Page 176 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 149

It can be noted that by definition the shear stresses are not influenced by adding a
volumetric component in the stress tensor formulation (4.13).

4.5.3 Principal stresses

A feature of the stress tensor, similar to the strain tensor, is that at every material point
of any coordinate system there exist three mutually perpendicular planes, called princi-
pal planes, along which zero shear stresses are observed. The normal stresses on these
planes are called principal stresses.
The stress vector acting on a principal plane is characterised by only the normal
component. Therefore considering Eq. (4.10) and by indicating with n i the unit vector
of a principal plane characterised by direction cosines n x ; n y and n z in a rectangular
Cartesian coordinate system x; y; z, and with λ the modulus of the corresponding
component, the stress vector can be expressed as

ð4:17Þ
σ ji n j 5 λ n i
Eq. (4.17) is equivalent to the following eigenvalue problem

σ ij 2 λ δ ij n j 5 0 ð4:18Þ
that in extended form reads

8
ðσ xx 2 λ Þn x 1 σ xy n y 1 σ xz n z 5 0
>
<
σ xy n x 1 ðσ yy 2 λ Þn y 1 σ yz n z 5 0 ð4:19Þ
>
σ xz n x 1 σ zy n y 1 ðσ zz 2 λ Þn z 5 0
:
with
2
2
2
n 1 n 1 n 5 1 ð4:20Þ
x y z
Eqs (4.19) and (4.20) lead to a nontrivial solution for the direction cosines only if
the determinant of the coefficients is zero. This condition leads to the characteristic
cubic equation (also called characteristic polynomial)
3 2
λ 2 I 1 λ 1 I 2 λ 2 I 3 5 0 ð4:21Þ
where I 1 ; I 2 and I 3 are three coefficients that do not change for different coordinate
transformations, that is they are invariants. Any linear combination of invariants still is
an invariant.
The solution of the above equation is found for three real eigenvalues
λ 1 5 σ 1 ; λ 2 5 σ 2 and λ 3 5 σ 3 that represent the principal stresses. Differently to the
components of the stress tensor that change for different coordinate systems taken