AIRY STRESS FUNCTION FOR TWO DIMENSIONAL INCLUSION ...

epresentation
It is observed that these equations will be identically satisfied by choosing a
∂ φ
σ x =
∂x
∂ φ
σ y =
∂x
∂ φ
τ xy =
∂x∂y −
2
2 ,
2
2 ,
2
.
where φ = φ(
x, y) is called the Airy stress function.
be written as
7
(2.1.3)
The compatibility relationship, assuming no body forces, in terms of stress can
where ∇ is the Laplace operator.
(2.1.3), we get
2
∇ ( σ + σ ) = 0,
x y (2.1.4)
Now representing the relation in terms of the Airy stress function using relations
4
4 4
∂ φ 2∂
φ ∂ φ 4
4 + 2 2 + 4 = ∇ φ = 0.
(2.1.5)
∂x
∂x ∂y
∂y
This relation is called the biharmonic equation, and its solutions are known as
biharmonic functions[1]. Now that we have the problem of elasticity reduced to a
single equation in terms of the Airy stress function, φ , it has to be determined in the
two-dimensional region R bounded by the boundary S as shown in Figure 2-1.
Appropriate boundary conditions over S are necessary to complete the solution.