User:Inwind/stress tensor

Figure 1.1. Stress in a loaded deformable material body assumed as a continuum.

In general, however, the stress is not uniformly distributed over a cross section of a material body, and consequently the stress at a point on a given area is different than the average stress over the entire area. Therefore, it is necessary to define the stress not at a given area but at a specific point in the body (Figure 1.1). According to Cauchy, the stress at any point in an object, assumed to be a continuum, is completely defined by the nine components $\sigma _{ij}\,\!$ of a second order tensor known as the Cauchy stress tensor, ${\boldsymbol {\sigma }}\,\!$: The components of stress sij are commonly written in matrix format:

$[\sigma ]=\left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right]\,\!$

The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress.

The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations. For large deformations, also called finite deformations, other measures of stress are required, such as the first and second Piola-Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor.