Page 177 - Analysis and Design of Energy Geostructures

150 Analysis and Design of Energy Geostructures

with reference to the same point, the three principal stresses are invariants under coor-
dinate transformation. The substitution of each of the three eigenvalues in Eq. (4.19)
allows computing the eigenvectors n 1 ; n 2 and n 3 that represent the principal directions.
The three invariant coefficients expressed in Eq. (4.21) are the first, second and
third stress invariants (or invariants of the stress tensor) and are given by
8

> I 1 5 tr σ ij 5 σ ii 5 σ xx 1 σ yy 1 σ zz
>
>
< 1 2 2 2
I 2 5 σ ii σ jj 2 σ ij σ ij 5 σ xx σ yy 1 σ yy σ zz 1 σ zz σ xx 2 σ 2 σ 2 σ
2 xy yz zx ð4:22Þ
>
>
> 2 2 2
: I 3 5 det σ ij 5 σ xx σ yy σ zz 1 2σ xy σ yz σ zx 2 σ xx σ 2 σ yy σ 2 σ zz σ
yz zx xy
By setting equal to zero the off-diagonal components of the stress tensor in
Eq. (4.22), the relations between the invariants and the principal stresses can be found.
Alternative formulations of the stress invariants can be derived from the stress ten-
sor itself instead from the characteristic polynomial and read
8
I 1 5 tr σ ij 5 σ ii 5 I 1
>
>
> 1 1
> 2
>
< I 2 5 σ ij σ ij 5 I 2 I 2
1
2 2
ð4:23Þ
1 1
>
> 3
>
I 3 5 σ ik σ km σ mi 5 I 2 I 1 I 2 1 I 3
>
> 1
: 3 3
Invariants can also be expressed in similar forms for the deviatoric stress tensor, for
example, as
8
J 1 5 tr s ij 5 s ii 5 σ ii 2 σ nn 5 0
>
>
1 1 2
> h i
> 2 2 2 2 2
> J 2 5 s ij s ij 5
> ð ð
> σ xx 2pÞ 1 σ yy 2p 1 σ zz 2pÞ 1 2s 1 2s 1 2s
> 2 2 xy yz xz
>
>
<
2
5 I 2 2 I 1 ð4:24Þ
6
>
>
>
>
>
1 2 2
>
J 3 5 s ik s km s mi 5 I 3 2 I 1 I 2 1
> 3
>
>
> I 1
3 3 27
:
When a coordinate system is chosen such that the directions are parallel to the
principal directions, the stress tensor reduces to
2 3
σ 1 0 0
σ ij 5 4 0 σ 2 0 5 ð4:25Þ
0 0 σ 3