Page 222 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 195

σ yz 5 σ zy

σ zx 5 σ xz

i. To prove this statement, the equilibrium of a small portion of a contin-
uum body in the shape of a tetrahedron, such as that reported in the
following figure, is considered. If n i is the unit outward vector normal to
the surface ABC (whose components n x , n y , n z are its direction cosines
n
n
n
with respect to the coordinate axes), the components t , t , t acting on
x y z
this surface can be obtained by considering the tetrahedron equilibrium.
Z
A

n
i
σ
τ x
xy
σ (n)
z
σ τ yx τ xz
σ y (n)
y
(n)
σ x
τ y
yz
O
C
τ
zx
τ
zy
σ
z
B
x
n
ð
The force equilibrium in the x direction reads t area ABCÞ
x
ð
5 σ x area AOCð Þ 1 τ yx area AOBÞ 1 τ zx area BOCÞ 2 ρb x dV and similar
ð
equations are written in the y and z directions. In the previous equation,
3
the vector b [N/kg] represents the body force per unit mass, ρ [kg/m ]is
3
the density, V [m ] is the volume and the stresses are the average stresses
acting on the faces of the tetrahedron. The volume of the tetrahedron
2
can be expressed in the form dV 5 1=3ðh dSÞ, if dS [m ] is the area of the
surface ABC and h [m] is its distance from the point O.