Decomposition of Elasticity Tensor on Material Constants and Mesostructures of Metal Plates

Abstract

Most metal plates are orthorhombic aggregates of cubic crystallites. First, we discuss the representations of the stress tensor, the strain tensor, the elasticity tensor, and the rotation tensor under the Kelvin notation. Then, we give the decomposition of determining the material constants and the mesostructure tensors on the metal plate of cubic crystallites. Under the Voigt model and the Reuss model, we derive the volume average stiffness tensor and the volume average flexibility tensor’s inverse, respectively, of cubic crystallites based on the decomposition. The elasticity tensors of the Voigt model and the Reuss model are upper and lower bounds of the effective elasticity tensor, respectively. We make use of an FEM example to check the decomposition of the elasticity tensor on the material constants and the mesostructures. The results of our decomposition are consistent with the FEM simulation’s results.

1. Introduction

Let the metal plate Ω be an orthorhombic aggregates of cubic crystallites. The mean stress tensor ${\overline{\sigma }}_{\mathrm{ij}}$ and the mean strain tensor ${\overline{\epsilon }}_{ij}$ of the metal plate Ω are

$${\overline{\sigma }}_{ij}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{\sigma }_{ij}}(x)dx,{\overline{\epsilon }}_{ij}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{\epsilon }_{ij}}(x)dx$$

(1)

The local stress strain relations of the metal plate at $x\in \mathsf{\Omega }$ are

$${\sigma }_{ij}(x)=C_{ijkl}R(x){\epsilon }_{kl}(x),{\epsilon }_{ij}(x)=S_{ijkl}R(x){\sigma }_{kl}(x),$$

where $R(x)$ is a piecewise constant tensor function [1]. The effective elastic stiffness tensor $C_{ijkl}^{\mathrm{eff}}$ and the effective elastic flexibility tensor $S^{\mathrm{eff}}$ of the metal plate are defined by [2,3]

$${\overline{\sigma }}_{ij}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{C}_{ijkl}}(R(x)){\epsilon }_{kl}(x)dx\stackrel{\mathrm{def}}{=}{C}_{ijkl}^{\mathrm{eff}}{\overline{\epsilon }}_{kl},$$

(2)

$${\overline{\epsilon }}_{ij}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{S}_{ijkl}}(R(x)){\sigma }_{kl}(x)dx\stackrel{\mathrm{def}}{=}{S}_{ijkl}^{\mathrm{eff}}{\overline{\sigma }}_{kl}.$$

(3)

It is important to obtain the effective elastic stiffness tensor $C_{ijkl}^{\mathrm{eff}}$ and the effective elastic flexibility tensor $S_{ijkl}^{\mathrm{eff}}$ because the metal plate behaves under forces macroscopically like a homogeneous material. Of course, it may be possible to determine the effective elasticity tensor $C_{ijkl}^{\mathrm{eff}}$ directly by experiments and numerical simulations. The macroscopic elastic properties of the metal plate depend not only on the statistical data of grain orientations but also on the elastic constants of single crystals. However, if the relation among the microstructures, elastic constants, and its effective stiffness tensor of the metal plate is ascertained, then by suitably arranging the microstructures, we can make the metal plates we need.

The orthogonal machining process of metal sheets results in an orthotropic distribution of grain orientations in the metal material from a macroscopic statistical perspective [4].

For the Voigt model, all crystallites in a metal plate experience the same state of deformation (${\epsilon }_{ij}(x)={\overline{\epsilon }}_{ij}$). From the definition of $C_{ijkl}^{\mathrm{eff}}$ in (2), there is

$$C_{ijkl}^{\mathrm{eff}}\stackrel{\mathrm{def}}{=}{C}_{ijkl}^{\mathrm{Voigt}}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{C}_{ijkl}}(R(x))dx$$

(4)

under the Voigt model; hence, the compatibility condition is satisfied, but traction continuity across the crystallites may not be guaranteed. For the Reuss model, since the basic assumption is that all crystallites in the metal plate have the same state of stress (${\sigma }_{ij}(x)={\overline{\sigma }}_{ij}$). By the definition of $S_{ijkl}^{\mathrm{eff}}$ in (3), we have

$$S_{ijkl}^{\mathrm{eff}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{S}_{ijkl}^{\mathrm{Reuss}}={\displaystyle \frac{1}{\left|\mathrm{\Omega }\right|}}{\displaystyle {\int }_{\mathrm{\Omega }}{S}_{ijkl}}(R(x))dx$$

(5)

under the Reuss model; the condition of traction continuity at the boundaries of the crystallites is satisfied, while the compatibility of deformation between the crystallites may be violated. The Voigt model and the Reuss model provide an upper bound and a lower bound [5] on the effective elasticity tensors of polycrystalline aggregates, respectively.

The ODF was introduced independently by Bunge [6] and by Roe [7]. The orientation distribution function (ODF) is used to describe the probability density of finding a crystallite with orientation $R$ in SO(3). In linear elasticity the ODF was first introduced into the constitutive relation on the metal plate through the Voigt model and orientational averaging by Morris [8] and Sayers [9]. Morris [10], Man [11], and Man and Huang [12,13] presented the elasticity tensor of metal plates with the effect of the crystalline orientation distribution. Numerous studies have demonstrated that the internal composition of composite materials directly influences their performance in all aspects [14,15,16]. Therefore, research on the microstructural characteristics of materials is of significant importance.

In the study of material microstructure and mechanical properties, Jeff et al. [17] used resonant ultrasound spectroscopy with Bayesian inference to accurately estimate the elastic constants of anisotropic microstructures, including residual stresses. This method, therefore, enabled the precise determination of all 21 elastic constants, which in turn, aided in predicting their elastic behavior. Similarly, Napat et al. [18] proposed an optimization method based on integer approximation and the L1 norm, which effectively reduced data redundancy and thereby improved simulation efficiency. Furthermore, Glüge and Bucci [19] explored the physical meaning of convexity in yield surfaces of plastic materials, showing that, although convexity is important mathematically, it does not directly relate to physical properties. In addition, Oberman and Salvador [20] introduced a simplified partial differential equation obstacle problem to compute the visibility set from a given point to obstacles, improving the numerical efficiency of visibility analysis. Lou and Yoon [21] transformed stress invariants into an anisotropic yield function using a fourth-order tensor, which effectively modeled the anisotropic yielding behavior of BCC and FCC metals. Lastly, Cazacu [22] developed yield criteria for randomly oriented polycrystalline metals, providing a theoretical basis for predicting material behavior under complex stresses. Overall, these studies support the integration of microstructure analysis with macro-mechanical performance prediction, thereby laying a solid foundation for future research in material science.

The outline of this paper is as follows. In Section 2, assuming that the space X can be written as a direct sum of its irreducible parts $D_0,D_1,\cdot \cdot \cdot ,D_m$, by the representation theorem [23,24,25,26,27] of group, we study the irreducible decomposition and the characters of the elasticity tensor. In Section 3, the stress tensor, the strain tensor, the elasticity tensor, and the rotation tensor are presented under the Kelvin notation. In Section 4, based on the papers [28,29] of Man and Huang, a set of irreducible basis tensors is listed under the Kelvin notation. In Section 5, the elastic constants and the mesostructures of the elasticity tensor on cubic crystals are decomposed. In Section 6 and Section 7, the material constants and the mesostructure tensors of the elasticity tensor on the metal plate of cubic crystallites are decomposed. We give the decomposition methods of determining the material constants and the mesostructure tensors on the metal plate of cubic crystallites. We derive the volume average stiffness tensor and the volume average flexibility tensor’s inverse, respectively, of cubic crystallites by the decomposition method. The elasticity tensors of the metal plate under the Voigt model and under the Reuss model are an upper bound and a lower bound [5], respectively, of the effective elasticity tensor. In Section 8, an example is used to show that our decomposition is reliable by comparing the FEM simulation’s results with our expressions listed in

Table 1. $C_{ij}$ (GPa) for $\mathrm{\Omega }$ under the Kelvin notation.

Model	$C_{11}$	$C_{22}$	$C_{33}$	$C_{44}$	$C_{55}$	$C_{66}$	$C_{12}$	$C_{13}$	$C_{66}$
$C^{\mathrm{Voigt}}$	276.17	244.12	270.24	230.81	166.70	218.96	134.48	108.35	140.41
$C^{\mathrm{FEM}}$	275.29	243.87	269.31	230.82	164.56	218.58	134.57	109.14	140.55
${(S^{\mathrm{Reuss}})}^{-1}$	259.92	240.14	256.26	229.17	138.08	204.25	137.60	121.48	141.26

.

2. Irreducible Decomposition of Elasticity Tensors

Let us assume that a fixed spatial Cartesian coordinate system has been chosen and that $e_1={[1,0,0]}^T,e_2={[0,1,0]}^T$, and ${\mathrm{e}}_3={[0,0,1]}^T$ are unit vectors along three coordinate axes $x_i(i=1,2,3)$ of the system, respectively. The 2nd-order rotation tensor $R$ can be described by the Euler angles $(\psi ,\theta ,\varphi )$ as follows [7].

$$R(\psi ,\theta ,\varphi )=R\left(e_3,\psi \right)R\left(e_2,\theta \right)R\left(e_3,\varphi \right),$$

(6)

where

$$R\left(e_2,\theta \right)=\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right],R\left(e_3,\varphi \right)=\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]$$

(7)

Any function $F\left(R\right)\in L^2\left(SO\left(3\right)\right)$ (Lebesgue space) can be expanded as an infinite series in terms of the Wigner D-functions $D_{mn}^l$ [6]:

$$F\left(R\right)={\displaystyle \sum _{l=0}^{\infty }{\displaystyle \sum _{m=-l}^l{\displaystyle \sum _{n=-l}^l{f}_{mn}^l}}}{D}_{mn}^l\left(R\right),f_{mn}^l={\left(-1\right)}^{m+n}{\left(f_{\overline{m}\overline{n}}^l\right)}^{*},$$

(8)

where the complex number $f_{mn}^l$ are the expanded coefficients, ($f_{mn}^l$)* denotes the complex conjugate of $f_{mn}^l$. The Wigner D-function bases $D_{mn}^l$ are [24,26,27].

$$D_{mn}^l\left(R\left(\psi ,\theta ,\varphi \right)\right)=d_{mn}^l\left(\theta \right)e^{-i\left(m\psi +n\varphi \right)},$$

(9)

where

$$d_{mn}^l\left(\theta \right)={\displaystyle \sum _{k=\max \left(m-n,0\right)}^{\min \left(l-n,l+m\right)}{W}_{mn}^{lk}}{\left(\cos {\displaystyle \frac{\theta }{2}}\right)}^{2l-n+m-2k}{\left(\sin {\displaystyle \frac{\theta }{2}}\right)}^{2k-m+n},$$

