B. D. ANNIN

THE LAGRANGE–SYLVESTER FORMULA FOR A TENSOR FUNCTION DEPENDING ON TWO TENSORS

(Presented by Academician L. I. Sedov, 1 IV 1960)

Let \(H, T_1, T_2\) be symmetric tensors of second rank, where \(H\) is a tensor function of \(T_1\) and \(T_2\). Suppose that the tensors \(T_1\) and \(T_2\) cannot be simultaneously reduced to canonical form and that the eigenvalues \(\lambda_i,\ i=1,2,3,\) of the tensor \(T_1\) are distinct. In this case the functional dependence \(H=f(T_1,T_2)\) can be represented in the form (1)

\[ H=K_1G+K_2T_1+K_3T_1^2+K_4T_2+K_5(T_1T_2+T_2T_1)+K_6(T_1^2T_2+T_2T_1^2), \tag{1} \]

where \(G\) is the unit tensor; \(K_i,\ i=1,2,\ldots,6,\) are scalar functions of the joint invariants of the tensors \(G, T_1\), and \(T_2\).

In a rectangular Cartesian coordinate system coinciding with the principal axes of the tensor \(T_1\), the tensors have the form

\[ H=\|H_{ij}\|,\quad G=\|\delta_{ij}\|,\quad T_1=\|\delta_{ij}\lambda_j\|,\quad T_2=\|T_{ij}\|. \]

The tensor relation (1) is equivalent to 6 scalar equalities, which in the above-named system are written in the form

\[ H_{ij}=K_1\delta_{ij}+K_2\delta_{ij}\lambda_j+K_3\delta_{ij}\lambda_j^2+K_4T_{ij}+K_5T_{ij}(\lambda_i+\lambda_j)+K_6T_{ij}(\lambda_i^2+\lambda_j^2), \tag{2} \]

where the indices take the values \(ij=11,22,33,12,13,23\). The equalities (2) may be regarded as a system of equations with respect to the unknown quantities \(K_1,K_2,\ldots,K_6\). The determinant of this system, by virtue of the assumptions made, is nonzero.

Determining from (2) the quantities \(K_i,\ i=1,2,\ldots,6,\) and substituting them into (1), we obtain the Lagrange–Sylvester formula for a tensor function depending on two tensors:

\[ H=S_{T_1}^{\lambda_i}(\overline{H}_{11},\overline{H}_{22},\overline{H}_{33}) +T_2\cdot S_{T_1}^{\lambda_i}(\overline{H}_{23},\overline{H}_{13},\overline{H}_{12})+ \]

\[ +S_{T_1}^{\lambda_i}(\overline{H}_{23},\overline{H}_{13},\overline{H}_{12})\cdot T_2+ST_2; \tag{3} \]

here it is denoted

\[ S=-(\overline{H}_{23}+\overline{H}_{13}+\overline{H}_{12}), \]

\[ S_T^{\mu_i}(a,b,c)= \frac{(T-\mu_2G)(T-\mu_3G)}{(\mu_1-\mu_2)(\mu_1-\mu_3)}\,a + \frac{(T-\mu_3G)(T-\mu_1G)}{(\mu_2-\mu_3)(\mu_2-\mu_1)}\,b + \]

\[ +\frac{(T-\mu_1G)(T-\mu_2G)}{(\mu_3-\mu_1)(\mu_3-\mu_2)}\,C, \tag{4} \]

where \(\mu_1,\mu_2,\mu_3,a,b,c\) are arbitrary numbers, and \(T\) is a symmetric tensor of second rank,

\[ \begin{aligned} \overline H_{11} &= H_{11}-T_{11}\left(\frac{H_{12}}{T_{12}}+\frac{H_{13}}{T_{13}}-\frac{H_{23}}{T_{23}}\right),\\ \overline H_{22} &= H_{22}-T_{22}\left(\frac{H_{12}}{T_{12}}-\frac{H_{13}}{T_{13}}+\frac{H_{23}}{T_{23}}\right),\\ \overline H_{33} &= H_{33}-T_{33}\left(-\frac{H_{12}}{T_{12}}+\frac{H_{13}}{T_{13}}+\frac{H_{23}}{T_{23}}\right),\\ \overline H_{ij} &= -\,\frac{H_{ij}}{T_{ij}},\qquad i\ne j,\qquad i,j=1,2,3. \end{aligned} \tag{5} \]

In particular, for \(T_2=0\) the tensor function becomes isotropic: \(H=f(T_1)\); consequently, \(H_{ij}=0\) for \(i\ne j\), \(H_{ii}=H_i=f(\lambda_i)\); moreover, \(H_{ii}=H_i,\ i=1,2,3\), and from (3) there follows the usual Lagrange–Sylvester formula for an isotropic tensor function, which, using the notation introduced above, can be written as

\[ H=f(T_1)=S_{T_1}^{\lambda_i}(H_1,H_2,H_3). \tag{6} \]

Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR

Received
30 III 1960

CITED LITERATURE

1. L. I. Sedov, Foundations of Nonlinear Mechanics of Continuous Media, Part 1, 1959, p. 65.