theory proposed by Eshelby and equation (7), Lubarda [24] derived the elastic modulus of n^{th} crystallite with spherical geometry as:
$$\begin{array}{l}{C}_{ijkl}^{\text{sc,}n}={C}_{ijkl}^{n}+a\text{\hspace{0.17em}}[\left({c}_{11}{c}_{12}\right)\text{\hspace{0.17em}}{\delta}_{ij}{\delta}_{kl}+4{c}_{44}{I}_{ijkl}\hfill \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left[3\text{\hspace{0.17em}}\left({c}_{11}{c}_{12}\right)+4{c}_{44}\right]\text{\hspace{0.17em}}{T}_{ijkl}^{n}],\hfill \end{array}$$(10)
where the elasticity tensor ${C}_{ijkl}^{n}$ of n^{th} cubic crystallite in the laboratory reference frame reads:
$${C}_{ijkl}^{n}{c}_{12}\text{\hspace{0.17em}}\left({\delta}_{ij}{\delta}_{kl}\right)+{c}_{44}\text{\hspace{0.17em}}\left({\delta}_{ik}{\delta}_{jl}+{\delta}_{il}{\delta}_{jk}\right)+v{T}_{ijkl}^{n},$$(11)
in which v is the anisotropy coefficient of the single crystal, defined as v = c_{11} − c_{12} − 2c_{44}. The effective elasticity tensor of each polycrystal with total volume V and N_{g} grains can then be calculated using the volume average:
$$\langle {C}_{ijkl}^{\text{eff}}\rangle =\frac{1}{V}\text{\hspace{0.17em}}{\displaystyle \int {}_{\text{\Omega}}\text{\hspace{0.17em}}{C}_{ijkl}^{\text{sc,}n}\text{\hspace{0.17em}}\left(\text{x}\right)\text{dx\hspace{0.17em}=\hspace{0.17em}}\frac{1}{V}{\displaystyle \sum _{n=1}^{{N}_{g}}{C}_{ijkl}^{\text{sc,}n}{V}_{n}}},$$(12)
where the integration domain Ω is the whole volume of the polycrystal, V = ∫Ω dx, and V_{n} is the volume of n^{th} grain in polycrystal.
Fig. 4 Relative errors (normalized gradient) of the cumulative mean (left), and of the cumulative coefficient of variation (right) of the generated Euler angles with parameters CL for Poly1 over 200 realizations. 
3 Results and discussion
The statistical properties of effective elastic moduli have been investigated on different cubic materials, listed in Table 3. The normalized single crystal anisotropy factor is defined as A = 1 − A_{Z} , where A_{Z} = 2c_{44}/(c_{11} − c_{12}) is the socalled Zener anisotropy factor [25]. For the particular case of isotropy, A_{Z} equals 1 and thus A vanishes. Based on the values of A presented in Table 3, aluminum (Al) has the lowest normalized anisotropy factor, while lithium (Li) has the largest one.
The effective elastic moduli of the synthetic polycrystals are estimated using the aforementioned SC approach. Figure 5 presents the PDFs of the effective elasticity tensor component ${C}_{11}^{\text{eff}}$. These PDFs are estimated for iron (Fe) using 200 realizations of each polycrystal sample set Poly2, Poly4, and Poly6, considering the Euler angles as uncorrelated (case Uncorrelated) and correlated with a relatively long correlation (case CL). It clearly shows that increasing the number of grains (N_{g}) decreases the variance of the corresponding distributions of the effective elastic moduli, regardless of the microstructural details of the random polycrystals.
The standard deviations of the effective elastic moduli are subsequently calculated for better investigating the influence of correlated and uncorrelated Euler angles for the case Uncorrelated, CS and CL on the statistical properties:
$$\begin{array}{cc}\overline{Y}=\frac{1}{{N}_{r}}{\displaystyle \sum _{i=1}^{{N}_{r}}{y}_{i},}& \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\sigma}_{Y}=\sqrt{\frac{1}{{N}_{r}1}{\displaystyle \sum _{i=1}^{{N}_{r}}{\left({y}_{i}\overline{Y}\right)}^{2}}}\end{array},$$(13)
wherein N_{r} is the number of the realizations of each polycrystal model, being N_{r} = 200 in this study. Y is the quantity of interest, and $\overline{Y}$ and σ_{Y} are its expectation and standard deviation, respectively. Here, the random variable of interest is the component ${C}_{11}^{\text{eff}}$ of the effective macroscopic elasticity tensor.
Following [3], in order to obtain a master curve for all materials, the standard deviations of ${C}_{11}^{\text{eff}}\text{\hspace{0.17em}}\left({\sigma}_{{C}_{11}^{\text{eff}}\text{\hspace{0.17em}}}\right)$ are normalized by the absolute value of the anisotropy coefficient of the SC approach v_{sc}, which is defined as v_{sc} = v(1 − 3a) − 10a c_{44}, for all the cases in our study. It is obvious that ${\sigma}_{{C}_{11}^{\text{eff}}\text{\hspace{0.17em}}}$ is linearly proportional to v_{sc}. As shown in Figure 6, regardless of the parameters of Euler angles, the normalized standard deviations of ${C}_{11}^{\text{eff}}$for all materials with different N_{g} are dominated by three main linear curves in a logscale. Furthermore, an inverse relationship was observed between the normalized ${\sigma}_{{C}_{11}^{\text{eff}}\text{\hspace{0.17em}}}$ and N_{g}. The main curves, denoted as f_{1}, f_{2} and f_{3} in the legends of Figure 6, can be expressed as a function linking ${\sigma}_{{C}_{11}^{\text{eff}}}$ to N_{g,} each with different coefficients a and β corresponding to different cases:
$$\frac{{\sigma}_{{C}_{11}^{e\text{ff}}}}{\left{v}_{\text{sc}}\right}=\alpha \times {N}_{g}^{\beta}.$$(14)
In Table 4, the estimated values of coefficients a and β for different cases are summarized. It is worth mentioning that the normalized variabilities of the effective elastic moduli for other components, e.g., ${C}_{12}^{\text{eff}}$ and ${C}_{44}^{\text{eff}}$, have similar trends as those observed for the component ${C}_{11}^{\text{eff}}$. However, the values oº the parameters a and β are different.
The effect of correlated Euler angles, with varying correlation length parameters, on the normalized standard deviations of effective elastic moduli ${C}_{11}^{\text{eff}}$, is compared to the uncorrelated case and presented in Figure 7. As expected, the polycrystals with uncorrelated Euler angles have the lowest standard deviation of effective elastic moduli, compared with the polycrystals whose Euler angles are correlated. This could be explained as the uncorrelated Euler angles of the grains in one realization of polycrystal could have more variabilities than the correlated ones. As a result, more variabilities of the Euler angles reduce the statistical anisotropy level of the polycrystal, hence the normalized standard deviation of effective elastic moduli decreases. If the polycrystals are close to theoretical isotropy, then the standard deviations of effective elastic moduli between each realization will tend towards 0.
