CONTINUUM MECHANICS for ENGINEERS

6.7 Airy Stress Function
As stated in Section 6.5, the underlying equations for two-dimensional problems
in isotropic elasticity consist of the equilibrium relations, Eq 6.5-2, the
compatibility condition, Eq 6.5-4 and Hooke’s law, either in the form of
Eq 6.5-5 (plane stress), or as Eq 6.5-14 (plane strain). When body forces in
Eq 6.5-2 are conservative with a potential function V = V(x 1, x 2) such that b i =
–V i, we may introduce the Airy stress function, φ = φ(x 1, x 2) in terms of which
the stresses are given by
σ 11 = φ ,22 + ρV; σ 22 = φ ,11 + ρV; σ 12 = –φ ,12 (6.7-1)
Note that by using this definition the equilibrium equations are satisfied
identically.
For the case of plane stress we insert Eq 6.5-5 into Eq 6.5-4 to obtain
which in terms of φ becomes
σ 11,22 + σ 22,11 – ν(σ 11,11 + σ 22,22) = 2(1 + ν)σ 12,12
(6.7-2)
φ ,1111 + 2φ ,1212 + φ ,2222 = –(1-ν) ρ (V ,11 + V ,22) (6.7-3)
Similarly, for the case of plane strain, when Eq 6.5-14 is introduced into
Eq 6.5-4 the result is
or in terms of φ
(1 – ν) (σ 11,22 + σ 22,11) – ν(σ 11,11 + σ 22,22) = 2σ 12,12
(6.7-4)
φ ,1111 + 2φ ,1212 + φ ,2222 = –(1 – 2ν) ρ (V ,11 – V ,22)/(1 – ν) (6.7-5)
If the body forces consist of gravitational forces only, or if they are constant
forces, the right-hand sides of both Eqs 6.7-3 and 6.7-5 reduce to zero and φ
must then satisfy the bi-harmonic equation
φ ,1111 + 2φ ,1212 + φ ,2222 = 4 φ = 0 (6.7-6)
In each case, of course, boundary conditions on the stresses must be satisfied
to complete the solution to a particular problem. For bodies having a rectangular
geometry, stress functions in the form of polynomials in x 1 and x 2
are especially useful as shown by the examples that follow.