Reports of the Academy of Sciences of the USSR
1962. Volume 146, No. 3

THEORY OF ELASTICITY

S. D. VOLKOV, N. A. KLINSKIKH

ON THE DISTRIBUTION OF ELASTIC CONSTANTS IN QUASI-ISOTROPIC POLYCRYSTALS*

(Presented by Academician P. A. Rebinder on 11 IV 1962)

1. The distribution of stresses and strains in a quasi-isotropic polycrystalline body, under prescribed conditions on the boundary of the body, is characterized by the distribution of the moments of the components of the stress and strain tensors (^1). In an elastic medium, finding the moments as functions of the coordinates of a point of the body is a direct generalization of the classical boundary-value problem of the theory of elasticity, in which, owing to the microscopic homogeneity of the medium (absence of structure), moments of the second and higher orders vanish and the distribution of only the first-order moments (the macroscopic components of the stress and strain tensors) is sought. To solve the above-mentioned generalized problem in the general case, it is first necessary to find the distribution, in a macroscopically elementary neighborhood \(W\) of a point of the body, of the components of the tensor of elastic constants (^1, ^2). Below, a new general method for solving this problem is considered, and data are given for a crystal lattice of cubic symmetry.

2. Let the elastic medium be quasi-isotropic (isotropic on the large scale and anisotropic on the small scale (^2)). Then the components \(a_{ij}\) (\(b_{ij}\)) of the elastic constants, referred to a fixed Cartesian coordinate system \((x, y, z)\) and defined for a microelementary neighborhood of a point of the medium \(V\) (for a single crystal), will be random quantities. Let us further assume that the polycrystal is single-phase. Then the characteristic values \(a'_{ij}\) (\(b'_{ij}\)) of the components \(a_{ij}\) (\(b_{ij}\)) in the crystallographic rectangular coordinate system \((x', y', z')\) will be deterministic quantities, determined experimentally. We shall denote by \(\alpha_{ks}\) \((k, s = 1, 2, 3)\) the direction cosines of the crystallographic coordinate axes \((x', y', z')\) with respect to the fixed coordinate system \((x, y, z)\). Owing to the chaotic orientation of the axes \((x', y', z')\) in a quasi-isotropic polycrystal, the quantities \(\alpha_{ks}\) are random.

The transformation formulas for the elastic constants are obtained from the condition of invariance of the potential energy of elastic strains (^3):

\[ a_{ij} = \sum_{m,n=1}^{6} a'_{mn} q_{mi} q_{nj} \qquad (i, j = 1, 2, \ldots, 6), \tag{1} \]

where \(q_{mi}\) and \(q_{nj}\) are known functions of the direction cosines. Owing to the orthogonality of the coordinate axes, among the 9 quantities \(\alpha_{ks}\) there exist 6 orthogonality relations. Consequently, only 3 quantities are independent. We choose as the independent quantities the 3 Euler angles \(\varphi, \psi, \theta\). Then, using the known transformation formulas for the direction cosines \(\alpha_{ks} = \alpha_{ks}(\varphi, \psi, \theta)\), we can write formulas (1) in the form

\[ a_{ij} = a_{ij}(\varphi, \psi, \theta) \qquad (i, j = 1, 2, \ldots, 6). \tag{2} \]

* Reported at the First All-Union Congress on Theoretical and Applied Mechanics in January 1960 in Moscow.

1. In order to find the moments of the distribution of the elastic constants (2), we first compute the density of the joint distribution of the Euler angles.

By the multiplication theorem for distribution laws, we write the density of the joint distribution in the form

\[ \rho(\theta,\psi,\varphi)=\rho(\theta)\rho\left(\frac{\psi,\varphi}{\theta}\right) =\rho(\theta)\rho\left(\frac{\psi}{\theta}\right)\rho\left(\frac{\varphi}{\theta,\psi}\right), \tag{3} \]

where \(\rho(\theta)\) is the density of the marginal distribution of the angle \(\theta\); \(\rho\left(\frac{\psi}{\theta}\right)\) is the density of the conditional marginal distribution of the angle \(\psi\) for fixed angle \(\theta\); \(\rho\left(\frac{\psi}{\theta,\psi}\right)\) is the density of the conditional marginal distribution of the angle \(\varphi\) for fixed \(\theta\) and \(\varphi\).