(10)

$$W_{mn}^{lk}={\displaystyle \frac{{\left(-1\right)}^k\sqrt{\left(l+m\right)!\left(l-m\right)!\left(l+n\right)!\left(l-n\right)!}}{k!\left(l-n-k\right)!\left(l+m-k\right)!\left(k-m+n\right)!}}$$

and

$$D_{\overline{m}\overline{n}}^l={\left(-1\right)}^{m+n}{\left(D_{mn}^l\right)}^{*},D_{mn}^l\left(R^{-1}\right)={\left(D_{nm}^l\left(R\right)\right)}^{*},$$

(11)

$$D_{mn}^l\left(R_1{R}_2\right)={\displaystyle \sum _{K=-l}^l{D}_{mk}^l}\left(R_1\right)D_{kn}^l\left(R_2\right)$$

where $\overline{m}=-m$ and $\overline{n}=-n$, along with the properties of $d_{mn}^l\left(\theta \right)$:

$$d_{mn}^l\left(-\theta \right)=d_{nm}^l\left(\theta \right),d_{\overline{m}\overline{n}}^l\left(\theta \right)=d_{mn}^l\left(-\theta \right),d_{mn}^l\left(\theta \right)={\left(-1\right)}^{m-n}{d}_{nm}^l\left(\theta \right)$$

(12)

$$d_{mn}^l\left(\pi -\theta \right)={\left(-1\right)}^{l+m}{d}_{m\overline{n}}^l\left(\theta \right),d_{mn}^l\left(0\right)={\delta }_{mn},d_{00}^0\left(\theta \right)=1.$$

As for the Wigner D-functions, Man [30] provides a detailed explanation of the Wigner D-functions and also explains the irreducible basis tensors. The Wigner D-functions $D_{mn}^l$ constitute an orthogonal basis whose inner products satisfy

$$〈D_{mn}^l,D_{m^{\prime }{n}^{\prime }}^{l^{\prime }}〉={\displaystyle {\int }_{SO\left(3\right)}{D}_{mn}^l\left(R\right)}{\left(D_{m^{\prime }{n}^{\prime }}^{l^{\prime }}\left(R\right)\right)}^{*}dg={\displaystyle \frac{8{\pi }^2}{2l+1}}{\delta }_{l{l}^{\prime }}{\delta }_{m{m}^{\prime }}{\delta }_{n{n}^{\prime }},$$

(13)

where ${\left(D_{m^{\prime }{n}^{\prime }}^{l^{\prime }}\left(R\right)\right)}^{*}$ denote the complex conjugate of the complex number $\left(D_{m^{\prime }{n}^{\prime }}^{l^{\prime }}\left(R\right)\right)$. If the rotation group SO(3) is parametrized by the Euler angles, there is

$${\displaystyle {\int }_{SO\left(3\right)}(\cdot )}dg\stackrel{\mathrm{def}}{=}{\displaystyle {\int }_0^{2\pi }}{\displaystyle {\int }_0^{\pi }}{\displaystyle {\int }_0^{2\pi }}(\cdot )\sin \theta d\psi d\theta d\varphi ,$$

(14)

where SO(3) denotes the group of all rotations in R³.

Assume that the space X can be written as a direct sum of its irreducible parts D₀, D₁, …, D_m [23,25]:

$$X=D_0^{\oplus r_0^{(h)}}\oplus D_1^{\oplus r_1^{(h)}}\cdot \cdot \cdot D_h^{\oplus r_h^{(h)}}$$

(15)

and

$$D_l=\left\{D_{mn}^l\left(R\right):\left|m\right|,\left|n\right|\le l\right\},$$

(16)

$D_{mn}^l\left(R\right)$ are given in (9). Each rotation $R$ induces a linear transformation $R^{\otimes h}$ on $X$ as defined by

$${\left(R^{\otimes h}C\right)}_{i_1{i}_2\cdot \cdot \cdot i_h}=R_{i_1{j}_1}{R}_{i_2{j}_2}\cdot \cdot \cdot R_{i_h{j}_h}{C}_{j_1{j}_2\cdot \cdot \cdot j_h}$$

(17)

with the $h^{\mathrm{th}}$-tensor $C\in X$.

Now we proceed to determine the values $r_l^{(h)}\left(l=0,1,\cdot \cdot \cdot ,h\right)$ in (15). Let

$${\chi }_l=\mathrm{tr}\left({\mathcal{D}}_l\right)={\displaystyle \sum _{m=-l}^l{D}_{mm}^l}(R(\psi ,\theta ,\varphi ))$$

(18)

be the character of the representation $D_l$. The inner product $〈{\chi }_l,{\chi }_{l^{\prime }}〉$ of ${\chi }_l$ and ${\chi }_{l^{\prime }}$ is defined by

$$\begin{array}{cc}〈{\chi }_l,{\chi }_{l^{\prime }}〉\hfill & ={\displaystyle \sum _{m=-l}^l{\displaystyle \sum _{n^{\prime }=-l^{\prime }}^{l^{\prime }}\frac{1}{8{\pi }^2}}}{\displaystyle {\int }_{SO\left(3\right)}}{D}_{mm}^l\left(R\right){\left(D_{m^{\prime }{m}^{\prime }}^{l^{\prime }}\left(R\right)\right)}^{*}dg\hfill \\ & ={\delta }_{l{l}^{\prime }}{\displaystyle \sum _{m=-l}^l{\displaystyle \sum _{m^{\prime }=-l^{\prime }}^{l^{\prime }}\frac{{\delta }_{m{m}^{\prime }}}{2l+1}}}={\delta }_{l{l}^{\prime }}{\displaystyle \frac{{\delta }_{m{m}^{\prime }}}{2l+1}}={\delta }_{l{l}^{\prime }}\hfill \end{array}$$

(19)

where the relation (13) is employed.

The rotation tensor $R\left(t,\omega \right)$ is described by the axis-angle parameters $\left(t,\omega \right)$ where $t=\overline{R}{e}_3$. Using Euler’s theorem $R\left(\overline{R}{e}_3,\omega \right)=\overline{R}R\left(e_3,\omega \right){\overline{R}}^T$ where $\overline{R}$ is a rotation tensor, one can obtain the character of representation $D^l$ [26,28].

$$\begin{array}{cc}{\chi }_l\left(\omega \right)\hfill & =\mathrm{tr}\left(D_l\left(R\left(t,\omega \right)\right)\right)=\mathrm{tr}\left(D^l\left(\overline{R}R\left(e_3,\omega \right){\overline{R}}^{-1}\right)\right)=\mathrm{tr}\left(D^l\left(R\left(e_3,\omega \right)\right)\right)\hfill \\ & ={\displaystyle \sum _{m=-l}^l{D}_{mm}^l\left(\omega ,0,0\right)}={\displaystyle \sum _{m=-l}^l{e}^{-im\omega }}={\displaystyle \frac{\sin \left(l+\frac{1}{2}\right)\omega }{\sin \frac{1}{2}\omega }}\hfill \end{array}$$

(20)

By (9) and $d_{mn}^l\left(0\right)={\delta }_{mn}$. The character of the representation on the space X can be expressed as

$$\begin{array}{cc}{\left.\chi \right|}_X\hfill & =\mathrm{tr}\left(X\right)=r_0^{\left(h\right)}\mathrm{tr}\left(D_0\right)+r_1^{\left(h\right)}\mathrm{tr}\left(D_1\right)\cdot \cdot \cdot +r_h^{\left(h\right)}\mathrm{tr}\left(D_h\right)\hfill \\ & =r_0^{\left(h\right)}{\chi }_0+r_1^{\left(h\right)}{\chi }_1\cdot \cdot \cdot +r_h^{\left(h\right)}{\chi }_h\hfill \end{array}$$

(21)

Combining (23), (20), and (21) with (19), one has the multiplicities $r_l^{\left(h\right)}$ in (15):

$$\begin{array}{cc}{r}_l^{\left(h\right)}\hfill & ={\displaystyle \frac{1}{8{\pi }^2}}{\displaystyle {\int }_0^{2\pi }}{\displaystyle {\int }_0^{\pi }}{\displaystyle {\int }_0^{\pi }}{\left.\chi \right|}_X{\left({\chi }_l\left(\omega \right)\right)}^{*}4{\sin }^2{\displaystyle \frac{\omega }{2}}\sin \mathsf{\Theta }d\omega d\mathsf{\Theta }d\mathsf{\Phi }\hfill \\ & ={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\pi }}{\left.\chi \right|}_X{\left({\chi }_l\left(\omega \right)\right)}^{*}{\sin }^2{\displaystyle \frac{\omega }{2}}d\omega \hfill \end{array}$$

(22)

by integral on SO(3) in terms of the axis-angle parameters [27], where

$${\displaystyle {\int }_{SO\left(3\right)}(\cdot )}dg={\displaystyle {\int }_0^{2\pi }}{\displaystyle {\int }_0^{\pi }}{\displaystyle {\int }_0^{\pi }}(\cdot )4{\sin }^2{\displaystyle \frac{\omega }{2}}\sin \mathsf{\Theta }d\omega d\mathsf{\Theta }d\mathsf{\Phi }.$$

(23)

Because det $\left(\left[R_{ij}\left(e_3,\omega \right)-\lambda {\delta }_{ij}\right]\right)=0$ has three distinct eigenvalues

$${\lambda }_1=e^{i\omega },{\lambda }_2=e^{-i\omega },{\lambda }_3=1,$$

(24)

the corresponding unitary eigenvectors $b_i\left(i=1,2,3\right)$ constitute a matrix $B=\left[b_1,b_2,b_3\right]$ and satisfy

$$B^{-1}R\left(e_3,\omega \right)B=\Lambda ,\Lambda =\mathrm{diag}\left({\lambda }_1,{\lambda }_2,{\lambda }_3\right).$$

(25)

Hence the character ${\left.\chi \right|}_V$ of the tensor representation for $R\mapsto \left.R\right|V$ is [23,26,27].

$$\begin{array}{cc}{\left.\chi \right|}_V\hfill & =\mathrm{tr}\left(R\right)=\mathrm{tr}\left(BR\left(e_3,\omega \right)B^{-1}\right)\hfill \\ & =\mathrm{tr}\left(\Lambda \right)={\mathsf{\Lambda }}_{i_1{i}_1}={\displaystyle \sum _{i=1,2,3}{\lambda }_i}=1+2\cos \omega .\hfill \end{array}$$

(26)

Example 1.

When $X=V^h$, the components of the rotation tensor product $R^{\otimes h}$ are

$${\left(R^{\otimes h}\right)}_{i_1{j}_1{i}_2{j}_2\cdot \cdot \cdot i_h{j}_h}=R_{i_1{j}_1}{R}_{i_2{j}_2}\cdot \cdot \cdot R_{i_h{j}_h},$$

(27)

whose character ${\chi }_{\left|V^h\right.}$ for the tensor product $R^{\otimes h}$ is

$${\left.\chi \right|}_{V^h}=R_{i_1{i}_1}{R}_{i_2{i}_2}\cdot \cdot R_{i_h{i}_h}={\mathsf{\Lambda }}_{i_1{i}_1}{\mathsf{\Lambda }}_{i_2{i}_2}\cdot \cdot \cdot {\mathsf{\Lambda }}_{i_h{i}_h}={\left(1+2\cos \omega \right)}^h$$

(28)

by (25) and (26).

