This model implements the rate dependent viscoplasticity (Lemaitre & Chaboche, 1990) with multiple internal variables. First the viscoplastic potential is represented as, $$$$\begin{eqnarray} \Omega = \int_V \tilde{\Omega} \left[J_2(\sigma-X)-\sigma_y + \left(\frac{D}{2C}J_2^2(X)-\frac{2DC}{9}J_2^2(\alpha)\right)\right] dV, \end{eqnarray}$$$$

where, $$$J_2(\sigma)=\sqrt{\frac{3}{2}(\sigma':\sigma')}$$$ with $$$\sigma'$$$ being the deviatoric stress. $$$X$$$ is the back stress, calculated as, $$$$\begin{eqnarray} X=\frac{2}{3}C\alpha. \end{eqnarray}$$$$

Here, $$$\alpha$$$ is the internal variable corresponding to kinematic hardening.

The flow rule is defined as,

$$$$\begin{eqnarray} \dot{\epsilon}^p=\frac{\partial \Omega}{\partial \sigma}=\frac{3}{2}\dot{p}\frac{\sigma'-X'}{J_2(\sigma-X)} \end{eqnarray}$$$$

Plastic strain rate $$$\dot{\epsilon}^p$$$ is a RankTwoTensor and $$$\dot{p}$$$ is the rate of change in internal variable due to isotropic hardening.

Isotropic Hardening

Internal parameter for isotropic hardening represents the cumulative plastic strain and is defined as,

$$$$\begin{eqnarray} \dot{p}=\bigg\langle {\frac{J_2(\sigma-X)-R_0-R}{K}}\bigg\rangle^N \end{eqnarray}$$$$

Here, $$$R_0$$$ is the yield stress and $$$R$$$ is the isotropic hardening represented by increase in the size of the elasticity domain and $$$K$$$ is the drag force. This has been further simplified as,

$$$$\begin{eqnarray} \dot{p}=\bigg\langle {\frac{J_2(\sigma-X)-\sigma_y}{\sigma_y}}\bigg\rangle^N \end{eqnarray}$$$$

Kinematic Hardening

Internal parameter for kinematic hardening is defined as,

$$$$\begin{eqnarray} \dot{\alpha}=\dot{\epsilon}^p-D\alpha\dot{p} \end{eqnarray}$$$$

Internal variable $$$\alpha$$$ is a RankTwoTensor.

$$$C,D$$$ are temperature dependent material parameters

Note: This model is currently being implemented in TensorMechanics. Further details about the model will be added once the implementation is complete.