Associated flow rule

Remarks

Then $\dot\varepsilon^p$ is an outward normal to the yield surface at $\sigma$. At a corner of a non-smooth yield surface, such as those of the Tresca hexagon, the rule says that $\dot\varepsilon^p$ lies in the cone of outward normals at that point. Example: for $f(\sigma) = \sqrt{3J_2} - \sigma_y$, with $s$ the deviatoric stress and $J_2 = \tfrac12 s:s$, one has $\partial f / \partial \sigma = \frac{3 s}{2 \sqrt{3 J_2}}$. So $\dot\varepsilon^p$ is parallel to $s$, and $\tr \dot\varepsilon^p = \tr s = 0$: plastic flow does not change volume. Non-example: for soils with the Drucker–Prager yield function, models often take a plastic potential $g \ne f$ with a smaller pressure term, to reduce the dilatancy; such a flow rule is non-associated.