Page 223 - Analysis and Design of Energy Geostructures

196 Analysis and Design of Energy Geostructures

If the limit for h-0 is considered, the term corresponding to body
force vanishes and the average stresses reduce to the value reached at the
point O. This value is expressed by the following limit

dI

lim 5 t i
dA-0 dA
where the vector dI applied at the point O is the reduction of the actions
on the small element of surface dA, oriented by the unit outward normal
vector n i . The vector t i is called traction or stress vector.
Therefore since:
area AOC
n x 5
area ABC

area AOB
n y 5
area ABC

area BOC
n z 5
area ABC

the equilibrium equations can be written in the following matrix form:
n 2
8 9 38 9
x
< t = σ x τ xy τ xz < n x =
t n 5 τ yx σ y τ yz 5 n y
4
y
t τ zx τ zy σ z n z
: n ; : ;
z
and in tensor notation
n
t 5 σ ij n j
i
j. The stress tensor in terms of its deviatoric and spherical components
reads:
σ ij 5 pδ ij 1 s ij

where σ ij [Pa] is the total stress tensor, p is the mean stress, δ ij [-] is the
Kronecker delta and s ij [Pa] is the deviatoric stress tensor.
In matrix form, the stress tensor reads: