In mechanics of continuous media, the stress tensor, also called stress tensor or stress tensor is the tensor that accounts for the distribution of internal stresses and forces in the continuous medium.

Cauchy tension tensor

Cauchy's theorem on the stresses of a body establishes that given a distribution of internal stresses on the geometry of a deformed continuous medium, which satisfies the conditions of Cauchy's principle, there is a symmetric tensor field T defined on the deformed geometry with the following properties:

The third property means that this tensor will be given over the coordinates specified by a symmetric matrix. It should be noted that in a mechanical problem a priori it is difficult to know the Cauchy tension tensor since it is defined on the geometry of the body once deformed, and this is not known in advance. Therefore, it is first necessary to find the deformed shape to know exactly the Cauchy tensor. However, when the deformations are small, in engineering and practical applications this tensor is used, although defined on the coordinates of the undeformed body (which does not lead to excessive calculation errors if all maximum deformations are less than 0.01).

Given an orthogonal reference system, the Cauchy tension tensor is given by a symmetric matrix, whose components are:

The third form is the common way of calling the components of the tension tensor in engineering.

First Piola-Kirchhoff tension tensor

Piola-Kirchhoff tensors are introduced to avoid the difficulty of having to work with a tensor defined on the already deformed geometry (which is usually not known in advance). The so-called first Piola-Kirchhoff tensor () and the so-called second Piola-Kirchhoff tensor () are defined. The relationship between both tensors is given by:.

Where is the deformation gradient tensor. This tensor, however, has the problem that it is not symmetric (see second Piola-Kirchhoff tension tensor).

Second Piola-Kirchhoff tension tensor

This tensor is introduced to achieve a tensor defined on the geometry prior to deformation and which is also symmetrical, unlike the first Piola-Kirchhoff tensor which does not have to be symmetrical. The second Piola-Kirchhoff tension tensor is given by:.

Kirchhoff tension tensor

The tensor given by:.