AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

Sz
b b
χ ( z)
= (log z) b .
z z
−
2
2 3
+ 1 − − 2
4 2
A plate with a hole is better manipulated using the polar coordinates.
21
(3.2.2)
Substituting z re iθ
= in (3.2.1) and (3.2.2) gives the Airy stress function in terms
of the polar coordinates as
1 4 2
φ ( r,
θ) = 2 ( Sr sin θ − b3 cos2θ + b1r log r
2r
2 2
− 2b cosθ + 2a r cos2θ + 2a cos3θ + 2a cos 4θ).
2 1 2 3
(3.2.3)
Applying (2.1.7) to the above Airy stress function, we get the stresses in polar
form as follows:
1 4 4
2
σ rr = 4 ( Sr + Sr cos2θ + 2b1r + 4b2r
cosθ
2r
2
+ 6b cos2θ − 8a r cos2θ − 20a r cos 3θ
),
3 1
1 4 4
2
σ θθ = 4 ( Sr − Sr cos2θ − 2b1r − 4b2r
cosθ
2r
− 6b cos2θ + 4a r cos 3θ
),
3 2
1 4
τ rθ
= 4 ( − Sr sin 2θ + 4b2r sin θ + 6b3 sin2θ
2r
2
− 4a r sin 2θ − 12a r sin 3θ
).
1
The displacements can be determined using (2.2.11) as
u
r
1
4
4 4
= 3 ( − Sr + Sκ 1r
− 2Sr cos2θ
− 4b1r
8µ
r
1
2
− 4b r cosθ − 4b cos2θ + 4a r cos2θ
2 3 1
2
+ 4κ a r cos2θ + 8a r cos3θ + 4κ a r cos 3θ
),
1 1
2
2 1 2
2
2
(3.2.4)
(3.2.5)
(3.2.6)
(3.2.7)