AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

can be clubbed with the finite simply connected domain, by maintaining the equilibrium
and continuity of stresses and displacements at the boundary of the two phases.
3.3.1 Stress Field Inside the Two Dimensional Circular Inclusion
The two-dimensional circular disc is similar to a finite simply bounded region.
Complex potentials are assumed to be
∑
n
γ ( z) = c z ,
n=
0
∑
n
ψ ( z) = d z .
2
2
n=
0
24
n
n
(3.3.1)
Since the disc is going to be clubbed with the infinite plate, the same boundary
conditions apply to it. Therefore we take the first three terms.
2
γ ( z) = c + c z + c z ,
0 1 2
2
ψ ( z) = d + d z + d z ,
0 1 2
2 3
d1z d 2z
χ ( z) = d0z + + .
2 3
The Airy stress function in polar form, z re iθ
= , is expressed as
2 2
2 2
3
φ ( r, θ) = c r cosθ + c r cos θ + c r sin θ + c r cosθ cos2θ
0 1
3
+ c r sin θ sin2θ + d r cosθ cos2θ + d r sin θ sin 2θ
2
1 2
1 2
+ d1r cosθ cos3θ + d1r sin θ sin3θ
2
2
1 3
1 3
+ d2r cosθ cos4θ + d2r sin θ sin 4θ.
3
3
1
0 0
The stresses in polar form using (2.1.6) are expressed as
2
(3.3.2)
(3.3.3)