Prandtl–Reuss equations

Remarks

The elastic part is the rate form of Hooke's law. The plastic part is the associated flow rule for the von Mises yield function, so the equations form a flow plasticity model. Since $\tr s = 0$, plastic flow does not change the volume. In perfect plasticity, the deviatoric part of the equations gives $\dot s = 2G(\dot e - \dot\lambda s)$, where $\dot e$ is the deviatoric part of $\dot\varepsilon$. The consistency condition $s : \dot s = 0$ during plastic loading then gives $\dot\lambda = \dfrac{s : \dot\varepsilon}{s : s}$. Example: in pure shear with $\sigma_{12} = \tau$ at yield, $\tau = \sigma_y / \sqrt3$, and $\dot\varepsilon_{12} = \dot\tau / (2G) + \dot\lambda \tau$. Non-example: the Lévy–Mises equations $\dot\varepsilon = \dot\lambda s$ drop the elastic part, so they describe a rigid perfectly plastic material and are not the Prandtl–Reuss equations.