Example 2.

Let $X=[{[V^2]}^{\frac{h}{2}}]\left(h=2,4,6\right)$ be the space of the major and minor symmetry tensors. The character of the tensor representation $[V^2]$ is

$$\begin{array}{cc}{\left.\chi \right|}_{[V^2]}\left(\omega \right)\hfill & ={\displaystyle \frac{1}{2!}}\left({\mathsf{\Lambda }}_{i_1{i}_1}{\mathsf{\Lambda }}_{j_1{j}_1}+{\mathsf{\Lambda }}_{i_1{j}_1}{\mathsf{\Lambda }}_{j_1{i}_1}\right)\hfill \\ & ={\displaystyle \frac{1}{2!}}\left[{\left({\lambda }_1+{\lambda }_2+{\lambda }_3\right)}^2+{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2\right]\hfill \\ & ={\displaystyle \sum _{j_1\ge i_1}{\lambda }_{i_1}{\lambda }_{j_1}}.\hfill \end{array}$$

(29)

Taking

$$\begin{array}{l}{\mathsf{\Gamma }}_1={\lambda }_1{\lambda }_1,{\mathsf{\Gamma }}_2={\lambda }_1{\lambda }_2,{\mathsf{\Gamma }}_3={\lambda }_1{\lambda }_3,\\ {\mathsf{\Gamma }}_4={\lambda }_2{\lambda }_2,{\mathsf{\Gamma }}_5={\lambda }_2{\lambda }_3,{\mathsf{\Gamma }}_6={\lambda }_3{\lambda }_3,\end{array}$$

(30)

We have

$${\left.\chi \right|}_{{[V^2]}^{\frac{h}{2}}]}={\displaystyle \sum _{i_1=1}^6{\displaystyle \sum _{i_2=i_1}^6\cdot \cdot \cdot }}{\displaystyle \sum _{i_{\frac{h}{2}}=i_{\frac{h}{2}-1}}^6{\mathsf{\Gamma }}_{i_1}{\mathsf{\Gamma }}_{i_2}\cdot \cdot \cdot {\mathsf{\Gamma }}_{i_{\frac{h}{2}}}}.$$

(31)

Putting (24) and (30) into (31), we obtain the characters of the tensor representation $[{[V^2]}^{\frac{h}{2}}](h=2,4,6)$.

$${\left.\chi \right|}_{\left[V^2\right]}={\displaystyle \sum _{i=1}^6\mathsf{\Gamma }_i=2\cos 2\omega +2\cos \omega +2,}$$

(32)

$$\begin{array}{cc}{\left.\chi \right|}_{\left[{\left[V^2\right]}^2\right]}\hfill & ={\displaystyle \sum _{i=1}^6\sum _{j=i}^6\mathsf{\Gamma }_i{\mathsf{\Gamma }}_j=2\cos 4\omega +2\cos 3\omega }\hfill \\ & +6\cos 2\omega +6\cos \omega +5,\hfill \end{array}$$

$$\begin{array}{cc}{\left.\chi \right|}_{\left[{\left[V^2\right]}^3\right]}\hfill & ={\displaystyle \sum _{i=1}^6{\displaystyle \sum _{j=i}^6{\displaystyle \sum _{k=j}^6\mathsf{\Gamma }_i\mathsf{\Gamma }_j\mathsf{\Gamma }_k}}=2\cos 6\omega +2\cos 5\omega }\hfill \\ & +6\cos 4\omega +8\cos 3\omega +14\cos 2\omega +14\cos \omega +10.\hfill \end{array}$$

The substitution of (20) and (32) into (22) leads to

$$r_1^{\left(2\right)}=0,r_0^{\left(2\right)}=r_2^{\left(2\right)}=1,$$

(33)

$$r_1^{\left(4\right)}=r_3^{\left(4\right)}=0,r_0^{\left(4\right)}=r_2^{\left(4\right)}=2,r_4^{\left(4\right)}=1,$$

$$r_1^{\left(6\right)}=r_5^{\left(6\right)}=0,r_0^{\left(6\right)}=r_2^{\left(6\right)}=3,r_4^{\left(6\right)}=2,r_3^{\left(6\right)}=r_6^{\left(6\right)}=1.$$

Combining (15) with (33), one obtains the direct sums of the irreducible parts on $X=[{[V^2]}^{\frac{h}{2}}](h=2,4,6)$.

$$[V^2]=D_0\oplus D_2,[{[V^2]}^2]=D_0^{\oplus 2}\oplus D_2^{\oplus 2}\oplus D_4,$$

(34)

$$[{[V^2]}^3]=D_0^{\oplus 3}\oplus D_2^{\oplus 3}\oplus D_3\oplus D_4^{\oplus 2}\oplus D_6.$$

However, in this paper, we are concerned with the direct sums of the irreducible parts of the elasticity tensors $C\in [{[V^2]}^2]$.

3. Stress Tensor, Strain Tensor, Elasticity Tensor, and Rotation Tensor Under Kelvin Notation

3.1. Representation of Stress Tensor and Strain Tensor Under Kelvin Notation

Use $\stackrel{⌢}{\sigma }$ and $\stackrel{⌢}{\epsilon }$ denote the stress tensor and the strain tensor of a reference crystallite ${\rm B}$, respectively.

$$\stackrel{⌢}{\sigma }=\left[\begin{array}{ccc}{\stackrel{⌢}{T}}_{11}& {\stackrel{⌢}{T}}_{12}& {\stackrel{⌢}{T}}_{13}\\ {\stackrel{⌢}{T}}_{12}& {\stackrel{⌢}{T}}_{22}& {\stackrel{⌢}{T}}_{23}\\ {\stackrel{⌢}{T}}_{13}& {\stackrel{⌢}{T}}_{23}& {\stackrel{⌢}{T}}_{33}\end{array}\right],\stackrel{⌢}{\epsilon }=\left[\begin{array}{ccc}{\stackrel{⌢}{E}}_{11}& {\stackrel{⌢}{E}}_{12}& {\stackrel{⌢}{E}}_{13}\\ {\stackrel{⌢}{E}}_{12}& {\stackrel{⌢}{E}}_{22}& {\stackrel{⌢}{E}}_{23}\\ {\stackrel{⌢}{E}}_{13}& {\stackrel{⌢}{E}}_{23}& {\stackrel{⌢}{E}}_{33}\end{array}\right]$$

(35)

The 2nd-order stress tensor $\stackrel{⌢}{\sigma }$ and the 2nd-order strain tensor $\stackrel{⌢}{\epsilon }$ on the crystallite ${\rm B}$ are rewritten as

$$\stackrel{⌢}{T}={\left[\begin{array}{cccccc}{\stackrel{⌢}{T}}_{11}& {\stackrel{⌢}{T}}_{22}& {\stackrel{⌢}{T}}_{33}& \sqrt{2}{\stackrel{⌢}{T}}_{23}& \sqrt{2}{\stackrel{⌢}{T}}_{13}& \sqrt{2}{\stackrel{⌢}{T}}_{12}\end{array}\right]}^T,$$

(36)

$$\stackrel{⌢}{E}={\left[\begin{array}{cccccc}{\stackrel{⌢}{E}}_{11}& {\stackrel{⌢}{E}}_{22}& {\stackrel{⌢}{E}}_{33}& \sqrt{2}{\stackrel{⌢}{E}}_{23}& \sqrt{2}{\stackrel{⌢}{E}}_{13}& \sqrt{2}{\stackrel{⌢}{E}}_{12}\end{array}\right]}^T$$

under the Kelvin notation, respectively, which makes the deformation energy density $u$ satisfies $u={\displaystyle \frac{1}{2}}{\stackrel{⌢}{T}}_{ij}{\stackrel{⌢}{E}}_{ij}={\displaystyle \frac{1}{2}}{\stackrel{⌢}{T}}^T\stackrel{⌢}{E}$ of ${\rm B}$ in (35) and (36).

3.2. Representation of Elasticity Tensor Under Kelvin Notation

The 4th-order elasticity tensor ${\stackrel{⌢}{C}}_{ijkl}$ constitutes the stress strain relation of the reference crystallite ${\rm B}$ as follows:

$${\stackrel{⌢}{T}}_{ij}={\stackrel{⌢}{C}}_{ijkl}{\stackrel{⌢}{E}}_{kl}.$$

(37)

The 4th-order elasticity tensor ${\stackrel{⌢}{C}}_{ijkl}$ is often expressed in the following matrix form

$$\left[\begin{array}{c}{\stackrel{⌢}{T}}_{11}\\ {\stackrel{⌢}{T}}_{22}\\ {\stackrel{⌢}{T}}_{33}\\ {\stackrel{⌢}{T}}_{23}\\ {\stackrel{⌢}{T}}_{13}\\ {\stackrel{⌢}{T}}_{12}\end{array}\right]=\left[\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& c_{14}& c_{15}& c_{16}\\ c_{12}& c_{22}& c_{23}& c_{24}& c_{25}& c_{26}\\ c_{13}& c_{23}& c_{33}& c_{34}& c_{35}& c_{36}\\ c_{14}& c_{24}& c_{34}& c_{44}& c_{45}& c_{46}\\ c_{15}& c_{25}& c_{35}& c_{45}& c_{55}& c_{56}\\ c_{16}& c_{26}& c_{36}& c_{46}& c_{56}& c_{66}\end{array}\right]\left[\begin{array}{c}{\stackrel{⌢}{E}}_{11}\\ {\stackrel{⌢}{E}}_{22}\\ {\stackrel{⌢}{E}}_{33}\\ 2{\stackrel{⌢}{E}}_{23}\\ 2{\stackrel{⌢}{E}}_{13}\\ 2{\stackrel{⌢}{E}}_{12}\end{array}\right]$$

(38)

under the Voigt notation, where $c_{IJ}={\stackrel{⌢}{C}}_{ijkl}$ (i.e., the indexes $ij$ and $kl$ in ${\stackrel{⌢}{C}}_{ijkl}$ are denoted by indexes $I$ and $J$ of $c_{IJ}$, respectively, with conventions $11\to 1$, $22\to 2$, $33\to 3$, $23\to 4$, $13\to 5$, $12\to 6$; for instance, ${\stackrel{⌢}{C}}_{1122}=c_{12},$ ${\stackrel{⌢}{C}}_{1123}=c_{14}$).

