Deformation tensor in canonical form

For a state of plane deformation it can be shown that there exists a Cartesian coordinate system XYZ, with the OZ axis coinciding with the barycentric axis of the elastic prism, in which the body deformation tensor has the form:

Using the same coordinates the tension tensor will take the form:.

Flat elastic problem

The approach of the elastic problem for a state of flat elasticity in the absence of volume forces leads to somewhat simpler equations than usual, so that both the Airy stress function method and the complex variable methods can be applied to find a solution. First, the equilibrium equations are reduced to just two independent equations:

While the compatibility equations are reduced to a single independent equation:

In fact, under the same previous conditions and for a linear elastic solid, it can be shown that the system formed by () and () is equivalent to the system formed by () and ():.

Airy stress function

In the previous conditions it can be proven that the solution of the system formed by () and () on a certain flat domain, simply connected, can be expressed in terms of a biharmonic function on said domain, the tensions being expressible in terms of this unique biharmonic function:[1]​.

Furthermore, it is true that:

It can be proven that every real function that satisfies the above equations can be expressed in terms of two holomorphic harmonic functions:

Deformation tensor in canonical form

For a state of plane deformation it can be shown that there exists a Cartesian coordinate system XYZ, with the OZ axis coinciding with the barycentric axis of the elastic prism, in which the body deformation tensor has the form:

Using the same coordinates the tension tensor will take the form:.

Flat elastic problem

The approach of the elastic problem for a state of flat elasticity in the absence of volume forces leads to somewhat simpler equations than usual, so that both the Airy stress function method and the complex variable methods can be applied to find a solution. First, the equilibrium equations are reduced to just two independent equations:

While the compatibility equations are reduced to a single independent equation:

In fact, under the same previous conditions and for a linear elastic solid, it can be shown that the system formed by () and () is equivalent to the system formed by () and ():.

Airy stress function

In the previous conditions it can be proven that the solution of the system formed by () and () on a certain flat domain, simply connected, can be expressed in terms of a biharmonic function on said domain, the tensions being expressible in terms of this unique biharmonic function:[1]​.

Furthermore, it is true that:

It can be proven that every real function that satisfies the above equations can be expressed in terms of two holomorphic harmonic functions: