AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

σ
σ
x
y
( ± l, y) = S,
( x, ± c)
= 0,
τ ( ± l, y) = τ ( x, ± c)
= 0,
xy xy
σ r ( ± a, ± a) − τ rθ
( ± a, ± a)
= 0.
Solving for the Airy stress function, we get
2 2 2 2 2
2 2
2 2
S( x − y )( − 6a + x + y ) + 12b0 x( a − y ) + 12a0
x( a − y )
φ ( x, y)
=
2 2
.
12(
a − y )
19
(3.1.4)
(3.1.5)
From the above result we can see that the Airy stress function is an indefinite
value. Therefore, we can conclude that for a finite plate with a central hole infinite
series having appropriate boundary conditions has to be taken into consideration to get a
valid solution.
3.2 Infinite Plate with a Hole Subjected to Tensile Loading
Consider an infinite plate with a central hole subjected to uniform tensile far
∞
field loading σ x = S in the x direction. From (2.3.4), we get the complex potentials to
be
m
∑
Fk
∞ ∞
σ σ
k
x +
= 1
y **
γ ( z)
= − log z + z + γ ( z),
2π ( 1 + κ ) 4
m
∑
κ Fk
∞ ∞ ∞
σ σ iτ
k
y − x + 2
= 1
xy **
ψ ( z)
= log z +
z + ψ ( z),
2π ( 1 + k)
2
where Fk is the resultant force on the central hole, but the logarithmic part of the
equations is omitted as it corresponds to discontinuity in the displacements or