Page 27 - Applied Petroleum Geomechanics

Stresses and strains 17

1.4.4 Plane stress and plane strain in isotropic thermal
rocks
1.4.4.1 Plane stress state
Plane stress and plane strain states can simplify 3-D stressestrain relations into
the corresponding 2-D forms. For a plane stress (biaxial stress) deformation
state, it has s z ¼ s yz ¼ s zx ¼ 0; therefore, substituting this condition into
Eq. (1.23) the strainestress relations have the following forms:
1
ε x ¼ ðs x ns y Þ a T DT
E
1
ε y ¼ ðs y ns x Þ a T DT
E (1.29)
n
ε z ¼ ðs x þ s y Þ a T DT
E
1
ε xy ¼ s xy
2G
and from Eq. (1.24) stressestrain relations can be expressed as follows:
E Ea T DT
s x ¼ ðε x þ nε y Þ þ
1 n 2 1 n
E Ea T DT (1.30)
s y ¼ 2 ðε y þ nε x Þþ
1 n 1 n
s xy ¼ 2Gε xy
This case occurs when a thin plate is stressed in its own plane. It also
occurs in the analysis at any free surface, if the x- and y-axes are taken in the
surface ( Jeager and Cook, 1979).

1.4.4.2 Plane strain state
For a plane strain (biaxial strain) deformation state, ε z ¼ ε yz ¼ ε zx ¼ 0;
substituting this relation into Eq. (1.23) the strainestress relations can be
expressed as follows:
1 þ n
ε x ¼ ½ð1 nÞs x ns y Š ð1 þ nÞa T DT
E
1 þ n
ε y ¼ ½ð1 nÞs y ns x Š ð1 þ nÞa T DT (1.31)
E
1
ε xy ¼ s xy
2G