Page 188 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 161

In general, it is convenient to explicitly express stresses in terms of the strains. In
compact form, this reads

ð
σ ij 5 D ijkl ε kl 1 β T 2 T 0 Þ ð4:52Þ
kl
In matrix form, Eq. (4.52) reads
2 3 2 ν ν 3
ð 1 2 νÞ 0 0 0
σ xx
ν ð 1 2 νÞ ν 0 0 0
6 σ yy 7 6 7
ν ν ð 1 2 νÞ 0 0 0
6 7 6 7
6 σ zz 7 6 7
0 0 0 ð 1 2 2νÞ 0 0
6 7 5 E 06 7
6 σ xy 7 6 7
0 0 0 0 ð 1 2 2νÞ 0
4 5 4 5
σ yz
0 0 0 0 0 ð 1 2 2νÞ
σ zx
8 9 ð4:53Þ
2 3 2 α 3
> ε xx >
> >
> >
> 6 ε yy 7 6 α 7 >
> >
< 6 7 6 7 =
6 ε zz 7 6 α 7
7 T 2 T 0 Þ
ð
6 7 1 6
> 6 ε xy 7 6 0 7 >
> >
0
> 4 5 4 5 >
> >
> ε yz >
: ;
0
ε zx
where E 0 5 E= 1 1 νð ½ Þ 1 2 2νފ.
ð
In extended form, Eq. (4.52) reads
σ xx 5 2Gε xx 1 νE ε xx 1 ε yy 1 ε zz 1 3Kα T 2 T 0 Þ

ð
ð
ð 1 1 νÞ 1 2 2νÞ

σ yy 5 2Gε yy 1 νE ε xx 1 ε yy 1 ε zz 1 3Kα T 2 T 0 Þ
ð
ð 1 1 νÞ 1 2 2νÞ
ð

σ zz 5 2Gε zz 1 νE ε xx 1 ε yy 1 ε zz 1 3Kα T 2 T 0 Þ ð4:54Þ
ð
ð 1 1 νÞ 1 2 2νÞ
ð
σ xy 5 2Gε xy
σ yz 5 2Gε yz
σ zx 5 2Gε zx
4.9.3 Separation of stresses caused by mechanical and thermal loads
Eq. (4.52) expresses that stresses can arise in materials characterised by a thermoelastic behav-
iour because of two causes: mechanical loads and thermal loads (cf. Fig. 4.12). Stresses arise
in a material because of the application of force fields or the prescription of displace-
ments to ensure equilibrium and continuity of the material. Stresses arise in a material