The Kelvin notation’s form of (37) or (38) is [28]

$$\stackrel{⌢}{T}=C\left(I\right)\stackrel{⌢}{E},$$

(39)

where $\stackrel{⌢}{T}$ and $\stackrel{⌢}{E}$ are given in (36), and

$$C\left(I\right)=\left[\begin{array}{cccccc}{c}_{11}& c_{12}& c_{13}& \sqrt{2}{c}_{14}& \sqrt{2}{c}_{15}& \sqrt{2}{c}_{16}\\ c_{12}& c_{22}& c_{23}& \sqrt{2}{c}_{24}& \sqrt{2}{c}_{25}& \sqrt{2}{c}_{26}\\ c_{13}& c_{23}& c_{33}& \sqrt{2}{c}_{34}& \sqrt{2}{c}_{35}& \sqrt{2}{c}_{36}\\ \sqrt{2}{c}_{14}& \sqrt{2}{c}_{24}& \sqrt{2}{c}_{34}& 2c_{44}& 2c_{45}& 2c_{46}\\ \sqrt{2}{c}_{15}& \sqrt{2}{c}_{25}& \sqrt{2}{c}_{35}& 2c_{45}& 2c_{55}& 2c_{56}\\ \sqrt{2}{c}_{16}& \sqrt{2}{c}_{26}& \sqrt{2}{c}_{36}& 2c_{46}& 2c_{56}& 2c_{66}\end{array}\right]$$

(40)

The matrix forms $C\left(I\right)$ in (40) can be taken as a 4th-order elasticity tensor under the Kelvin notation. The 6 linear equations on (39) are the same as those of (38).

3.3. Representation of 4th-Order Rotation Tensor Under Kelvin Notation

After the reference crystallite ${\rm B}$ with its acting force has a rotation $R$, ${\rm B}$ becomes ${{\rm B}}_R$, where $R$ is shown in (6) and (7). The stress tensor $\sigma$ and the strain tensor $\epsilon$ of ${{\rm B}}_R$ are given by

$$\sigma =\left[\begin{array}{ccc}{T}_{11}& T_{12}& T_{13}\\ T_{12}& T_{22}& T_{23}\\ T_{13}& T_{23}& T_{33}\end{array}\right]=R\stackrel{⌢}{\sigma }{R}^T,\epsilon =\left[\begin{array}{ccc}{E}_{11}& E_{12}& E_{13}\\ E_{12}& E_{22}& E_{23}\\ E_{13}& E_{23}& E_{33}\end{array}\right]=R\stackrel{⌢}{\epsilon }{R}^T.$$

(41)

$\sigma$ and $\epsilon$ of ${{\rm B}}_R$ can be denoted by the stress tensor $T$ and the strain tensor $E$

$$T={\left[\begin{array}{cccccc}{T}_{11}& T_{22}& T_{33}& \sqrt{2}{T}_{23}& \sqrt{2}{T}_{13}& \sqrt{2}{T}_{12}\end{array}\right]}^T,$$

(42)

$$E={\left[\begin{array}{cccccc}{E}_{11}& E_{22}& E_{33}& \sqrt{2}{E}_{23}& \sqrt{2}{E}_{13}& \sqrt{2}{E}_{12}\end{array}\right]}^T$$

under the Kelvin notation.

Because the components of the tensors $\sigma$ and $\stackrel{⌢}{\sigma }$ are symmetric ($T_{ij}=T_{ji}$ and ${\stackrel{⌢}{T}}_{ij}={\stackrel{⌢}{T}}_{ji}$), the relation $\sigma =R\stackrel{⌢}{\sigma }{R}^T$ (or $\epsilon =R\stackrel{⌢}{\epsilon }{R}^T$) constitutes the 6 linear equations between $T_{ij}$ and ${\stackrel{⌢}{T}}_{ij}$. The 6 linear equations can be written in the following form [19].

$$T=Q\stackrel{⌢}{T}\mathrm{or}E=Q\stackrel{⌢}{E},$$

(43)

where $Q$ can be taken as a 4th-order rotation tensor under the Kelvin notation and $Q$ can be described by the Euler angles $\left(\psi ,\theta ,\varphi \right)$ as follows:

$$Q(\psi ,\theta ,\varphi )=Q\left(e_3,\psi \right)Q\left(e_2,\theta \right)Q\left(e_3,\varphi \right),I\stackrel{\mathrm{def}}{=}Q(0,0,0),$$

(44)

$$Q\left(e_2,\theta \right)=\left[\begin{array}{cccccc}{\cos }^2\theta & 0& {\sin }^2\theta & 0& {\displaystyle \frac{1}{\sqrt{2}}}\sin 2\theta & 0\\ 0& 1& 0& 0& 0& 0\\ {\sin }^2\theta & 0& {\cos }^2\theta & 0& {\displaystyle \frac{-1}{\sqrt{2}}}\sin 2\theta & 0\\ 0& 0& 0& \cos \theta & 0& -\sin \theta \\ {\displaystyle \frac{-1}{\sqrt{2}}}\sin 2\theta & 0& {\displaystyle \frac{1}{\sqrt{2}}}\sin 2\theta & 0& \cos 2\theta & 0\\ 0& 0& 0& \sin \theta & 0& \cos \theta \end{array}\right],$$

(45)

$$Q\left(e_3,\beta \right)=\left[\begin{array}{cccccc}{\cos }^2\beta & {\sin }^2\beta & 0& 0& 0& {\displaystyle \frac{-1}{\sqrt{2}}}\sin 2\beta \\ {\sin }^2\beta & {\cos }^2\beta & 0& 0& 0& {\displaystyle \frac{1}{\sqrt{2}}}\sin 2\beta \\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& \cos \beta & \sin \beta & 0\\ 0& 0& 0& -\sin \beta & \cos \beta & 0\\ {\displaystyle \frac{1}{\sqrt{2}}}\sin 2\beta & {\displaystyle \frac{-1}{\sqrt{2}}}\sin 2\beta & 0& 0& 0& \cos 2\beta \end{array}\right],$$

(46)

where $\beta =\psi \mathrm{o}\mathrm{r}\varphi$. It is easy to show that $Q$ satisfies

$$Q^T=Q^{-1}$$

(47)

Putting (43) and (47) into (39), we obtain the stress strain relation of $B_Q$ (or $B_R$) under the Kelvin notation

$$T=C\left(Q\right)E,C\left(Q\right)=QC\left(I\right)Q^T,$$

(48)

where $C\left(I\right)$ is given in (40).

The rotation relations in (43), (48), and (44) show that $T$, $E$, $C(Q)$, and $Q$ can be taken as the 2nd-order stress tensor, the 2nd-order strain tensor, the 4th-order elasticity tensor, and the 4th-order rotation tensor, respectively, under the Kelvin notation. Although the stress–strain relation (38) under the Voigt notation is often used in engineering mechanics, the elasticity tensor (40) under the Kelvin notation has some advantages because the rotation relations of (43) and (48) exist. Under the Kelvin notation, it is easier to complete the computation of $C\left(Q\right)=QC\left(I\right)Q^T$ by matrix multiplications.

The stress–strain constitutive relationship C_ijkl belongs to a fourth-order tensor. According to the principle of objectivity, after material rotation R_ij, the material’s constitutive relationship becomes R_imR_jnR_kpR_lqC_mnpq. Considering that C_ijkl has minor symmetry, the fourth-order tensor can be represented by a 6 × 6 matrix using Voigt notations or Kelvin notations, which is convenient for engineering operations. Kelvin notations and Voigt notations have significant differences. Voigt notations are merely a convention in engineering, but the 6 × 6 matrix $C$ under Voigt notations is not a tensor. The 6 × 6 matrix $C$ under Kelvin notations is a tensor with relation (48).

4. Representation of Elasticity Tensor and Its Irreducible Basis Tensors

The dot product of two 4th-order complex tensors $A$ and $B$ under the Kelvin notation is defined by

$$〈A,B〉=A_{IJ}{B}_{IJ}^{*},\left(1\le I,J\le 6\right)$$

(49)

The elasticity tensor $C\left(I\right)\in D_0^{\oplus 2}\oplus D_2^{\oplus 2}{D}_4$ in (40) with (15) and (34) can be decomposed into [23,28].

$$\begin{array}{cc}C\left(I\right)\hfill & ={\displaystyle \sum _{l=0,2,4}{\displaystyle \sum _{p=1}^{r_l^{\left(4\right)}}{\displaystyle \sum _{m=-l}^l{a}_m^{l,p}{H}_m^{l,p}}}}\hfill \\ & ={\displaystyle \sum _{p=1}^2{a}_0^{0,p}{H}_0^{0,P}}+{\displaystyle \sum _{P=1}^2{\displaystyle \sum _{m=-2}^2{a}_m^{2,p}{H}_m^{l,p}}}+{\displaystyle \sum _{m=-4}^4{a}_m^{4,1}{H}_m^{4,1}}\hfill \end{array}$$

(50)

where $H_m^{l,p}$ is a set of irreducible basis tensors with the properties

$$H_m^{l,p}\in D_l,a_m^{l,p}=〈C\left(I\right),H_m^{l,p}〉,$$

(51)

$$H_m^{l,p}={\left(-1\right)}^m{\left(H_{\overline{m}}^{l,p}\right)}^{*},a_m^{l,p}={\left(-1\right)}^m{\left(a_{\overline{m}}^{l,p}\right)}^{*},$$

and the irreducible basis tensors $H_m^{l,p}$ of $C\left(I\right)$ in (40) can be obtained by the following integrations

$$B_m^l={\displaystyle {\int }_{\mathrm{SO}(3)}QC}(I)Q^T{D}_{m0}^l(Q)dg(Q)={\displaystyle \sum _{p=1}^{r_l^{(h)}}{c}_p}{H}_m^{l,p}.$$

(52)

with the requirement

$$〈H_m^{l,p},H_{m^{\prime }}^{l^{\prime },p^{\prime }}〉={\delta }_{l{l}^{\prime }}{\delta }_{m{m}^{\prime }}{\delta }_{p{p}^{\prime }}.$$

(53)

Because ${{\rm B}}_m^l$ in (52) have the following properties

$$\begin{array}{l}Q{B}_m^l{Q}^T=Q\left({\displaystyle {\int }_{\mathrm{SO}(3)}{Q}_1}C{\left(Q_1\right)}^T{D}_{m0}^l\left(Q_1\right)dg\right)Q^T\\ \stackrel{Q{Q}_1=Q_2}{=}{\displaystyle {\int }_{\mathrm{SO}(3)}{Q}_2}C{Q}_2^T{D}_{m0}^l\left(Q^T{Q}_2\right)dg\\ ={\displaystyle \sum _{k=-l}^l{D}_{mk}^l}\left(Q^T\right){\displaystyle {\int }_{\mathrm{SO}(3)}{Q}_2}C{Q}_2^T{D}_{k0}^l\left(Q_2\right)dg\\ ={\displaystyle \sum _{k=-l}^l{D}_{mk}^l}\left(Q^T\right)B_k^l\end{array}$$

(54)

by (11), the irreducible basis tensors $H_m^{l,p}$ of ${{\rm B}}_m^l$ in (52) satisfy

$$Q{H}_m^{l,p}{Q}^T={\displaystyle \sum _{k=-l}^l{D}_{mk}^l}\left(Q^T\right)H_k^{l,p}$$

(55)

By (53), (15), and (33), Man and Huang [28] completed the integrations for (0, 2, 4) in (52) to give some of the irreducible basis tensors $H_m^{l,p}$ on $C\left(I\right)\in D_0^{\oplus 2}\oplus D_2^{\oplus 2}{D}_4$ in (40) under the Kelvin notation

$$H_0^{0,1}={\displaystyle \frac{1}{\sqrt{30}}}\left(\begin{array}{cccccc}-1& -2& -2& 0& 0& 0\\ -2& -1& -2& 0& 0& 0\\ -2& -2& -1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)$$

