Drucker stability

Remarks

Here $A : B = \sum_{i,j} A_{ij} B_{ij}$. Take a plasticity model with a smooth yield function $f$. Drucker stability implies that the elastic domain is convex. It also implies that $d\varepsilon^p$ is a nonnegative multiple of $\partial f / \partial \sigma$, normal to the yield surface. Example: a bar with linear hardening, Young's modulus $E > 0$ and plastic modulus $H > 0$, under uniaxial plastic loading has $d\sigma = \frac{EH}{E + H} d\varepsilon$ and $d\varepsilon^p = d\sigma / H$, so $d\sigma\,d\varepsilon > 0$ and $d\sigma\,d\varepsilon^p \ge 0$. Non-example: a softening bar, with $d\sigma = -k\,d\varepsilon$ for some $k > 0$ during plastic loading, has $d\sigma\,d\varepsilon < 0$, so clause 1 fails. A perfectly plastic material satisfies clause 1 only with $\ge$ in place of $>$; it is said to be stable in the weaker, limiting sense.