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Airy's Stress Function (Optional Section) (§ 2.6.0)
The solution to the stress field ahead of a crack tip problem must satisfy all equilibrium requirements (equilibrium and compatibility). Since we are dealing with partial differential equations, finding such a solution can be quite difficult. A particular technique used in a differential equations course is to construct a general function that satisfies the partial differential equations by definition. This same technique is used in the solution of this stress field problem, except the general function is called Airy's Stress Function (named after Sir George Biddell Airy who first used this methodology). According to Airy's stress function, Equations 43, 44, and 45 hold true:
Remember the equilibrium condition (covered in § below:
By plugging in the equations associated with Airy's Stress Function into the equilibrium condition, we can illustrate that the functions do indeed satisfy equilibrium. You'll find that the expressions in red are parts of Airy's Stress Function (Equations 43-45):
δ
δx | ( | δ2ψ
δy2 | ) + | δ
δy | (- | δ2ψ
δxδy | ) = 0 |
| (47) |
| | δ3ψ
δxδy2 | - | δ3ψ
δxδy2 | = 0 |
| (48) |
Next, we can investigate the relationship between the constitutive relations and Airy's Stress Function. Remember our compatibility condition:
| | δ2εxx
δy2 | + | δ2εyy
δx2 | - | δ2γxy
δxδy | = 0 |
| (49) |
To begin, we will substitute our constitutive relations (found in § 2.1.3: Linear Elastic Behavior of this tutorial) into our compatibility equation:
δ2
δy2 | ( | 1
E | (σxx -νσyy) | ) | + | δ2
δx2 | ( | 1
E | (σyy - νσxx) | ) - | δ2
δxδy | ( | (2 + 2ν)
E | τxy) = 0 |
| (50) |
Like we did above with our equilibrium conditions above, we'll substitute in the components of Airy's Stress Function (in red below) into our compatibility condition and simplify.
| δ2
δy2 | ( | 1
E | ( | δ2ψ
δy2 | - | νδ2ψ
δx2 | ) | ) | + | δ2
δx2 | ( | 1
E | ( | δ2ψ
δx2 | - | νδ2ψ
δy2 | ) | ) | - | δ2
δxδy | ( | (2+2ν)
E | ( | -δ2ψ
δxδy | ) | ) = 0 |
| (51) |
This simplifies to:
1
E | ( | δ4 ψ
δ y4 | - ν | δ4
δy2x2 | + | δ4ψ
δx4 | -ν | δ4ψ
δx2y2 | ) + | 2
E | ( | δ4ψ
δx2y2 | + ν | δ4ψ
δx2y2 | ) | = 0 |
| (52) |
| | δ4ψ
δx4 | + 2 | δ4ψ
δx2δy2 | + | δ4ψ
δy4 | = 0 |
| (53) |
Equation 53 is also known in mathematics as the Biharmonic Equation and can be expressed compactly as:
where ∇2 is the Laplacian Operator.
This equation is the counterpart of the PDE, Equation 34, developed before the introduction of Airy's Stress Function.
Next: Complex Stress Functions (Optional) (§ 2.7.0)
Up: Tutorial Contents
Previous: Boundary Conditions (§ 2.5.0)
August 9, 2004