(56)

$$H_0^{0,2}={\displaystyle \frac{1}{\sqrt{6}}}\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)$$

$$H_0^{2,1}={\displaystyle \frac{\sqrt{14}}{42}}\left(\begin{array}{cccccc}2& -4& 2& 0& 0& 0\\ -4& 2& 2& 0& 0& 0\\ 2& 2& -4& 0& 0& 0\\ 0& 0& 0& -3& 0& 0\\ 0& 0& 0& 0& -3& 0\\ 0& 0& 0& 0& 0& 6\end{array}\right)$$

(57)

$$H_2^{2,1}={\displaystyle \frac{\sqrt{21}}{42}}\left(\begin{array}{cccccc}-2& 0& 2& 0& 0& \sqrt{2}i\\ 0& 2& -2& 0& 0& \sqrt{2}i\\ 2& -2& 0& 0& 0& -2\sqrt{2}i\\ 0& 0& 0& 3& 3i& 0\\ 0& 0& 0& 3i& -3& 0\\ \sqrt{2}i& \sqrt{2}i& -2\sqrt{2}i& 0& 0& 0\end{array}\right)$$

$$H_0^{2,2}={\displaystyle \frac{1}{6}}\left(\begin{array}{cccccc}2& 2& -1& 0& 0& 0\\ 2& 2& -1& 0& 0& 0\\ -1& -1& -4& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$

(58)

$$H_2^{2,2}={\displaystyle \frac{\sqrt{6}}{12}}\left(\begin{array}{cccccc}-2& 0& -1& 0& 0& \sqrt{2}i\\ 0& 2& 1& 0& 0& \sqrt{2}i\\ -1& 1& 0& 0& 0& \sqrt{2}i\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ \sqrt{2}i& \sqrt{2}i& \sqrt{2}i& 0& 0& 0\end{array}\right)$$

$$H_0^{4,1}={\displaystyle \frac{\sqrt{70}}{140}}\left(\begin{array}{cccccc}3& 1& -4& 0& 0& 0\\ 1& 3& -4& 0& 0& 0\\ -4& -4& 8& 0& 0& 0\\ 0& 0& 0& -8& 0& 0\\ 0& 0& 0& 0& -8& 0\\ 0& 0& 0& 0& 0& 2\end{array}\right)$$

(59)

$$H_2^{4,1}={\displaystyle \frac{\sqrt{7}}{28}}\left(\begin{array}{cccccc}-2& 0& 2& 0& 0& \sqrt{2}i\\ 0& 2& -2& 0& 0& \sqrt{2}i\\ 2& -2& 0& 0& 0& -2\sqrt{2}i\\ 0& 0& 0& -4& -4i& 0\\ 0& 0& 0& -4i& 4& 0\\ \sqrt{2}i& \sqrt{2}i& -2\sqrt{2}i& 0& 0& 0\end{array}\right)$$

$$H_4^{4,1}={\displaystyle \frac{1}{4}}\left(\begin{array}{cccccc}1& -1& 0& 0& 0& -\sqrt{2}i\\ -1& 1& 0& 0& 0& \sqrt{2}i\\ 0& 0& 0& 0& 0& -2\sqrt{2}i\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ -\sqrt{2}i& \sqrt{2}i& 0& 0& 0& -2\end{array}\right)$$

where $i=\sqrt{-1}$.

5. Elastic Constants and Mesostructure Tensors on Elasticity Tensor of Cubic Crystals

5.1. Decomposition of Elasticity Tensor on Elastic Constants and Mesostructure Tensors of Crystals

From (49) and (50), we know the elastic constants (${\kappa }_{l,p}^c,{\varsigma }_{l,p}^c$) of the stiffness tensor $C^{\mathrm{c}}(I)$ and the flexibility tensor $S^{\mathrm{c}}(I)={(C^{\mathrm{c}})}^{-1}$ respectively on the reference crystal ${\rm B}$ as follows:

$${\kappa }_{l,p}^{\mathrm{c}}=〈C^{\mathrm{c}}(I),H_0^{l,p}〉,{\varsigma }_{l,p}^{\mathrm{c}}=〈S^{\mathrm{c}}(I),H_0^{l,p}〉,$$

(60)

where ${\kappa }_{l,p}^c$ is the elastic constant of $C^{\mathrm{c}}(I)$, and ${\varsigma }_{l,p}^c$ is the flexibility constant of $S^{\mathrm{c}}(I)$ on the reference crystallite ${\rm B}$. Let

$$N=\{\left.(l,p)\right|{\kappa }_{l,p}^c\ne 0 \mathrm{o}\mathrm{r} {\varsigma }_{l,p}^c\ne 0\}.$$

(61)

Define

$$c_m^{l,p}(I)\stackrel{def}{=}{\displaystyle \frac{1}{{\kappa }_{l,p}^c}}〈C^{\mathrm{c}}(I),H_m^{l,p}〉\stackrel{or}{=}{\displaystyle \frac{1}{{\varsigma }_{l,p}^c}}〈S^{\mathrm{c}}(I),H_m^{l,p}〉\mathrm{for}(l,p)\in N$$

(62)

to be the mesostructure coefficients of the reference crystallite ${\rm B}$. Obviously, there are

$$c_0^{l,p}(I)=1, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} (l,p)\in N.$$

(63)

From $C(I)\in D_0^{\oplus 2}\oplus D_2^{\oplus 2}\oplus D_4^{\oplus 2}$ in (40), the stiffness tensor and the flexibility tensor of the reference crystallite B can be expressed as

$$C^{\mathrm{c}}(I)={\displaystyle \sum _{(l,p)\in N}{\kappa }_{l,p}^c{\mathbf{H}}^{l,p}}(I),S^{\mathrm{c}}(I)={\displaystyle \sum _{(l,p)\in N}{\varsigma }_{l,p}^c{H}^{l,p}(I)}$$

(64)

basis tensors $H_m^{l,p}$, where

$$H^{l,p}(I)={\displaystyle \sum _{m=-l}^l{c}_m^{l,p}\left(I\right)H_m^{l,p}}\mathrm{for}(l,p)\in N$$

(65)

are the mesostructure tensors of the reference crystallite ${\rm B}$.

After the reference cubic crystallite ${\rm B}$, with its acting force, has a rotation $\mathbf{Q}$, ${\rm B}$ becomes $B_{\mathbf{Q}}$. The stiffness tensor $C^{\mathrm{c}}(Q)$ of $B_{\mathbf{Q}}$ is

$$C^{\mathrm{c}}(Q)={\displaystyle \sum _{(l,p)\in N}}{\kappa }_{l,p}^c{H}^{l,p}(Q),$$

(66)

where the mesostructure tensors $H^{l,p}\left(Q\right)$ of $B_{\mathbf{Q}}$ is

$$H^{l,p}(Q)=Q{H}^{l,p}(I)Q^T={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}}(I){\displaystyle \sum _{m=-l}^l{D}_{km}^l}\left(Q^T\right)H_m^{l,p}={\displaystyle \sum _{m=-l}^l{c}_m^{l,p}}(Q)H_m^{l,p}$$

(67)

by (55) and (65), and

$$c_m^{l,p}(Q)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}(I)D_{km}^l(Q^T)}$$

(68)

is the mesostructure coefficient of $B_{\mathbf{Q}}$. The above relation (68) is obtained by

$$\begin{array}{ll}{c}_m^{l,p}\left(Q_1{Q}_2\right)\hfill & ={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}}(I)D_{km}^l\left(Q_2^T{Q}_1^T\right)={\displaystyle \sum _{s=-l}^l{\displaystyle \sum _{k=-l}^l{c}_k^{l,p}}}(I)D_{ks}^l\left(Q_2^T\right)D_{sm}^l\left(Q_1^T\right)\hfill \\ & ={\displaystyle \sum _{k=-l}^l{D}_{km}^l}\left(Q_1^T\right){\displaystyle \sum _{s=-l}^l{c}_s^{l,p}}(I)D_{sk}^l\left(Q_2^T\right)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}}\left(Q_2\right)D_{km}^l\left(Q_1^T\right)\hfill \end{array}$$

(69)

because of (11).

5.2. Elastic Constants and Mesostructure Tensors on Elasticity Tensor of Cubic Crystals

Let the three four-fold symmetric axes of the reference cubic crystal coincide with the fixed coordinate axes. The elasticity stiffness tensor $C^{\mathrm{c}}(I)$ and the elasticity flexibility tensor $S^{\mathrm{c}}(I)$ of the reference cubic crystal ${\rm B}$ have the following form [10]

$$C^c(I)=\left(\begin{array}{cccccc}{c}_{11}& c_{12}& c_{12}& 0& 0& 0\\ c_{12}& c_{11}& c_{12}& 0& 0& 0\\ c_{12}& c_{12}& c_{11}& 0& 0& 0\\ 0& 0& 0& 2c_{44}& 0& 0\\ 0& 0& 0& 0& 2c_{44}& 0\\ 0& 0& 0& 0& 0& 2c_{44}\end{array}\right),S^{\mathrm{c}}(I)={\left(C^{\mathrm{c}}(I)\right)}^{-1}$$

(70)

under the Kelvin notation. Putting (70) into (60) and (62), we have the elastic constants ${\kappa }_{l,p}^c$ and the mesostructure coefficients $c_m^{l,p}(I)$ of the elastic stiffness tensor on ${\rm B}$.

$$N=\{(0,1),(0,2),(4,1)\},$$

(71)

$$\begin{array}{l}{\kappa }_{0,1}^c=-{\displaystyle \frac{\sqrt{30}}{10}}(c_{11}+4c_{12}-2c_{44}),\\ {\kappa }_{0,2}^c={\displaystyle \frac{\sqrt{6}}{2}}(c_{11}+2c_{44}),\\ {\kappa }_{4,1}^c={\displaystyle \frac{\sqrt{70}}{10}}(c_{11}-c_{12}-2c_{44})\end{array}$$

and

$$c_0^{0,1}(I)=c_0^{0,2}(I)\stackrel{\mathrm{def}}{=}{c}_0^0(I)=c_0^0=1,$$

(72)

$$c_0^{4,1}(I)=1\stackrel{\mathrm{def}}{=}{c}_0^4(I)=c_0^4,$$

$$c_4^{4,1}(I)=c_{\overline{4}}^{4,1}(I)\stackrel{\mathrm{def}}{=}{c}_4^4(I)=c_4^4={\displaystyle \frac{1}{14}}\sqrt{70},$$

all others $c_m^{l,p}(I)=0$, which shows that

$$C^{\mathrm{c}}(I)\in D_0^{\oplus 2}\oplus D_4$$

(73)

and that the stiffness tensor of ${\rm B}$ can be expressed as

$$C^{\mathrm{c}}(I)={\kappa }_{0,1}^c{c}_0^0{H}_0^{0,1}+{\kappa }_{0,2}^c{c}_0^0{H}_0^{0,2}+{\kappa }_{4,1}^c{\mathbf{H}}^{4,1}(I)$$

