CONTINUUM MECHANICS for ENGINEERS

have a multiplicity of two, and determine the principal directions.
Solution
Here Eq 2.6-6 is given by
for which the characteristic equation becomes
or
For 8 , Eq 2.6-5 yields
λ( 1)
=
5−λ1 2
1 5−λ 2 = 0
2 2 6−λ
3 2
λ − 16λ + 80λ− 128 = 0
2
( λ−8) ( λ−4)
= 0
− 3n + n + 2n = 0
1 2 3
n − 3n + 2n = 0
1 2 3
2n + 2n − 2n = 0
1 2 3
From the first two of these equations n = n , and from the second and third
1 2
equations n3 = 2n2.
Therefore, using, nn i i = 1 , we have
2 2
2
( 2)
+ ( 2)
+ ( 2)
=
n n 2n 1
and so n and , from which the unit vector in the
1 = n2
= ± 1/ 2 n3 =± 1 2
principal direction associated with λ (the so-called normalized eigen-
( 1)
8
vector) is
=
ˆn 2 ( )
nˆ eˆ eˆ eˆ e ˆ*
1 ( ) 1
= ( 1+ 2 + 2 3)= 3
2
ˆn 1 ( )
For
being
, we choose any unit vector perpendicular to ; an obvious choice