Kirchhoff stress tensor

The most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several other measures of stress can be defined. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are:

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 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.

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