(74)

by (64) and (56), where $H_0^{0,1}$ and $H_0^{0,2}$ are isotropic tensors. From (71), (72), and (65), we obtain the mesostructure tensor $H^{4,1}(I)$ of ${\rm B}$.

$$H^{4,1}(I)=c_0^4{H}_0^{4,1}+c_4^4(H_4^{4,1}+H_{\overline{4}}^{4,1}).$$

(75)

After the cubic crystal ${\rm B}$ with its acting force has a rotation $Q$, ${\rm B}$ becomes $B_{\mathbf{Q}}$. The stiffness tensor $C^{\mathrm{c}}(Q)\in D_0^{\oplus 2}\oplus D_4$ of $B_{\mathbf{Q}}$ is

$$C^{\mathrm{c}}(Q)=Q{C}^{\mathrm{c}}(I)Q^T={\kappa }_{0,1}^c{H}_0^{0,1}+{\kappa }_{0,2}^c{H}_0^{0,2}+{\kappa }_{4,1}^c{\mathbf{H}}^{4,1}(Q),$$

(76)

where

$$H^{4,1}(Q)=Q{H}^{4,1}(I)Q^T={\displaystyle \sum _{m=-4}^4\left({\displaystyle \sum _{k=-l}^l{c}_k^l}{D}_{km}^l\left(Q^T\right)\right)}{H}_m^{4,1}={\displaystyle \sum _{m=-4}^4{c}_m^4}(Q)H_m^{4,1},$$

(77)

$$c_m^4(Q)=D_{0m}^4(Q^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{4m}^4(Q^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{\overline{4}m}^4(Q^T)$$

by combining (72) with (68) and (67).

By the relation between $S^{\mathrm{c}}(I)$ and $C^{\mathrm{c}}(I)$, the flexibility tensor $S^{\mathrm{c}}(I)$ of the reference cubic crystal ${\rm B}$ can be expressed as

$$S^{\mathrm{c}}(I)={(C^{\mathrm{c}}(I))}^{-1}=\left(\begin{array}{cccccc}{s}_{11}& s_{12}& s_{12}& 0& 0& 0\\ s_{12}& s_{11}& s_{12}& 0& 0& 0\\ s_{12}& s_{12}& s_{11}& 0& 0& 0\\ 0& 0& 0& 2s_{44}& 0& 0\\ 0& 0& 0& 0& 2s_{44}& 0\\ 0& 0& 0& 0& 0& 2s_{44}\end{array}\right),$$

(78)

where

$$s_{11}={\displaystyle \frac{c_{11}+c_{12}}{c_{11}^2+c_{11}{c}_{12}-2c_{12}^2}},s_{12}=-{\displaystyle \frac{c_{12}}{c_{11}^2+c_{11}{c}_{12}-2c_{12}^2}},s_{44}={\displaystyle \frac{1}{4c_{44}}}.$$

(79)

Similar to (76) and (71), the elastic constants and the mesostructure tensors of the elastic flexibility tensor $S^{\mathrm{c}}(Q)\in D_0^{\oplus 2}\oplus D_4$ for the cubic crystal $B_{\mathbf{Q}}$ can be expressed as

$$S^{\mathrm{c}}(Q)=Q{S}^{\mathrm{c}}(I)Q^T={\varsigma }_{0,1}^c{H}_0^{0,1}+{\varsigma }_{0,2}^c{H}_0^{0,2}+{\varsigma }_{4,1}^c{H}_0^{4,1}(Q),$$

(80)

where the mesostructure tensor $H^{4,1}(Q)$ of $B_{\mathbf{Q}}$ is given in (77) and the elastic constants ${\varsigma }_{l,p}^c$ of the flexibility tensor on $B_{\mathbf{Q}}$ (or ${\rm B}$) are given by

$${\varsigma }_{0,1}^c=-{\displaystyle \frac{\sqrt{30}}{10}}(s_{11}+4s_{12}-2s_{44}),$$

(81)

$$\begin{array}{l}{\varsigma }_{0,2}^c=-{\displaystyle \frac{\sqrt{6}}{2}}(s_{11}+4s_{12}),\\ {\varsigma }_{4,1}^c={\displaystyle \frac{\sqrt{70}}{10}}(s_{11}-s_{12}-2s_{44}).\end{array}$$

6. Elastic Constants and Mesostructures on an Orthorhombic Set ${\mathsf{\Omega }}_s$ of Four Cubic Crystallites’ Orientations

To describe the orientation of a crystallite, we pick a reference crystallite ${\rm B}$. The crystalline orientation of any point $x\in \mathsf{\Omega }$ can be described by the crystallite orientation function $Q(x)$ [1].

$$Q(x)={\displaystyle \sum _{s=1}^N{\displaystyle \sum _{q=1}^4{Q}_{s,q}}}{\chi }_{s,q}(x) \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}.\mathrm{e}. \mathbf{x}\in \mathsf{\Omega }$$

(82)

$${\chi }_{s,q}(x)=1 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \mathbf{x}\in {\mathsf{\Omega }}_{s,q} \mathrm{a}\mathrm{n}\mathrm{d} {\chi }_{s,q}(x)=0 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \mathbf{x}\notin {\mathsf{\Omega }}_{s,q},$$

where ${\chi }_{s,q}(x)$ is the characteristic function of ${\mathsf{\Omega }}_{s,q}$. ${\mathsf{\Omega }}_{s,q}$ is $s=1,2,\cdot \cdot \cdot ,N$ set of crystallites with orientation ${\mathbf{Q}}_{s,q}$ with respect to the reference crystallite ${\rm B}$.

Let ${\mathsf{\Omega }}_s={\cup }_{q=1}^4{\mathsf{\Omega }}_{s,q}$ be an orthorhombic set of four crystalline orientations ${\mathsf{\Omega }}_{s,q}$ ($q=1,2,3,4$) with $\left|{\mathsf{\Omega }}_{s,q}\right|=\left|{\mathsf{\Omega }}_s\right|/4$ and $\left|{\mathsf{\Omega }}_s\right|$ being the volume of ${\mathsf{\Omega }}_s$. The metal plate $\mathsf{\Omega }={\cup }_{s=1}^N{\mathsf{\Omega }}_s$ is an aggregate of all orthorhombic sets ${\mathsf{\Omega }}_s$ (s = 1, 2, ···, N).

$${\mathsf{\Omega }}_s={\cup }_{q=1}^4{\mathsf{\Omega }}_{s,q},\left|{\mathsf{\Omega }}_{s,q}\right|={\displaystyle \frac{1}{4}}\left|{\mathsf{\Omega }}_s\right|$$

(83)

and

$$Q_{s,1}=Q\left(e_1,0\right)Q_s,Q_{s,2}=Q\left(e_1,\pi \right)Q_s,$$

(84)

$$Q_{s,3}=Q\left(e_2,\pi \right)Q_s,Q_{s,4}=Q\left(e_3,\pi \right)Q_s,$$

where $\left|{\mathsf{\Omega }}_s\right|$ denotes the volume of ${\mathsf{\Omega }}_s$, and $Q_s=Q\left(e_3,{\psi }_s\right)Q\left(e_2,{\theta }_s\right)Q\left(e_3,{\varphi }_s\right)$ is given in (44).

The volume average of the elastic stiffness tensor $C^{\mathrm{c}}\left(Q_s\right)$ in (66) on the orthorhombic set ${\mathsf{\Omega }}_s$ of the four crystalline orientations is

$${\tilde{C}}^{\mathrm{c}}\left(Q_s\right)={\displaystyle \frac{1}{\left|{\mathsf{\Omega }}_s\right|}}{\displaystyle \sum _{q=1}^4\left|{\mathsf{\Omega }}_{s,q}\right|}{C}^{\mathrm{c}}\left(Q_{s,q}\right)={\displaystyle \frac{1}{4}}{\displaystyle \sum _{q=1}^4{C}^{\mathrm{c}}}\left(Q_{s,q}\right)={\displaystyle \sum _{(l,p)\in \mathcal{N}}{\kappa }_{l,p}^c}{\tilde{H}}^{l,p}\left(Q_s\right),$$

(85)

where $N=\{(0,1),(0,2),(4,1)\}$ in (71), ${\tilde{H}}^{l,p}(Q_s)$ are called the mesostructure tensors of ${\mathsf{\Omega }}_s$.

$${\tilde{H}}^{l,p}(Q_{\mathrm{s}})={\displaystyle \sum _{m=-l}^l{\tilde{c}}_m^{l,p}(Q_s)}{H}_m^{l,p},{\tilde{c}}_m^{l,p}(Q_s)={\displaystyle \frac{1}{4}}{\displaystyle \sum _{q=1}^4{c}_m^{l,p}(Q_{s,q})}$$

(86)

because of (83). Since there are

$$D_{km}^l(Q(e_1,0))={\delta }_{km},\left.D_{km}^l\right|(Q(e_1,\pi ))=d_{km}^l(\pi )e^{-im\pi }$$

(87)

$$=(-1{)}^{l+k}{d}_{k\overline{m}}^l(0)e^{-im\pi }=(-1{)}^{l+k+m}{\delta }_{k\overline{m}}^l,$$

$$D_{km}^l(Q(e_2,0))=d_{km}^l(\pi )={(-1)}^{l+k}{d}_{km}^l(0)={(-1)}^{l+k}{\delta }_{k\overline{m}},$$

$$D_{km}^l(Q(e_3,0))=d_{km}^l(0)e^{-im\pi }={(-1)}^m{\delta }_{km}$$

in (9), we have

$$c_m^{l,p}(Q(e_1,0)Q_s)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}(Q_s){\delta }_{km}=c_m^{l,p}}(Q_s),$$

(88)

$$c_m^{l,p}(Q(e_1,\pi )Q_s)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}(Q_s)(-1{)}^{l+k+m}{\delta }_{k\overline{m}}=c_{\overline{m}}^{l,p}}(Q_s){(-1)}^l,$$

$$c_m^{l,p}(Q(e_2,\pi )Q_s)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}(Q_s)(-1{)}^{l+k}{\delta }_{k\overline{m}}=c_{\overline{m}}^{l,p}}(Q_s){(-1)}^{l+m},$$

$$c_m^{l,p}(Q(e_3,\pi )Q_s)={\displaystyle \sum _{k=-l}^l{c}_k^{l,p}(Q_s)e^{-ik\pi }=c_m^{l,p}}(Q_s){(-1)}^m$$

by (69). Since there is $l$$\in$ even in (85), we obtain the mesostructure coefficients ${\tilde{c}}_m^{l,p}(Q_s)$ of ${\mathsf{\Omega }}_s$.

$${\tilde{c}}_m^{l,p}(Q_s)={\displaystyle \frac{1}{4}}[c_m^{l,p}(Q_s)+c_m^{l,p}(Q_s){(-1)}^m+c_{\overline{m}}^{l,p}(Q_s)+c_{\overline{m}}^{l,p}(Q_s){(-1)}^m]$$

