Stress Function

A stress function is an auxiliary function whose partial derivatives represent components of a stress tensor. Such representations can satisfy the equilibrium equations identically, leaving compatibility equations and boundary conditions to determine the function.

For a two-dimensional stress field with no body forces, an Airy stress function gives the components of the symmetric stress tensor by

			(1)
			(2)
			(3)

The equilibrium equations

			(4)
			(5)

follow by cancellation of mixed partial derivatives. Conversely, every sufficiently differentiable symmetric two-dimensional stress field satisfying these equations has an Airy stress function locally, and globally on a simply connected domain. Adding an affine function,

(6)

does not change the stress tensor.

For homogeneous, isotropic, linear elasticity with no body forces or imposed strains, the compatibility equations for the strain tensor require the biharmonic equation

(7)

This equation applies to both plane stress () and plane strain () (Timoshenko and Goodier 1951, pp. 24-27). It is a compatibility requirement in addition to equilibrium. On domains with holes, these local equations need not ensure single-valued displacement.

For example, the quadratic polynomial

(8)

gives the uniform stress tensor

(9)

It satisfies the biharmonic equation for arbitrary constants , , and (Timoshenko and Goodier 1951, p. 29).

The Prandtl stress function is a different construction. It uses first partial derivatives to represent the shear stresses of a uniformly twisted shaft and is determined by a Poisson problem. Thus the differential equation satisfied by a stress function depends on the representation and the elasticity problem.