Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 345

r 1 A 12 A 13
r 2 A 22 A 23
r 3 A 32 A 33

A 11 r 1 A 13 A 11 A 12 r 1

A 21 r 2 A 23 A 21 A 22 r 2

A 31 r 3 A 33 A 31 A 32 r 3

t 1 ¼ ; t 2 ¼ ; t 3 ¼ : (A.51)

Det A Det A Det A

Considering the case n ¼ 3 and applying Cramer's rule to the system of

equations (A.49) we find that

0 A 12 A 13 A 11 l 0 A 13

0 A 22 l A 23 A 21 0 A 23

0 A 32 A 33 l A 31 0 A 33 l

t 1 ¼ ; t 2 ¼ ; t 3

Det ½ A l 1 Det ½ A l 1

A 11 l A 12 0

A 21 A 22 l 0

A 31 A 32

0

¼

Det ½ A l 1

which shows, due to the column of zeros in each numerator determinant, that the
only solution is that t

¼

[0, 0, 0], unless Det[ A

l 1 ] ¼

0. If Det[ A

l 1 ] ¼

0,
the values of t 1 , t 2 , and t 3 are all of the form 0/0 and therefore undefined. In this
case Cramer's rule provides no information. In order to avoid the trivial solution
t ¼ [0, 0, 0] the value of l is selected so that Det[ Al1 ] ¼ 0. While the argument
was specialized to n ¼ 3 in order to conserve page space, the result

Det ½ A l 1 ¼ 0 (A.52)

holds for all n . This condition forces the matrix [ A 1 ] to be singular and forces the

system of equations (A.48) to be linearly dependent. The further solution of (A.52)
is explored retaining the assumption of n

l

3 for convenience, but it should noted
that all the manipulations can be accomplished for any n including the values of n of
interest here, 2, 3, and 6. In the case of n

¼

¼ 3, (A.52) is written in the form

A 11 l A 12 A 13

A 21 A 22 l A 23 ¼ 0 ; (A.53)

A 31 A 32 A 33 l

l

and, when the determinant is expanded, one obtains a cubic equation for :

3 2

l I A l þ II A l III A ¼ 0 (A.54)

where

X

k¼ 3

I A ¼ tr A

¼ A kk ¼ A kk ¼ A 11 þ A 22 þ A 33 ;

(A.55)

k¼ 1