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 three-dimensional 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
"«iWVfcliil-ili))^!
1 f*i; + <Ai
/
Y
-
= ηΙ-
7
^δ(Χ
ι
-Χ
ιί
)άΧ
1
U
(4^)
=
^"
(ili)
(2-84)
^1
v
^1
/ V^l