Page 175 - Analysis and Design of Energy Geostructures

148 Analysis and Design of Energy Geostructures

components, σ kl , act tangential to the coordinate planes and are called shear stresses.
More generally, when considering each component of the stress tensor, the first sub-
script indicates the coordinate plane on which the stress component acts and the sec-
ond subscript identifies the direction along which it acts.
The symmetric character of the stress tensor can be proven considering the princi-
ple of balance of angular momentum (cf. Section 4.5.3). As a result, the complemen-
tary components of the shear stress are equal

ð4:12Þ
σ xy 5 σ yx
σ yz 5 σ zy
σ zx 5 σ xz
The above implies that there are only six independent stress components: three
normal stresses σ xx ; σ yy ; σ zz and three shear stresses σ xy ; σ yz ; σ xz .
Relation (4.10) expresses that when the components σ ij acting on any three mutu-
ally perpendicular planes through a point H are known, the stress vector on any plane
through H can be determined. This fact is formalised by the Cauchy’s theorem
(Cauchy, 1823) [for the full development, see, e.g. Lancellotta (2008)].

4.5.2 Volumetric and deviatoric stresses
In many cases, it is useful to decompose the stress tensor in a volumetric (i.e. spherical)
part and a deviatoric (i.e. distortional) part. The above can be mathematically expressed as

ð4:13Þ
σ ij 5 pδ ij 1 s ij
where p is a scalar quantity called mean stress and s ij is a tensor characterised by zero
trace called deviatoric stress tensor (or stress deviator). The mean stress can be written
in rectangular Cartesian coordinates as
1 1 1
p 5 σ kk 5 tr σ ij 5 ð4:14Þ
3 3 3 σ xx 1 σ yy 1 σ zz
The deviatoric stress tensor in rectangular Cartesian coordinates reads

2 3
s xx
s ij 5 4 σ yx σ xy σ xz 5 ð4:15Þ
σ yz
s yy
where s ii 5 σ ii 2 p. σ zx σ zy s zz
Eq. (4.13) can therefore be rewritten in matrix form as

2 3 2 3 2 3
00
p s xx σ xy σ xz
σ xz
σ xx
σ xy
5 0 0 1
4 σ yx σ yy σ yz 5 4 p 5 4 σ yx s yy σ yz 5 ð4:16Þ
0 0 p σ zx σ zy s zz
σ zx
σ zz
σ zy