Page 272 - Intro to Tensor Calculus

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We will consider these equations together with the compatibility equations (2.4.65). The equations
(2.4.66) will be automatically satisfied if we introduce a scalar function φ = φ(x, y) and assume that the
stresses are derivable from this function and the potential function V according to the rules:

2
2
2
∂ φ ∂ φ ∂ φ
σ xx = + V σ xy = − σ yy = + V. (2.4.67)
∂y 2 ∂x∂y ∂x 2
The function φ = φ(x, y) is called the Airy stress function after the English astronomer and mathematician
Sir George Airy (1801–1892). Since the equations (2.4.67) satisfy the equilibrium equations we need only
consider the compatibility equation(s).
For a state of plane strain we substitute the relations (2.4.63) into the compatibility equation (2.4.65)
and write the compatibility equation in terms of stresses. We then substitute the relations (2.4.67) and
express the compatibility equation in terms of the Airy stress function φ. These substitutions are left as
exercises. After all these substitutions the compatibility equation, for a state of plane strain, reduces to the
form
4
4
4
2
2
∂ φ ∂ φ ∂ φ 1 − 2ν ∂ V ∂ V
+2 + + + =0. (2.4.68)
2
∂x 4 ∂x ∂y 2 ∂y 4 1 − ν ∂x 2 ∂y 2
In the special case where there are no body forces we have V = 0 and equation (2.4.68) is further simplified
to the biharmonic equation.
4
4
4
∂ φ ∂ φ ∂ φ
4
∇ φ = +2 + =0. (2.4.69)
2
∂x 4 ∂x ∂y 2 ∂y 4
In polar coordinates the biharmonic equation is written
∂ 1 ∂ 1 ∂ ∂ φ 1 ∂φ 1 ∂ φ
2 2 2 2
4
2
2
∇ φ = ∇ (∇ φ)= + + + + =0.
2
2
∂r 2 r ∂r r ∂θ 2 ∂r 2 r ∂r r ∂θ 2
For conditions of plane stress, we can again introduce an Airy stress function using the equations (2.4.67).
However, an exact solution of the plane stress problem which satisfies all the compatibility equations is
difficult to obtain. By removing the assumptions that σ xx ,σ yy ,σ xy are independent of z, and neglecting
body forces, it can be shown that for symmetrically distributed external loads the stress function φ can be
represented in the form
νz 2 2
φ = ψ − ∇ ψ (2.4.70)
2(1 + ν)
4
where ψ is a solution of the biharmonic equation ∇ ψ =0. Observe that if z is very small, (the condition
of a thin plate), then equation (2.4.70) gives the approximation φ ≈ ψ. Under these conditions, we obtain
the approximate solution by using only the compatibility equation (2.4.65) together with the stress function
defined by equations (2.4.67) with V =0. Note that the solution we obtain from equation (2.4.69) does not
satisfy all the compatibility equations, however, it does give an excellent first approximation to the solution
in the case where the plate is very thin.
In general, for plane strain or plane stress problems, the equation (2.4.68) or (2.4.69) must be solved for
the Airy stress function φ which is defined over some region R. In addition to specifying a region of the x, y
plane, there are certain boundary conditions which must be satisfied. The boundary conditions specified for
the stress will translate through the equations (2.4.67) to boundary conditions being specified for φ. In the
special case where there are no body forces, both the problems for plane stress and plane strain are governed
by the biharmonic differential equation with appropriate boundary conditions.