Model: Viscoplasticity

The classical Perzyna-type viscoplasticity models are not suitable for rate-dependent viscoplastic models with nonsmooth multisurface, because of using unclearly defined nested viscoplastic loading surfaces. As an alternative, the Duvaut-Lions viscoplastic theory, precludes these difficulties by excluding the concept of nested viscoplastic loading surfaces. The viscoplastic constitutive equation that relates the stress $\boldsymbol{\sigma}$ and the viscoplastic strain rate $\dot{\boldsymbol{\epsilon}^{vp}}$ is given by:

$\boldsymbol{\sigma} - \bar{\boldsymbol{\sigma}} = \frac{1}{t_*}\tensor{c}:\dot{\boldsymbol{\epsilon}^{vp}}$

Here, $\bar{\boldsymbol{\sigma}}$ represents the inviscid stress, which is the rate-independent elasto-plastic stress part that can be solved by using elasto-plastic solvers (such as Drucker-Prager, CamClay, etc.). $\tensor{c}$ is the tangent stiffness tensor and $t_*$ is the relaxation time, which is measured in units of time. The viscoplastic strain rate $\dot{\boldsymbol{\epsilon}}^{vp}$ can be approximated using the following finite difference formula:

$\dot{\boldsymbol{\epsilon}^{vp}} = \frac{1}{\Delta t}(\Delta \boldsymbol{\epsilon} - \Delta \boldsymbol{\epsilon}^{elas})$

Here, $\Delta t$ is the time increment, $\Delta \boldsymbol{\epsilon}$ is the total strain increment, and $\Delta \boldsymbol{\epsilon}^{elas}$ is the elastic part of the strain increment. Note that the elastic strain increment is related to the stress increment through Hook’s law.

$\Delta \boldsymbol{\sigma} = \tensor{c}:\Delta \boldsymbol{\epsilon}^{elas}$

With some arrangements, we can obtain the following formula to update the stress tensor of the Duvaut-Lions elasto-viscoplastic materials:

$\boldsymbol{\sigma} = r_t \hat{\boldsymbol{\sigma}} + (1-r_t) \bar{\boldsymbol{\sigma}}$

Here, the time ratio $r_t$ is calculated from the relaxation time $t_*$ and the time increment $\Delta t$ as:

$r_t = \frac{1}{1+\Delta t/t_*}$

Assuming elastic behavior, the trial stress tensor $\hat{\boldsymbol{\sigma}}$ is computed using the strain increment:

$\hat{\boldsymbol{\sigma}}^{t+\Delta t} = \boldsymbol{\sigma}^t + \tensor{c}^{t+\Delta t}:\Delta \boldsymbol{\epsilon}^{t+\Delta t}$

The tangent stiffness tensor is updated using the following equivalent approximation:

$\tensor{c}^{t+\Delta t} = r_t \tensor{c}^e + (1-r_t) \tensor{c}^{t}$

Here, $\tensor{c}^e$ is the elastic stiffness tensor.

The name of the viscoplastic solver is formed by adding the prefix Visco to the name of the elasto-plastic solver used to compute the inviscid stress $\overline{\boldsymbol{\sigma}}$ . For example, the solver Visco Drucker-Prager corresponds to cases where the inviscid stress is computed by the Drucker-Prager solver. It is interesting to note that equivalent viscoelastic solutions can also be obtained using the Duvault-Lions algorithm by updating the inviscid stress with an elastic solver.