AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

Since they are single-valued within the region R, they can be expressed as the following
series.
γ
ψ
*
n
( z) = a z ,
*
∞
∑
n=
0
∞
∑
n
n
( z) = b z .
n=
0
Substituting the above in the complex potential functions we get
γ ( z)
= −
ψ ( z)
=
χ ( z)
=
n
∑
k = 1
n
∑
k = 1
∫
⎛
⎜
⎝
18
n
Fk
2 3
log( z − zk ) + a0 + a1z + a2z + a3z + ...,
2π ( 1 + κ )
κFk
2 3
log( z − zk ) + b0 + b1z + b2z + b3z + ...,
2π ( 1 + κ )
n
∑
k = 1
κFk
⎞
2 3
log( z − zk ) + b0 + b1z + b2z + b3z + ... ⎟dz,
2π ( 1 + κ ) ⎠
(3.1.2)
(3.1.3)
where k=1 as there is only one internal boundary, zk = 0 as the center of the internal
boundary which is a circle is the origin (0, 0). In fact, the logarithmic part of the
equations is omitted as it corresponds to discontinuity in the displacements or
dislocation which does not exist in this case as it is an elastic plate. Therefore, the Airy
stress function of (2.2.8) is
∞
∞
⎡
n
n ⎤
φ ( z) = Re ⎢z
∑ anz + ∫ ∑ bnz dz⎥.
⎣ n=
0 n=
0 ⎦
For the given tensile loading the boundary conditions for the above plate will be