Page 24 - Applied Petroleum Geomechanics

14 Applied Petroleum Geomechanics

For linear elastic isotropic dry materials, the stress and strain follow Hooke’s
law, which can be expressed as follows in the three-dimensional condition:
1
ε x ¼ ½s x nðs y þ s z ފ
E
1
ε y ¼ ½s y nðs x þ s z ފ
E
1
ε z ¼ ½s z nðs x þ s y ފ
E
(1.20)
1
ε xy ¼ s xy
2G
1
ε xz ¼ s xz
2G
1
ε yz ¼ s yz
2G
where s and s are the normal and shear stresses, respectively; ε xy , ε xz , ε yz
are the shear strains; n is Poisson’s ratio; E and G are Young’s and shear
moduli, respectively.
When any two of the moduli of E, G, n, l, and K are defined, the
remaining ones are fixed by certain relations. Some useful relations are listed
in Table 1.1.
A very useful stressestrain state is the uniaxial strain condition, where
the lateral strains are constrained (i.e., ε x ¼ ε y ¼ 0), and only the vertical

Table 1.1 Relations to convert elastic moduli and Poisson’s ratio.
E, Young’s modulus K, bulk modulus l, Lamé constant
E ¼ 3K (1 2n) 1 þ n l 2n
K ¼ l ¼
3n G 1 2n
2 1 þ n l
E ¼ 2G (1þn) K ¼ G ¼ 2n
3 1 2n l þ G
2 G
9KG K ¼ l þ G ¼ 1 2n
E ¼ 3 l þ G
3K þ G
GE l þ 2G
3l þ 2G K ¼
E ¼ G 9G 3E l þ G ¼ 2ð1 nÞ
l þ G
E 3l þ 2G
l G ¼ ¼ 2ð1 þ nÞ
E ¼ ð1 þ nÞð1 2nÞ 2ð1 þ nÞ l þ G
n