# Stress Rates

${\dot{\sigma}}_{ij}$ is not simply the time derivative of the Cauchy stress tensor as Cauchy stress components are associated with spatial directions in the current configuration. So, the derivatives will be nonzero in the case of a pure rigid body rotation, even if from the constitutive point of view the material is unchanged. The stress rate is a function of element average rigid body rotation and of strain rate.

For this reason, it is necessary to
separate
${\dot{\sigma}}_{ij}$
into two parts; one related to the rigid body motion and the
remainder associated with the rate form of the stress-strain law. Objective stress rate is used,
meaning that the stress tensor follows the rigid body rotation of the material. ^{1}

- ${\dot{\sigma}}^{v}{}_{ij}$
- Jaumann objective stress tensor derivative
- ${\dot{\sigma}}^{r}{}_{ij}$
- Stress rate due to the rigid body rotational velocity

and ${\Omega}_{kj}$ defined in Kinematic Description, Equation 14 (Isotropic Linear Elastic Stress Calculation).

^{1}Halphen B., “On the velocity field in thermoplasticity finished”, Laboratoire de Mécanique des Solides, Ecole Polytechnique, International Journal of Solids and Structures, Vol.11, pp 947-960, 1975.