Fundamentals - Preventive Biomechanics - page 50

Fig. 3.19 Transformation of the principal axis of the tress tensor

and in matrix notation,

2
4

3
5 h e i e i i¼

2
4

3
5 h n Hi n Hi i:

r 11 r 12 r 13 r H1 00

0 r H2 0
00r H3

½ S ¼ r 12 r 22 r 23 ð 3 : 101 Þ

r 13 r 23 r 33

In ( 3.100 ) and ( 3.101 ) the r Hi are the principal direct stress components
(eigen-values) and the n Hi are the corresponding (orthonormal) principal directions
(eigen-directions). Scalar (e.g. right) multiplication of ( 3.100 ) with n Hk using n Hi
n Hk ¼ d ik yields (note that it must not be summed over k on the right-hand side)

! n Hk ¼ X

S n Hk ¼ X

3 3

r Hi n Hi n Hi r Hi ð n Hi n Hj Þ

|{z}
d ik

n Hk ¼ r Hk n Hk

i ¼ 1 i ¼ 1

thus the eigen-value problem results in

S n Hj ¼ r Hj n Hj ð j ¼ 1 ; 2 ; 3 Þ ð 3 : 102 Þ

where (direction) vectors n Hj are sought which are mapped by the operator S into a
multiple of r Hj n Hj . Using ( 3.102 ), the identical transformation S n Hj r Hj n Hj
S n Hj r Hj I n Hj ¼ 0 holds, where the following algebraic, homogenous linear
system of equations for the three eigen-vectors n Hj is obtained

n Hj ¼ 0

S r Hj I ð j ¼ 1 ; 2 ; 3 Þ ð 3 : 103 Þ

or in full (whereas the n iHj (i = 1, 2, 3; j = 1, 2, 3) are the respective coordinates
of the three eigen-vectors n Hj with respect to a OBS)