Fig. 1

Since in a quasi-isotropic medium all directions of the axis \(z'\) are equiprobable, the unit vector of the axis \(z'\) describes a sphere of unit radius with a uniform density of distribution of points—the endpoints of this vector on the sphere (\(^2\)). Therefore the probability \(\mathbf{P}(\theta<\theta_0)\), where \(\theta_0\) is a fixed value of the quantity \(\theta\), is proportional to the surface area of the spherical segment described by the endpoint of the unit vector of the axis \(z'\) in the neighborhood of the fixed point—the endpoint of the unit vector of the axis \(z\) (see Fig. 1):

\[ \mathbf{P}(\theta<\theta_0)=c\cdot 2\pi(1-\cos\theta_0), \tag{4} \]

where the proportionality coefficient \(c\) is determined from the condition \(\mathbf{P}(\theta<\pi)=1\). Hence \(c=(4\pi)^{-1}\).

We now find the distribution function \(F(\theta_0)=\mathbf{P}(\theta<\theta_0)\), omitting for simplicity of notation the subscript of \(\theta_0\):

\[ F(\theta)=\tfrac{1}{2}(1-\cos\theta). \tag{5} \]

Differentiating (5), we find the required density of the marginal distribution of the angle \(\theta\):

\[ \rho(\theta)=\tfrac{1}{2}\sin\theta. \tag{6} \]

If we introduce unit vectors of the axes \(x'\) and \(y'\) and carry out elementary calculations analogous to the preceding ones, it is not difficult to obtain the densities of the conditional marginal distributions

\[ \rho\left(\frac{\psi}{\theta}\right)=\frac{1}{2\pi},\qquad \rho\left(\frac{\varphi}{\theta,\psi}\right)=\frac{1}{2\pi}. \tag{7} \]

Substitution of (6) and (7) into (3) gives:

\[ \rho(\theta,\psi,\varphi)=\frac{1}{8\pi^2}\sin\theta. \tag{8} \]

Formula (8) was used to compute the first-order moments of the distributions of elastic constants in work (\(^6\)).

1. We shall find the moments of the distributions of the components \(a_{ij}\) for quasi-isotropic polycrystals with cubic symmetry of the crystal lattice (copper, aluminum, \(\alpha\)-iron, etc.). In this case the transformation formulas (1) can be reduced to the form

\[ a_{ij}=a'_{ij}+A\gamma_{ij}\qquad (i,j=1,2,\ldots,6), \tag{9} \]

where \(A=2(a'_{11}-a'_{12})-a'_{44}\); \(\gamma_{ij}=\gamma_{ij}(a_{ks})\) are functions of the direction cosines, subdivided into 6 groups (\(^4\)):

I. $\gamma_{11}, \gamma_{22}, \gamma_{33}$.
II. $\gamma_{44}, \gamma_{55}, \gamma_{66}$.
III. $\gamma_{12}=\gamma_{21}, \gamma_{23}=\gamma_{32}, \gamma_{31}=\gamma_{13}$.
IV. $\gamma_{14}=\gamma_{41}, \gamma_{25}=\gamma_{52}, \gamma_{36}=\gamma_{63}$.
V. $\gamma_{56}=\gamma_{65}, \gamma_{64}=\gamma_{46}, \gamma_{45}=\gamma_{54}$.
VI$^{\mathrm{a}}$. $\gamma_{15}=\gamma_{51}, \gamma_{26}=\gamma_{62}, \gamma_{34}=\gamma_{43}$.
VI$^{\mathrm{b}}$. $\gamma_{16}=\gamma_{61}, \gamma_{24}=\gamma_{42}, \gamma_{35}=\gamma_{53}$.

