Kinematic hardening

The KinematicHardening class manages the kinematic hardening evolution laws.

Drucker-Prager linear kinematic hardening.

$$\dot{X}_{ij}^{dp} = \dfrac{2}{3}\, h\,D_{ij}^{vp}$$

Armstrong-Frederick kinematic hardening including dynamic restoration.

$$\dot{X}_{ij}^{af} = \dfrac{2}{3}\, h\,D_{ij}^{vp} - b\, \dot{\bar{\varepsilon}}\, X_{ij}^{af}$$

Chaboche kinematic hardening including static restoration.

$$\dot{X}_{ij}^{cf} = \dfrac{2}{3}\, h\,D_{ij}^{vp} - b\, \dot{\bar{\varepsilon}}\, X_{ij}^{ch} - \dfrac{h}{M} \left(\dfrac{J_2\left(\mathbf{X}^{ch}\right)}{M}\right)^{m-1} X_{ij}^{ch}$$

Asaro kinematic hardening.

$$X_{ij}^{as} = \dfrac{h_s}{b_s}\, \tanh \left(b_s\left|\left|E_{ij}^{vp}\right|\right|\right) \dfrac{E_{ij}^{vp}}{\left|\left|E_{ij}^{vp}\right|\right|}$$