AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

CHAPTER 3
ADVANCED APPLICATIONS OF THE AIRY STRESS FUNCTION
3.1 Finite Plate with a Hole Subjected to Tensile Loading
Applying the same approach as the finite multiply connected domains in
Chapter 2 for the plate shown below, the respective potential functions would be the
same as discussed in (2.3.2).
Figure 3-1 Finite Plate with a Hole Subjected to Tensile Loading.
Assume a finite plate of length 2l and width 2c with a central hole of radius a
subjected to uniform tension S along the x-axis. This is the case of a finite multiply
connected region, whose complex potential functions can be expressed as (2.3.2).
Sinceψ ( z) = χ ′ ( z)
, integrating the second complex function in (2.3.2) yields
n ⎛ κFk
⎞ *
χ ( z)
= ∫ ⎜ ∑ log( z − zk ) + ψ ( z) ⎟dz,
⎝ 2π ( 1 + κ ) ⎠
k = 1
17
(3.1.1)
where k = 1 since there is only one internal boundary and zk is 0 since the center of the
circle is taken as the origin (0,0) and γ * (z) and ψ * (z) are arbitrary analytic functions.
2 l
Y
a
X
2c
S