(89)

or

$${\tilde{c}}_m^{l,p}(Q_s)={\tilde{c}}_{\overline{m}}^{l,p}(Q_s)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}[c_m^{l,p}(Q_s)+c_{\overline{m}}^{l,p}(Q_s)],m\in \mathrm{even}\\ 0,m\in \mathrm{odd}\end{array}\right..$$

(90)

For cubic crystallites, the volume average stiffness tensor ${\tilde{C}}^{\mathrm{c}}\left(Q_s\right)$ and the volume average flexibility tensor ${\tilde{S}}^{\mathrm{c}}\left(Q_s\right)$ on an orthorhombic set ${\mathsf{\Omega }}_s$ of four crystallites’ orientations have the following form, respectively

$${\tilde{C}}^c(Q_s)={\kappa }_{0,1}^c{H}^{0,1}+{\kappa }_{0,2}^c{H}^{0,2}+{\kappa }_{4,1}^c{\tilde{H}}^{4,1}(Q_s),$$

(91)

$${\tilde{S}}^{\mathrm{c}}(Q_s)={\varsigma }_{0,1}^c{H}^{0,1}+{\varsigma }_{0,2}^c{H}^{0,2}+{\varsigma }_{4,1}^c{\tilde{H}}^{4,1}(Q_s),$$

where ${\kappa }_{l,p}^c$ and ${\varsigma }_{l,p}^c$ are given in (71) and (81). The mesostructure tensor ${\tilde{H}}^{4,1}\left({\mathbf{Q}}_s\right)$ of ${\mathsf{\Omega }}_s$ is

$${\tilde{H}}^{4,1}(Q_s)={\displaystyle \sum _{m=-4}^4{\tilde{c}}_m^{4,1}(Q_s)}{H}_m^{4,1}$$

(92)

$$={\tilde{c}}_0^{4,1}({\mathbf{Q}}_s)H_0^{4,1}+2{\tilde{c}}_2^{4,1}(Q_s)\mathrm{Re}{H}_2^{4,1}+2{\tilde{c}}_2^{4,1}(Q_s)\mathrm{Re}{H}_4^{4,1}$$

by $H_m^{l,p}={(-1)}^m{(H_{\overline{m}}^{l,p})}^{*}$, where the mesostructure coefficients ${\tilde{c}}_m^4(Q_s)$ of ${\mathsf{\Omega }}_s$ in (77) and (90) are

$${\tilde{c}}_m^4(Q_s)={\displaystyle \frac{1}{2}}[D_{0m}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{4m}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{\overline{4}m}^4(Q_s^T)]$$

(93)

$$\begin{array}{l}+{\displaystyle \frac{1}{2}}[D_{0\overline{m}}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{4\overline{m}}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}{D}_{\overline{4}\overline{m}}^4(Q_s^T)]\\ =\mathrm{Re}{D}_{0m}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}\mathrm{Re}{D}_{4m}^4(Q_s^T)+{\displaystyle \frac{\sqrt{70}}{14}}\mathrm{Re}{D}_{\overline{4}m}^4(Q_s^T)\end{array}$$

by $D_{\overline{m}\overline{n}}^l={(-1)}^{m+n}{(D_{mn}^l)}^{*}$ and $m\in \mathrm{even}.$

Because there is

$$Q_s^T=Q(e_3,-{\varphi }_s)Q(e_2,-{\theta }_s)Q(e_3,-{\psi }_s)$$

(94)

in (44), we have

$$\mathrm{Re}{D}_{mn}^l(Q_s^T)=\mathrm{Re}{D}_{mn}^l(R(-{\varphi }_s,-{\theta }_s,-{\psi }_s))$$

(95)

$$=d_{mn}^l(-{\theta }_s)\mathrm{Re}{e}^{i(m{\varphi }_s+n{\psi }_s)}={(-1)}^{m+n}{d}_{mn}^l({\theta }_s)\cos (m{\varphi }_s+n{\psi }_s)$$

$$=d_{mn}^l({\theta }_s)\cos (m{\varphi }_s+n{\psi }_s);\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} m,n\in \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},$$

where

$$d_{\overline{4}0}^4({\theta }_s)=d_{40}^4({\theta }_s)=d_{04}^4({\theta }_s)={\displaystyle \frac{\sqrt{70}}{128}}(\cos 4{\theta }_s-4\cos 2{\theta }_s+3),$$

(96)

$$\begin{array}{l}{d}_{42}^4({\theta }_s)={\displaystyle \frac{\sqrt{7}}{64}}(-\cos 4{\theta }_s-4\cos 3{\theta }_s-4\cos 2{\theta }_s+4\cos {\theta }_s+5),\\ d_{\overline{4}2}^4({\theta }_s)={\displaystyle \frac{\sqrt{7}}{64}}(-\cos 4{\theta }_s+4\cos 3{\theta }_s-4\cos 2{\theta }_s-4\cos {\theta }_s+5),\\ d_{44}^4({\theta }_s)={\displaystyle \frac{1}{128}}(\cos 4{\theta }_s+8\cos 3{\theta }_s+28\cos 2{\theta }_s+56\cos {\theta }_s+35),\\ d_{\overline{4}4}^4({\theta }_s)={\displaystyle \frac{1}{128}}(\cos 4{\theta }_s-8\cos 3{\theta }_s+28\cos 2{\theta }_s-56\cos {\theta }_s+35)\end{array}$$

by (10). From (95), we have the mesostructure coefficients of the orthorhombic set ${\mathsf{\Omega }}_s={\cup }_{q=1}^4{\mathsf{\Omega }}_{s,q}{\mathsf{\Omega }}_s$.

$${\tilde{c}}_m^4(Q_s)=d_{0m}^4({\theta }_s)\cos m{\psi }_s+{\displaystyle \frac{\sqrt{70}}{14}}{d}_{4m}^4({\theta }_s)\cos (4{\varphi }_s+m{\psi }_s)$$

(97)

$$+{\displaystyle \frac{\sqrt{70}}{14}}{d}_{\overline{4}m}^4({\theta }_s)\cos (-4{\varphi }_s+m{\psi }_s),$$

where $d_{mn}^4(\theta )$ are given in (96).

7. Upper Bound and Lower Bound of Elasticity Tensor on Metal Plate of Cubic Crystallites

The bounds for the effective stiffness tensor $C^{\mathrm{eff}}$ are expressed in terms of inequalities between tensors under the Kelvin notation:

$$E^T{C}^{\mathrm{lb}}E\le E^T{C}^{\mathrm{eff}}E\le E^T{C}^{\mathrm{ub}}E;\forall E\in {\mathrm{R}}^6,$$

(98)

where the forms of $E$ and $C$ are given in (42) and (40). Then $C^{\mathrm{ub}}$ and $C^{\mathrm{lb}}$ are called the upper bound and the lower bound of $C^{\mathrm{eff}}$, respectively. The following notation

$$C^{\mathrm{lb}}\le C^{\mathrm{eff}}\le C^{\mathrm{ub}}$$

(99)

is used to represent relation (98) for simplicity.

The metal plate $\mathsf{\Omega }={\cup }_{s=1}^N{\mathsf{\Omega }}_s$ is an aggregate of all orthorhombic sets ${\mathsf{\Omega }}_s$. The volume average stiffness tensor $C^{\mathrm{Voigt}}$ under the Voigt model and the volume average flexibility tensor $S^{\mathrm{Reuss}}$ under the Reuss model on orthorhombic aggregates of cubic crystallites $\mathsf{\Omega }={\cup }_{s=1}^N{\mathsf{\Omega }}_s$ with ${\mathsf{\Omega }}_s={\cup }_{s=1}^4{\mathsf{\Omega }}_{s,q}$ are, respectively

$$C^{\mathrm{Voigt}}={\displaystyle \frac{1}{\left|\mathsf{\Omega }\right|}}{\displaystyle \sum _{s=1}^N\left|{\mathsf{\Omega }}_s\right|}{\tilde{C}}^c(Q_s)={\kappa }_{0,1}^c{H}_0^{0,1}+{\kappa }_{0,2}^c{H}_0^{0,2}+{\kappa }_{4,1}^c{\overline{H}}_0^{4,1}({\overline{c}}_m^4),$$

(100)

$$S^{\mathrm{Reuss}}={\displaystyle \frac{1}{\left|\mathsf{\Omega }\right|}}{\displaystyle \sum _{s=1}^N\left|{\mathsf{\Omega }}_s\right|}{\tilde{S}}^c(Q_s)={\varsigma }_{0,1}^c{H}_0^{0,1}+{\varsigma }_{0,2}^c{H}_0^{0,2}+{\varsigma }_{4,1}^c{\overline{H}}_0^{4,1}({\overline{c}}_m^4),$$

where the material constants ${\kappa }_{0,1}^c$, ${\kappa }_{0,2}^c$, and ${\kappa }_{4,1}^c$ (${\varsigma }_{0,1}^c$, ${\varsigma }_{0,2}^c$, and ${\varsigma }_{4,1}^c$) of $\mathsf{\Omega }$ are given in (71) and (81), $H_0^{0,1}$ and $H_0^{0,2}$ are given in (56), the mesostructure tensor ${\overline{H}}^{4,1}$ of $\mathsf{\Omega }$ is

$${\overline{H}}^{4,1}\left({\overline{c}}_m^4\right)={\displaystyle \frac{1}{\left|\mathsf{\Omega }\right|}}{\displaystyle \sum _{s=1}^N\left|{\mathsf{\Omega }}_s\right|}{\tilde{H}}^{4,1}(Q_s)$$

(101)

$$={\overline{c}}_0^4{H}_0^{4,1}+2{\overline{c}}_2^4\mathrm{Re}{H}_2^{4,1}+2{\overline{c}}_4^4\mathrm{Re}{H}_4^{4,1}$$

$$=\left(\begin{array}{cccccc}-a_2-a_3& a_3& a_2& 0& 0& 0\\ a_3& -a_3-a_1& a_1& 0& 0& 0\\ a_2& a_1& -a_1-a_2& 0& 0& 0\\ 0& 0& 0& 2a_1& 0& 0\\ 0& 0& 0& 0& 2a_2& 0\\ 0& 0& 0& 0& 0& 2a_3\end{array}\right)$$

by (59) with

$$a_1=-{\displaystyle \frac{\sqrt{70}}{35}}{\overline{c}}_0^4-{\displaystyle \frac{\sqrt{7}}{7}}{\overline{c}}_2^4,$$

(102)

$$a_2=-{\displaystyle \frac{\sqrt{70}}{35}}{\overline{c}}_0^4+{\displaystyle \frac{\sqrt{7}}{7}}{\overline{c}}_2^4,$$

$$a_3={\displaystyle \frac{\sqrt{70}}{140}}{\overline{c}}_0^4-{\displaystyle \frac{1}{2}}{\overline{c}}_4^4,$$

and the mesostructure coefficients of Ω.

$${\overline{c}}_m^4={\displaystyle \frac{1}{\left|\mathsf{\Omega }\right|}}{\displaystyle \sum _{s=1}^N\left|{\mathsf{\Omega }}_s\right|}{\tilde{c}}_m^4(Q_s),$$

(103)

where ${\tilde{c}}_m^4(Q_s)$ are given in (93).

