Page 194 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 167

2. Four stress strain relations
1
8
ð
ð
> ε rr 5 ½ σ rr 2 νσ zz 1 σ θθ ފ 2 α T 2 T 0 Þ
>
>
>
E
>
>
1
>
>
>
ε zz 5
>
ð
ð
> ½ σ zz 2 νσ rr 1 σ θθ ފ 2 α T 2 T 0 Þ
>
<
E
ð4:65Þ
1
ε θθ 5
>
ð
ð
> ½ σ zz 2 νσ rr 1 σ zz ފ 2 α T 2 T 0 Þ
>
>
>
E
>
>
>
ε rz 5
>
> σ rz
>
>
:
2G
3. Four strain displacement relations
ε rr 52 @u
@r
ε zz 52 @w
@z
ð4:66Þ
ε θθ 52 u
r
!
ε rz 52 1 @u 1 @w
2 @z @x
4. Boundary conditions
In the two-dimensional case, traction boundary conditions, considering the sur-
face of the body as traction-free, take the following form
(
σ rr n r 1 σ rz n z 5 0
ð4:67Þ
σ rz n r 1 σ zz n z 5 0
For displacement boundary conditions

u 5 F 1 ðHÞ
ð4:68Þ
w 5 F 2 ðHÞ
4.9.6 One-dimensional thermoelastic modelling
Modelling thermoelastic problems with reference to one-dimensional conditions may
be of interest in some situations. A one-dimensional analysis approach is proposed in
the following by assuming that a temperature distribution of the form T 5 TðzÞ is
known.