Objective stress rate

Remarks

Example: the Jaumann rate $\overset{\nabla}{\sigma} = \dot\sigma - W \sigma + \sigma W$ is objective. Under a change of observer, $\dot\sigma^* = Q \dot\sigma Q^T + \dot Q \sigma Q^T + Q \sigma \dot Q^T$ and $W^* = Q W Q^T + \dot Q Q^T$. Substituting, the terms with $\dot Q$ cancel, because $\dot Q Q^T$ is skew-symmetric.

Non-example: the material derivative $\dot\sigma$ itself is not objective. The formula above shows that $\dot\sigma^*$ differs from $Q \dot\sigma Q^T$ by $\dot Q \sigma Q^T + Q \sigma \dot Q^T$, which is not zero in general when $Q$ depends on time. For instance, with $Q(t)$ a rotation about $e_3$ at constant angular speed and $\sigma = e_1 \otimes e_1$, this extra term is nonzero.

Other standard objective rates are the Truesdell rate $\dot\sigma - L \sigma - \sigma L^T + (\tr L) \sigma$ and the upper convected (Oldroyd) rate $\dot\sigma - L \sigma - \sigma L^T$. The Green–Naghdi rate has the form of the Jaumann rate with $\dot R R^T$ in place of $W$. Here $R$ is the rotation in the polar decomposition of the deformation gradient.