Page 26 - Applied Petroleum Geomechanics

16 Applied Petroleum Geomechanics

The inverse relationship can be expressed as:
2 3
6 1 n n n 0 0 0 7
6 7
2 3 6 n 1 n n 0 0 0 7
s x 6 7
6 7
6 n n 1 n 0 0 0
6 7 6 7
6 s y 7 7
6 7 6 7
6 s z 7 E 6 1 2n 7
6 7 ¼ 6 0 0 0 0 0 7
s 2
6 7 ð1 þ nÞð1 2nÞ 6 7
6 yz 7 6 7
6 7 6 1 2n 7
4 s xz 5 6 7
0 0 0 0 0
2
6 7
s xy 6 7
6 7
6 1 2n 7
0 0 0 0 0
4 5
2
2 3 2 3
ε x 1
1
ε
6 y 7 6 7
6
7
6 7
6 7 6 7
6 ε z
Ea T DT 6 1 7
7
6 7 þ 6 7
0
6 7 1 2n 6 7
6 2ε yz 7 6 7
6 7 6 7
4 2ε xz 5 4 0 5
2ε xy 0
(1.24)
This expression can be written in a much more convenient form using
index notation (Bower, 2010):
1 þ n n
ε ij ¼ s ij s kk d ij a T DTd ij (1.25)
E E
where, d ij is the Kronecker delta function; if i ¼ j, d ij ¼ 1; otherwise,
d ij ¼ 0.
The inverse relation is:
E h n i Ea T DT
s ij ¼ ε ij þ ε kk d ij þ d ij (1.26)
1 þ n 1 2n 1 2n
The stressestrain relations are often expressed using the elastic modulus
tensor C ijkl or the elastic compliance tensor S ijkl as follows:
s ij ¼ C ijkl ðε kl þ a T DTd kl Þ (1.27)
ε ij ¼ S ijkl s kl a T DTd ij (1.28)