In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined:

Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.

In the reference configuration ${\displaystyle \Omega _{0}}$, the outward normal to a surface element ${\displaystyle d\Gamma _{0}}$ is ${\displaystyle \mathbf {N} \equiv \mathbf {n} _{0}}$ and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is ${\displaystyle \mathbf {t} _{0}}$ leading to a force vector ${\displaystyle d\mathbf {f} _{0}}$. In the deformed configuration ${\displaystyle \Omega }$, the surface element changes to ${\displaystyle d\Gamma }$ with outward normal ${\displaystyle \mathbf {n} }$ and traction vector ${\displaystyle \mathbf {t} }$ leading to a force ${\displaystyle d\mathbf {f} }$. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity ${\displaystyle {\boldsymbol {F}}}$ is the deformation gradient tensor, ${\displaystyle J}$ is its determinant.

The Cauchy stress (or true stress) is a measure of the force acting on an element of area in the deformed configuration. This tensor is symmetric and is defined via

or

where ${\displaystyle \mathbf {t} }$ is the traction and ${\displaystyle \mathbf {n} }$ is the normal to the surface on which the traction acts.

The quantity,

is called the Kirchhoff stress tensor, with ${\displaystyle J}$ the determinant of ${\displaystyle {\boldsymbol {F}}}$. It is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation). It can be called weighted Cauchy stress tensor as well.