Stress Rates

In practice, the true stress (or Cauchy stress) for any time interval will be computed using the stress rate in an explicit time integration:(1)

σ_{i j} (t + δ t) = σ_{i j} (t) + {\dot{σ}}_{i j} δ t

${\dot{σ}}_{i j}$ 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 ${\dot{σ}}_{i j}$ 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

A stress law will be objective if it is independent of the space frame. To each definition of the rigid body rotation, corresponds a definition of the objective stress rate. The Jaumann objective stress tensor derivative will be associated with the rigid body rotation defined in Kinematic Description, Equation 14:(2)

{\dot{σ}}^{v}_{i j} = {\dot{σ}}_{i j} - {\dot{σ}}^{r}_{i j}

Where:

${\dot{σ}}^{v}_{i j}$: Jaumann objective stress tensor derivative
${\dot{σ}}^{r}_{i j}$: Stress rate due to the rigid body rotational velocity

The correction for stress rotation is given by:(3)

{\dot{σ}}^{r}_{i j} = σ_{i k} Ω_{k j} + σ_{j k} Ω_{k i}

and $Ω_{k j}$ 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.