Flow plasticity theory

Definition

Work with small strains at a material point, with strain $\varepsilon$, stress $\sigma$ and a vector $q$ of internal (hardening) variables, all functions of time. A rate-independent flow plasticity model consists of:

1. The additive split $\dot\varepsilon = \dot\varepsilon^e + \dot\varepsilon^p$ and the elastic law $\dot\sigma = C : \dot\varepsilon^e$, with $C$ an elasticity tensor.
2. A yield function $f(\sigma, q)$, whose zero set in $\sigma$ is the current yield surface, and the constraint $f(\sigma, q) \le 0$.
3. A flow rule $\dot\varepsilon^p = \dot\lambda \, \dfrac{\partial g}{\partial \sigma}(\sigma, q)$, where $g$ is a function called the plastic potential and $\dot\lambda$ is the plastic multiplier.
4. A hardening law $\dot q = \dot\lambda \, h(\sigma, q)$ for a given function $h$.
5. The loading and unloading conditions $\dot\lambda \ge 0$, $f(\sigma, q) \le 0$, $\dot\lambda \, f(\sigma, q) = 0$, and the consistency condition $\dot\lambda \, \dot f = 0$.