Page 183 - Analysis and Design of Energy Geostructures

156 Analysis and Design of Energy Geostructures

4.7.2 Traction boundary conditions
The boundary conditions for this case are expressed in terms of the stress components
in compact form as

ð4:43Þ
n
t 5 σ ji n i 5 p i
i
where p i is the vector of the prescribed stress components. Eq. (4.43) is equivalent to
the following equations written in extended form that need to be satisfied at every
point of the bounding surface:
8
σ xx n x 1 σ xy n y 1 σ xz n z 5 X
<
ð4:44Þ
σ yx n x 1 σ yy n y 1 σ yz n z 5 Y
:
σ zx n x 1 σ zy n y 1 σ zz n z 5 Z
where X, Y and Z are the components of the prescribed stress vector in the x; y; z
directions, respectively, and n x ; n y and n z are the direction cosines of the outward sur-
face normal. These formulas also give the components of the stress vector across any
interior surface.

4.7.3 Displacement boundary conditions
The boundary conditions for this case are expressed in compact form as

u i 5 F i ðHÞ ð4:45Þ

where u i is the displacement vector, F i is a vector containing prescribed displacement
functions and H is each of the points of the bounding surface at which Eq. (4.45)
needs to be satisfied. Eq. (4.45) is equivalent to the following equations in extended
form:
8
u 5 F 1 HðÞ
>
<
v 5 F 2 HðÞ ð4:46Þ
>
:
w 5 F 3 HðÞ
where F 1 ; F 2 and F 3 are prescribed functions.
In some cases, more complicated boundary conditions, which are generally defined
as mixed boundary conditions, may be encountered. For example the boundary con-
dition expressed in Eq. (4.44) may be specified over a portion of the bounding surface
while that expressed in Eq. (4.46) over the remainder of the surface (Boley and
Weiner, 1997). Another possible boundary condition may describe a support, in which
a functional relation exists between some of the displacement and some of the stress
vector components, as in the case of two materials in contact. The difficulties arising