Page 311 - Introduction to Continuum Mechanics

Constitutive Equations for Linearly Anisotropic Elastic Solid 295

(b) Since the diagonal elements are positive, the eigenvalues of C are all positive. Thus, the
determinant of C is positive (and nonzero) and the inverse of C exists.

(e) Consider the case where only EI and EI are not zero, then from Eq. (5.22.4)

That is, the sub-matrix is indeed positive definite. We note that since the inverse of this

where

Since both Cjjand Snare positive, therefore
Similarly, the positive definiteness of the submatrix

f^> jj
can be proved by considering the case where only EI and £3 are nonzero and the positive
definiteness of the matrix

can be proved by considering the case where only £" 1; EI and £3 are nonzero, etc.
Thus, we see that the determinant of C and of all submatrices whose diagonal elements
are diagonal elements of C are all positive definite, and similarly the determinant of S and
of all submatrices whose diagonal elements are diagonal elements of S are all positive
definite.