In each group the formulas are obtained by a cyclic permutation of the first indices of the direction cosines. We have:

\[ \begin{aligned} \gamma_{11}&=-\left(a_{11}^{2}a_{12}^{2}+a_{12}^{2}a_{13}^{2}+a_{13}^{2}a_{11}^{2}\right);\\ \gamma_{44}&=2\left(a_{21}^{2}a_{31}^{2}+a_{22}^{2}a_{32}^{2}+a_{23}^{2}a_{33}^{2}\right);\\ \gamma_{12}&=\frac12\left(a_{11}^{2}a_{21}^{2}+a_{12}^{2}a_{22}^{2}+a_{13}^{2}a_{23}^{2}\right); \end{aligned} \qquad \begin{aligned} \gamma_{14}&=a_{11}^{2}a_{21}a_{31}+a_{12}^{2}a_{22}a_{32}+a_{13}^{2}a_{23}a_{33};\\ \gamma_{56}&=2\left(a_{11}^{2}a_{21}a_{31}+a_{12}^{2}a_{22}a_{32}+a_{13}^{2}a_{23}a_{33}\right);\\ \gamma_{15}&=a_{11}^{3}a_{31}+a_{12}^{3}a_{32}+a_{13}^{3}a_{33};\\ \gamma_{16}&=a_{11}^{3}a_{21}+a_{12}^{3}a_{22}+a_{13}^{3}a_{23}. \end{aligned} \tag{10} \]

Taking into account that the quantity $A$ in formula (9) is determined, we find the matrix of initial first-order moments

\[ \left\|\bar a_{ij}\right\|=\left\|a'_{ij}\right\|+A\left\|\bar\gamma_{ij}\right\| \qquad (i,j=1,2,\ldots,6), \tag{11} \]

where $\left\|\bar\gamma_{ij}\right\|$ is the matrix of first-order moments of the quantities $\gamma_{ij}$.

For the first three groups it is sufficient to find the mean value of the product:

\[ \overline{a_{13}^{2}a_{23}^{2}} = \frac{1}{8\pi^{2}} \int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{\pi} \sin^{2}\varphi \cos^{2}\varphi \sin^{5}\theta \, d\psi\, d\varphi\, d\theta = \frac{1}{15}. \]

Then $\bar\gamma_{11}=\bar\gamma_{22}=\bar\gamma_{33}=-1/5$; $\bar\gamma_{44}=\bar\gamma_{55}=\bar\gamma_{66}=2/5$; $\bar\gamma_{12}=\bar\gamma_{23}=\bar\gamma_{13}=1/10$.

Determining the mean values of combinations of the form $a_{13}^{2}a_{23}a_{33}$ and $a_{13}^{3}a_{23}$, it is easy to establish that $\bar\gamma_{ij}$ for groups IV, V, and VI vanish. Thus,

\[ \left\|\bar\gamma_{ij}\right\| = 0.1 \left\| \begin{array}{rrrrrr} -2 & 1 & 1 & 0 & 0 & 0\\ 1 & -2 & 1 & 0 & 0 & 0\\ 1 & 1 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 4 & 0 & 0\\ 0 & 0 & 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 0 & 0 & 4 \end{array} \right\|. \tag{12} \]

Substitution of (12) into (11) gives

\[ \begin{aligned} \bar a_{11}=\bar a_{22}=\bar a_{33}&=a'_{11}-0.2A; \qquad &\bar a_{12}=\bar a_{23}=\bar a_{13}&=a'_{12}+0.1A;\\ \bar a_{44}=\bar a_{55}=\bar a_{66}&=a'_{44}+0.4A; \qquad &\bar a_{14}=\ldots=\bar a_{56}&=0. \end{aligned} \tag{13} \]

The first-order moments (13) were calculated earlier [5] by another method, practically unsuitable for calculating moments of second and higher orders because of the increasing complexity of the combinatorics. The method used in the present work is based on a preliminary calculation of the density of the joint distribution of the Euler angles. As a result, the calculation of moments of any order presents no fundamental difficulties.

Consider the second-order moments

\[ K_{ij,kl} = \overline{\Delta a_{ij}\Delta a_{kl}} = \overline{(a_{ij}-\bar a_{ij})(a_{kl}-\bar a_{kl})}. \tag{14} \]

Taking (9) into account, from (14) we find $\left\|K_{ij,kl}\right\|=A^{2}\left\|\Gamma_{ij,kl}\right\|$, where $\Gamma_{ij,kl}=\overline{\Delta\gamma_{ij}\Delta\gamma_{kl}}$, $\Delta\gamma_{ij}=\gamma_{ij}-\bar\gamma_{ij}$.

