Airy Stress Function
Solving 3 equations in 3 unknowns can be reduced to a simpler form. The 3 equations
can be reduced to a single equation containing some function φ(x, y), which we call the
Airy stress function.
The Airy stress function will satisfy the equilibrium equations when this is true:
∂ 2φ
σxx = 2 ∂y
∂ 2φ
σxy = −
∂x∂y ∂ 2φ
σyy = 2
∂x
NOTE: Eq. 8.55 in Pollard and Fletcher (2005), p.310, includes body forces.
But
€ Substituting, we get:
€ φ(x, y) must also satisfy
€ the compatibility equation.
∂ 4φ
∂ 4φ
∂ 4φ
4 +2
2
2 +
4 = 0 ∂x
∂x ∂y ∂y
This may also be written as:
∇ 4 φ = 0 (a fourth degree, biharmonic, partial differential equation) €
Thus, the solution of a 2D problem reduces to finding a solution for a single equation.
€
Any stress function which satisfies this equation is a valid stress function from which
stresses, strains, and displacements can be calculated.
The stresses must also satisfy the boundary conditions for any particular problem.
An interesting result is that the stress components are independent of the elastic moduli. So any elastic
material subject to the same loading with the same geometry will have the same state of stress. This is
why we can use an inexpensive or easy to prepare material to model elastic deformation in a more
expensive of difficult to prepare material.