Page 187 - Analysis and Design of Energy Geostructures

160 Analysis and Design of Energy Geostructures

In extended notation, Eq. (4.48) reads

1
ε xx 5 σ xx 2 νσ yy 1 σ zz 2 α T 2 T 0 Þ
ð
E
1
ε yy 5 σ yy 2 νσ zz 1 σ xx Þ 2 α T 2 T 0 Þ
ð
ð
E
1
ε zz 5 σ zz 2 νσ xx 1 σ yy 2 α T 2 T 0 Þ
ð
E
1 ð4:50Þ
ε xy 5 σ xy
2G
1
ε yz 5 σ yz
2G
1
ε zx 5 σ zx
2G
Based on Eq. (4.50), it can be remarked that strains caused by mechanical loads can
induce both a variation in size and a variation in shape of a material, while strains
caused by thermal loads can only cause a change in size. From Eq. (4.50) it can also be
obtained an expression that links the volumetric strain ε v 5 ε kk 5 ε xx 1 ε yy 1 ε zz to
the sum of the normal stresses σ kk 5 σ xx 1 σ yy 1 σ zz 5 3p. This formulation reads

ε v 5 p 2 3α T 2 T 0 Þ ð4:51Þ
ð
K
where K is the bulk modulus of the material.
The expressions of the stress strain relations written thus far can be formulated in
many other equivalent forms depending on the employed combination of the elastic
properties of the material. Table 4.2 presents typical relationships between the elastic
properties employed herein.

Table 4.2 Relationships between some elastic constants.
Parameter to define
Parameters Shear Young’s Poisson’s Bulk modulus,
available modulus, G modulus, E ratio, ν K
G; E E 2 2G GE
33G 2 EÞ
ð
2G
G; ν 2G 1 1 νÞ 2G 1 1 νÞ
ð
ð
ð
31 2 2νÞ
G; K 9KG 3K 2 2G
23K 1 GÞ
ð
3K 1 G
E; ν E E
ð
21 1 νÞ 31 2 2νÞ
ð
E; K 3KE 3K 2 E
9K 2 E 6K
ν; K 3K 1 2 2νÞ 3K 1 2 2νÞ
ð
ð
21 1 νÞ
ð