Crystal plasticity

Definition

Consider a single crystal under small strains, with Cauchy stress $\sigma$. Its infinitesimal strain is split as $\varepsilon = \varepsilon^e + \varepsilon^p$ into elastic and plastic parts. A slip system $\alpha$, for $\alpha = 1, \dots, N$, is a pair of orthogonal unit vectors $(s^\alpha, m^\alpha)$ fixed by the crystal lattice. Here $s^\alpha$ is the slip direction and $m^\alpha$ is the normal of the slip plane. A crystal plasticity model is a constitutive model with the following three parts.

1. Kinematics: the plastic strain rate is a sum of shears on the slip systems. It is $\dot\varepsilon^p = \sum_{\alpha=1}^N \dot\gamma^\alpha \tfrac12 (s^\alpha \otimes m^\alpha + m^\alpha \otimes s^\alpha)$, where $\dot\gamma^\alpha$ is the slip rate of system $\alpha$.
2. Driving force: the resolved shear stress on system $\alpha$ is $\tau^\alpha = s^\alpha \cdot \sigma m^\alpha$.
3. Flow rule: system $\alpha$ can slip only when $\abs{\tau^\alpha}$ reaches its critical resolved shear stress $\tau_c^\alpha > 0$ (Schmid's law). In rate-dependent versions, $\dot\gamma^\alpha$ is a given increasing function of $\tau^\alpha / \tau_c^\alpha$. Hardening laws describe how the $\tau_c^\alpha$ evolve.