To simplify the notation, we pass to a two-index system according to the following rule:

I. $11—1$; $22—2$; $33—3$.
II. $44—4$; $55—5$; $66—6$.
III. $23,32—7$; $31,13—8$; $12,21—9$.
IV. $14,41—10$; $25,52—11$; $36,63—12$.
V. $56,65—13$; $64,46—14$; $45,54—15$.
VI$^{\mathrm{a}}$. $34,43—16$; $15,51—17$; $26,62—18$.
VI$^{\mathrm{b}}$. $24,42—19$; $35,53—20$; $16,61—21$.

On the diagonal of the matrix \(\|K_{ij}\|\) stand the variances of the elastic constants. Calculating the elements of the matrix \(\|K_{ij}\|\), we find:

\[ \|K_{ij}\|=\frac{A^2}{2100} \begin{Vmatrix} & \mathrm{I} & & & \mathrm{II} & & & \mathrm{III} & & \\ \mathrm{I} & 16&6&6& 8&-32&-32& 2&-8&-8\\ & &16&6& -32&8&-32& -8&2&-8\\ & & &16& -32&-32&8& -8&-8&2\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \mathrm{II} & & & &144&-8&-8& 36&-4&-4\\ & & & & &144&-8& -4&36&-4\\ & & & & & &144& -4&-4&36\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \mathrm{III} & & & & & & &9&-1&-1\\ & & & & & & &9&-1\\ & & & & & & & &9 \end{Vmatrix} \tag{15} \]

\[ (i,j=1,2,\ldots,9); \]

\[ \|K_{ij}\|=0 \qquad (i=1,2,\ldots,9;\ j=10,11,\ldots,21); \tag{16} \]

\[ \|K_{ij}\|=\frac{A^2}{2100} \begin{Vmatrix} & \mathrm{IV} & & & \mathrm{V} & & & \mathrm{VI}^{a} & & & \mathrm{VI}^{b} & & \\ \mathrm{IV} & 20&0&0& 40&0&0& -10&0&0& -10&0&0\\ & &20&0& 0&40&0& 0&-10&0& 0&-10&0\\ & & &20& 0&0&40& 0&0&-10& 0&0&-10\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \mathrm{V} & & & &80&0&0& -20&0&0& -20&0&0\\ & & & & &80&0& 0&-20&0& 0&-20&0\\ & & & & & &80& 0&0&-20& 0&0&-20\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \mathrm{VI}^{a} & & & & & & &40&0&0& -30&0&0\\ & & & & & & &40&0& 0&-30&0\\ & & & & & & & &40& 0&0&-30\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \mathrm{VI}^{b} & & & & & & & & & &40&0&0\\ & & & & & & & & & &40&0\\ & & & & & & & & & & &40 \end{Vmatrix} \tag{17} \]

\[ (i,j=10,11,\ldots,21). \]

It is seen from (16) that the constants \(a_{ij}\) of groups I–III are not correlated with the constants \(a_{ij}\) of groups IV–VI, since the corresponding correlation moments are equal to zero. The elastic constants of groups I–III are mutually correlated, as is seen from (15). According to (17), the elastic constants of group IV are not correlated among themselves, but are correlated with some elastic constants of groups V and VI. The same holds for the constants of group V. The elastic constants of each of the two subgroups of group VI are not mutually correlated, but there is correlation between elastic constants from different subgroups.

Received
27 X 1961

CITED LITERATURE

1. S. D. Volkov, Izv. AN SSSR, OTN, No. 3, 65 (1958).
2. S. D. Volkov, Statistical Theory of Strength, 1960.
3. S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, 1950.
4. P. Bekhterev, “Analytical Study of the Generalized Hooke’s Law,” L., 1925.
5. A. Reuss, ZAMM, 9, H. 1 (1929).
6. E. Kröner, Zs. f. Phys., 151, No. 4 (1958).