48 POTENTIAL PROBLEMS
referred to the
X
l9
X
2
,
X
3
axes of orthotropy can be written as,
V£u
=
0 (2.77)
where for the threedimensional case,
Vl(
)
= ki
^x7
+k
>M7
+k
>^x7
(178)
or for two dimensions,
j
d
2
(
) d
2
( )
v
<
(
) =
fcl
^V
+
*
2
^7
(179)
The
k
t
terms define the material properties in the directions of
orthotropy.
If
we
now assume that a concentrated potential is acting at a point
T, the fundamental solution should satisfy the following equation:
V
k
2
w*h^
=
0 (2.80)
where S
t
can be written for three dimensions as,
δ(Χ
ι
Χ
ίί
)δ(Χ
2
Χ
2ί
)δ(Χ,Χ
3ί
)
(2.81)
The simplest way of finding the fundamental solution is to make the
following transformation:
We must now use the property that
^i^ii) = iU/Mii in)) = 4=5«!«!,) (2.83)
V
fc
i
which can be deduced by considering the integral
"«iWVfcliilili))^!
1 f*i; + <\Ai
/
Y
\

= ηΙ
7
^\δ(Χ
ι
Χ
ιί
)άΧ
1
U
(4^)
=
^"
(ili)
(284)
^1
\v
^1
/ V^l