Page 228 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 201

iii. The principal stresses and the principal directions can be written as
the eigenvalues and the eigenvectors of the stress tensor. By diagona-
lising the stress tensor in:
p
2 ffiffiffi 3
2 0 0
σ ij 5 4 0 1 0 5 ½kPaŠ
p ffiffiffi
0 0 2 2
The principal stresses are thus:
p ffiffiffi
σ 1 5 2 kPa

σ 2 5 1 kPa

p ffiffiffi
σ 3 52 2 kPa

u. If the compatibility equations are expressed in terms of stress compo-
nents, in absence of external restraints, body forces and displacement dis-
continuities, it results

1 1 ν @ T
2
2 2
ð 1 1 νÞr σ xx 1 αE r T 1 5 0
1 2 ν @x 2
1 1 ν @ T
2
2
2
ð 1 1 νÞr σ yy 1 αE r T 1 5 0
1 2 ν @y 2
1 1 ν @ T
2
2
2
ð 1 1 νÞr σ zz 1 αE r T 1 5 0
1 2 ν @z 2
2
@ T
2
ð 1 1 νÞr σ xz 1 αE 5 0
@x@z
2
@ T
2
ð 1 1 νÞr σ yx 1 αE 5 0
@x@y
2
@ T
2
ð 1 1 νÞr σ yz 1 αE 5 0
@y@z