Page 225 - Analysis and Design of Energy Geostructures

198 Analysis and Design of Energy Geostructures

ðT 2 T 0 Þ [ C] is the applied temperature variation. In matrix form, the

above reads:
2 3
1=E 2υ=E 2υ=E 0 0 0 α
2 3 2 3 2 3
ε xx σ xx
2υ=E 1=E 2υ=E 0 0 0 7
ε yy 6
6 7 7 σ yy 7 6 α 7
6
6
6 7 6 7 6 7
6 2υ=E 2υ=E 1=E 0 0 0 7
7 5 6 7ðT2T 0 Þ
6
6 ε zz 7 7 σ zz 7 6 α 7
6 6 726
6 0 0 0 1=ð2GÞ 0 0
6
6 ε xy 7 7 σ xy 7 6 0 7
6 7
4 5 4 5 4 5
ε yz 4 0 0 0 0 1=ð2GÞ 0 5 σ yz 0
ε zx 0 0 0 0 0 1=ð2GÞ σ zx 0
where for the considered material E [Pa] is the Young’s modulus, υ [ ]
is the Poisson’s ratio and G [Pa] is the shear modulus. In extended form,
the above reads:

ε xx 5 1 σ xx 2 υσ yy 1 σ zz 2 αðT 2 T 0 Þ
E

ε yy 5 1 σ yy 2 υσ zz 1 σ xx Þ 2 αðT 2 T 0 Þ
ð
E

ε zz 5 1 σ zz 2 υσ xx 1 σ yy 2 αðT 2 T 0 Þ
E
1
ε xy 5 σ xy
2G
1
ε yz 5 σ yz
2G

1
ε zx 5 σ zx
2G
q. If the stress tensor is expressed in terms of the strain tensor, it results

σ ij 5 D ijkl ½ε kl 1 β ðT 2 T 0 ފ
kl
Thus the thermally induced stress is proportional to the elastic stiff-
ness tensor of the material D ijkl .