The elasticity tensor $C^{\mathrm{Voigt}}$ and ($S^{\mathrm{Reuss}}$)⁻¹ of the metal plate under the Voigt model and under the Reuss model are an upper bound and a lower bound of the effective elasticity tensor, respectively; i.e., [5],

$$C^{\mathrm{ub}}=C^{\mathrm{Voigt}},C^{\mathrm{lb}}={(S^{\mathrm{Reuss}})}^{-1}$$

(104)

with (100).

8. Example and Discussion of Decomposition on Elasticity Tensor

8.1. Elastic Constants and Crystallite Orientation Distribution of Cubic Crystallites on Representation Element of Metal Plate

By Morris [10], the elastic constants of an iron single crystal in (70) is

$$c_{11}=237\mathrm{GPa},c_{12}=141\mathrm{GPa},c_{44}=116\mathrm{GPa}.$$

(105)

Take a representation element Ω in the iron plate. The representation element Ω is an orthorhombic aggregate of $4\times 53+4\times 71+4\times 57+4\times 69=1000$ iron cubic crystallites ${\overline{\omega }}_{s,q,p}$, where ${\mathsf{\Omega }}_1$, ${\mathsf{\Omega }}_2$, ${\mathsf{\Omega }}_3$, and ${\mathsf{\Omega }}_4$ are an orthorhombic aggregate of $53\times 4,71\times 4,57\times 4,$ and $69\times 4$ cubic crystallites, respectively; i.e.,

$$\mathsf{\Omega }={\cup }_{s=1}^4{\mathsf{\Omega }}_s,{\mathsf{\Omega }}_s={\cup }_{q=1}^4{\mathsf{\Omega }}_{s,q},{\mathsf{\Omega }}_{s,q}={\cup }_{p=1}^{m_s}{\overline{\omega }}_{s,q,p};$$

(106)

$$m_1=53,m_2=71,m_3=57,m_4=53.$$

Assume that each cubic crystallite ${\overline{\omega }}_{s,q,p}$ in $\mathsf{\Omega }$ has the same volume. The orientations $Q_{s,q}$ of cubic crystallites ${\overline{\omega }}_{s,q,p}(p=1,2,\cdot \cdot \cdot ,m_s)$ in the set ${\mathsf{\Omega }}_{s,q}$ are given by

$$Q_{s,1}=Q(e_1,0)Q_s,Q_{s,2}=Q(e_1,\pi )Q_s,$$

(107)

$$Q_{s,3}=Q(e_2,\pi )Q_s,Q_{s,4}=Q(e_3,\pi )Q_s$$

and

$$Q_1=Q(0^{\mathrm{o}},{60}^{\mathrm{o}},0^{\mathrm{o}}),Q_2=Q(5^{\mathrm{o}},{65}^{\mathrm{o}},0^{\mathrm{o}}),$$

(108)

$$Q_3=Q({10}^{\mathrm{o}},{70}^{\mathrm{o}},0^{\mathrm{o}}),Q_4=Q({15}^{\mathrm{o}},{75}^{\mathrm{o}},0^{\mathrm{o}}).$$

8.2. Effective Elasticity Tensor $C^{\mathrm{FEM}}$ Simulated by FEM on Representation Element $\mathsf{\Omega }$ in (106)

To check the expression (100), we compute the elasticity tensor of the representation element Ω in (106) by the finite element simulation. The cubic crystals and the finite element grids of Ω are shown in Figure 1. The represent element Ω in Figure 1 is an orthorhombic aggregate of cubic crystallites $\mathsf{\Omega }={\cup }_{s=1}^4{\cup }_{q=1}^4{\cup }_{p=1}^{m_s}{\overline{\omega }}_{s,q,p}$. The orientation of each crystallite is assigned according to (106)–(108). The position of each crystallite in Ω is random. The six boundary value problems for the representation element Ω consist of three uniaxial tensions and three pure shears. By the ANSYS program (version 2024 R2) we obtain the average stress ${\overline{T}}_i$ and the average strain ${\overline{E}}_i$ of Ω under the Kelvin notation for the six boundary value problems, respectively. Let $C^{\mathrm{FEM}}$ denote the effective elasticity tensor simulated by the finite element method on Ω. Using the relation ${\overline{T}}_i=C^{\mathrm{FEM}}{\overline{E}}_i$ of the six boundary value problems [31], we give the effective elasticity tensor $C^{\mathrm{FEM}}$ of Ω as follows:

$$C^{\mathrm{FEM}}=[{\overline{T}}_1,{\overline{T}}_2,\cdot \cdot \cdot ,{\overline{T}}_6]{[{\overline{E}}_1,{\overline{E}}_2,\cdot \cdot \cdot ,{\overline{E}}_6]}^{-1}$$

(109)

whose computation results are shown in Table 1.

8.3. Upper Bound and Lower Bound of Effective Elasticity Tensor Under Voigt Model and Under the Reuss Model, Respectively

8.3.1. Mesostructure Coefficients ${\overline{c}}_m^4$ and Mesostructure Tensor ${\overline{H}}^{4,1}({\overline{c}}_m^4)$ of Ω in (106)

Putting (107) and (108) into (96) and (97), we obtain the mesostructure coefficients of the orthorhombic sets ${\mathsf{\Omega }}_s(s=1,2,3,4)$ in the representation element Ω.

$${\tilde{c}}_0^4(Q_1)=0.0625,{\tilde{c}}_2^4(Q_1)=0.5929,{\tilde{c}}_2^4(Q_1)=0.4856,$$

(110)

$${\tilde{c}}_0^4(Q_2)=0.2665,{\tilde{c}}_2^4(Q_2)=0.4569,{\tilde{c}}_2^4(Q_2)=0.4792,$$

$${\tilde{c}}_0^4(Q_3)=0.4835,{\tilde{c}}_2^4(Q_3)=0.3069,{\tilde{c}}_2^4(Q_3)=0.4105,$$

$${\tilde{c}}_0^4(Q_4)=0.6875,{\tilde{c}}_2^4(Q_4)=0.1712,{\tilde{c}}_2^4(Q_4)=0.2801.$$

Since every cubic crystallite ${\varpi }_{s,q,p}$ in Ω has the same volume and since ${\mathsf{\Omega }}_s={\cup }_{q=1}^4{\cup }_{p=1}^{m_s}{\varpi }_{s,q,p}(s=1,2,3,4)$ are an orthorhombic aggregate of 53 × 4, 71 × 4, 57 × 4, and 69 × 4 cubic crystallites, respectively. From (103), the mesostructure coefficients of $\mathsf{\Omega }={\cup }_{s=1}^4{\mathsf{\Omega }}_s$ should be

$${\overline{c}}_m^4={\displaystyle \frac{53\times 4}{1000}}{\tilde{c}}_m^4(Q_1)+{\displaystyle \frac{71\times 4}{1000}}{\tilde{c}}_m^4(Q_2)+{\displaystyle \frac{57\times 4}{1000}}{\tilde{c}}_m^4(Q_2)+{\displaystyle \frac{69\times 4}{1000}}{\tilde{c}}_m^4(Q_2)$$

(111)

which with (110) leads to

$${\overline{c}}_0^4=0.3889,{\overline{c}}_2^4=0.3727,{\overline{c}}_4^4=0.4099.$$

(112)

Putting (112) into (101) and (102), we obtain the mesostructure tensor ${\overline{H}}^{4,1}({\overline{c}}_m^4)$ of $\mathsf{\Omega }$.

$${\overline{H}}^{4,1}({\overline{c}}_m^4)=\left(\begin{array}{cccccc}0.1338& -0.1817& 0.0479& 0& 0& 0\\ -0.1817& 0.4156& -0.2338& 0& 0& 0\\ 0.0479& -0.2338& 0.1859& 0& 0& 0\\ 0& 0& 0& -0.4677& 0& 0\\ 0& 0& 0& 0& 0.0958& 0\\ 0& 0& 0& 0& 0& -0.3635\end{array}\right)$$

(113)

8.3.2. Elastic Constants of Elasticity Tensor on $\mathsf{\Omega }$

Putting (105) into (71), (79), and (81), we obtain the elastic constants of the elasticity tensor in (100).

$${\kappa }_{0,1}^c=-311.65\mathrm{GPa},{\kappa }_{0,2}^c=574.41\mathrm{GPa},{\kappa }_{4,1}^c=-3113.79\mathrm{GPa},$$

(114)

$${\varsigma }_{0,1}^c=0.004406{(\mathrm{GPa})}^{-1},{\varsigma }_{0,2}^o=0.014571{(\mathrm{GPa})}^{-1},{\varsigma }_{4,1}^o=0.005109{(\mathrm{GPa})}^{-1}.$$

(115)

8.3.3. Upper Bound and Lower Bound of Effective Elasticity Tensor Under Voigt Model and Under the Reuss Model

Under the Voigt model, putting (114), (56), and (113) into (100)₁, we derive the effective elasticity tensor $C^{\mathrm{Voigt}}$ of the representation element Ω under the Voigt model. $C^{\mathrm{Voigt}}$ under the Voigt model is the upper bound of the effective elasticity tensor of the representation element Ω. The components of $C^{\mathrm{Voigt}}$ are shown in Table 1.

Under the Reuss model, putting (115), (56), and (113) into (100)₂, we obtain the inverse ${(S^{\mathrm{Reuss}})}^{-1}$ of the volume average flexibility tensor on $\mathsf{\Omega }$. The elasticity tensor ${(S^{\mathrm{Reuss}})}^{-1}$ under the Reuss model is the lower bound of the elasticity of the representation element $\mathsf{\Omega }$. The components of ${(S^{\mathrm{Reuss}})}^{-1}$ are shown in Table 1. In Table 1 there are ${(S^{\mathrm{Reuss}})}^{-1}<C^{\mathrm{FEM}}<C^{\mathrm{Voigt}}$. $C^{\mathrm{Voigt}}$ and ${(S^{\mathrm{Reuss}})}^{-1}$ are the upper and the lower bounds of the effective elasticity tensor $C^{\mathrm{FEM}}$, respectively, as shown in (98) and (99).

8.4. Discussion of Decomposition on Elasticity Tensor

In this paper, the stress tensor, the strain tensor, the elasticity tensor, and the rotation tensor are presented under the Kelvin notation. The elastic constants and the mesostructures of the elasticity tensor on cubic crystals are decomposed. The elasticity tensors of the metal plate under the Voigt model and under the Reuss model are an upper bound and a lower bound [5] of the effective elasticity tensor, respectively. In Section 8, an example is used to show that our decomposition is reliable by comparing the FEM simulation’s results with our expressions listed in Table 1. We make use of FEM examples to check the decomposition of material constants and mesostructures on iron plates. In fact, we have carried out many FEM calculations for different texture coefficients and different choices of boundary value problems and find that the forms of $C^{\mathrm{Voigt}}$ and $S^{\mathrm{Reuss}}$ in (100) is adequate and reasonable.