On the contrary, adding correlations to the Euler angles could reduce their variabilities in each realization and thus increase their anisotropy levels. As such, the standard deviations of effective elastic moduli will have a higher value. This conclusion is shown in Figure 7, where the values of the normalized ${\sigma}_{{C}_{11}^{\text{eff}}}$ for the polycrystals whose Euler angles have a longer correlation length (CL) are larger than those having a shorter correlation length (CS). In fact, a relatively long correlation length implies an increase in the characteristic size of heterogeneity. Considering; this heterogeneity as an “effective grain” with similar Euler angles and a larger size than the ones in CS and Uncorrelated cases, the “effective number of grains” is reduced, leading to increased variability of effective elastic moduli among the realizations of sample sets. Nevertheless, when comparing the CS case with the Uncorrelated one, the difference is minimal, particularly for polycrystals with a sufficient number of grains. Therefore, for polycrystals with Euler angles exhibiting relatively short correlation lengths comparable to the grain size, it is acceptable to simulate crystallographic orientations as uncorrelated to simplify the numerical model in order to expedite simulation speed.
In general, our study shows that spatially correlated Euler angles with correlation lengths relatively longer than the grain size can exert a nonnegligible impact on the statistical properties of effective elastic moduli in polycrystalline materials. Studying spatially correlated Euler angles of polycrystals holds significant implications for future simulation and experimental studies. By incorporating spatial correlation in simulations, a more realistic representation of polycrystal microstructure is achieved, leading to improved accuracy in predicting macroscopic behavior and mechanical properties. In experimental studies, using the EBSD technique to measure the spatial correlation enables better characterization of polycrystal microstructure, which is crucial for understanding material properties, deformation mechanisms, and failure behaviors. Additionally, including spatial correlation in crystallographic orientations ensures robust predictions, reducing uncertainties in simulations and experimental results. This enhances confidence in the reliability of study outcomes, which is essential for designing and developing materials for critical applications.
Fig. 5 PDFs of the component ${C}_{11}^{\text{eff}}$ of effective elasticity tensor of iron (Fe) using 200 realizations of Poly2 (100 grains per realization), Poly4 (1000 grains per realization) and Poly6 (10000 grains per realization) with Uncorrelated (left), and CL (right) types of Euler angles. 
Fig. 6 Normalized standard deviations of ${C}_{11}^{\text{eff}}$, in terms of the number of grains, for Euler angles with parameters Uncorrelated (top), CS (bottom left), and CL (bottom right). 
Coefficients a and β in equation (14), obtained from numerical results, with three different parameters of Euler angles.
Fig. 7 Main curves of the relation between normalized standard deviations of ${C}_{11}^{\text{eff}}$ and number of grains for Euler angles with Uncorrelated, CS, and CL parameters. 
4 Conclusion
The influence of the spatially correlated Euler angles on the statistical properties of effective elastic moduli has been investigated in this article based on the synthetic polycrystals of a variety of cubic materials having different anisotropy factors The random fields of the Euler angles, with an exponential kernel function with two different correlation lengths, are numerically simulated based on the Karhunen–Loève expansion. The effective elasticity tensors of the polycrystals are then estimated by using the classical EshelbyKröner SelfConsistent approach The normalized standard deviations of the effective elastic moduli ${C}_{11}^{\text{eff}}$ ( are calculated to investigate the statistical properties of the homogenized medium. Results show that apart from the inversely proportional relationship between the normalized standard deviations of ${C}_{11}^{\text{eff}}$ and the number of grains, the spatially correlated Euler angles also influence the statistical properties of effective elastic moduli. The correlation of the Euler angles will increase the standard deviations of the effective elastic properties compared with uncorrelated ones, especially when the correlation lengths exceed the mean grain size. These results highlight the necessity of considering spatially correlated Euler angles in the numerical models based on real polycrystals to enhance the reliability of future numerical investigations. Additionally, the influence of spatially correlated Euler angles on the statistical behaviors of ultrasonic phase velocities as well as the scatteringinduced attenuation, is currently under study.
Conflict of Interest
The authors certify that they have no financial conflict of interest in connection with this article.
Funding
This work is financed by Ministère de l’Education Nationale de l’Enseignement Supérieur, de la Recherche et de l’Innovation (MESRI). The support of Centre de Calcul Intensif des Pays de la Loire (CCIPL) is gratefully acknowledged.
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Cite this article as: N. Sheng, S. Khazaie, M. Chevreuil, S. Fréour, Statistical properties of effective elastic moduli of random cubic polycrystals, Mechanics & Industry 24, 33 (2023)
All Tables
Coefficients a and β in equation (14), obtained from numerical results, with three different parameters of Euler angles.
All Figures
Fig. 1 A realization of a polycrystal having 10000 grains (left), the PDF of its grain volumes (middle), and the PDF of its grain sizes D_{eq} (histogram, righf) compared with the target one (black solid line, right). 

