2DAiryStressfunktion

The 2D Airy stress function, also called the Airy stress function in two dimensions, is a scalar potential used in plane elasticity to formulate stress fields in a way that automatically satisfies equilibrium. It is especially useful for solving two-dimensional problems with complex geometries and boundary conditions, under the assumption of isotropic, homogeneous materials and no body forces.

If φ(x, y) denotes the Airy stress function, the in-plane stress components are given by

σ_xx = ∂^2φ/∂y^2,

σ_yy = ∂^2φ/∂x^2,

σ_xy = −∂^2φ/∂x∂y.

These relations ensure that the equilibrium equations ∂σ_xx/∂x + ∂σ_xy/∂y = 0 and ∂σ_yx/∂x + ∂σ_yy/∂y = 0 are identically satisfied, provided φ is sufficiently smooth.

In the absence of body forces, φ must satisfy the biharmonic equation

∇^4 φ = ∂^4φ/∂x^4 + 2∂^4φ/∂x^2∂y^2 + ∂^4φ/∂y^4 = 0.

This single scalar partial differential equation replaces the original set of coupled equilibrium and compatibility conditions, reducing the problem to finding a φ that meets boundary conditions on the domain.

Boundary conditions can be expressed in terms of φ through the traction vector t = [σ · n], where n is the outward normal. For example, the boundary tractions t_x and t_y relate to φ via t_x = φ_yy n_x − φ_xy n_y and t_y = −φ_xy n_x + φ_xx n_y. Alternatively, displacement or mixed boundary conditions can be incorporated via constitutive relations and strain-displacement formulas.

Applications include problems such as plates with holes, cracks, or complex shapes, where the Airy function method converts elasticity problems into biharmonic-function problems. The approach is connected to complex-variable methods and has a long historical role in linear elasticity.