In the text 
Fig. 2 PDFs of numerically simulated arid target uncorrelated Euler angles with statistical isotropy in 200 realizations of Poly6 model (left), and heir measured normalized relative entropy, as a function of the number of realizations (right). 

In the text 
Fig. 3 The nonGaussian random fields of Euler angles (Θ_{1},Θ,Θ_{2}) having CS parameters compared with the target PDFs in 200 realizations for the model Poly6. 

In the text 
Fig. 4 Relative errors (normalized gradient) of the cumulative mean (left), and of the cumulative coefficient of variation (right) of the generated Euler angles with parameters CL for Poly1 over 200 realizations. 

In the text 
Fig. 5 PDFs of the component ${C}_{11}^{\text{eff}}$ of effective elasticity tensor of iron (Fe) using 200 realizations of Poly2 (100 grains per realization), Poly4 (1000 grains per realization) and Poly6 (10000 grains per realization) with Uncorrelated (left), and CL (right) types of Euler angles. 

In the text 
Fig. 6 Normalized standard deviations of ${C}_{11}^{\text{eff}}$, in terms of the number of grains, for Euler angles with parameters Uncorrelated (top), CS (bottom left), and CL (bottom right). 

In the text 
Fig. 7 Main curves of the relation between normalized standard deviations of ${C}_{11}^{\text{eff}}$ and number of grains for Euler angles with Uncorrelated, CS, and CL parameters. 

In the text 
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