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 σ 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, Δt is the time increment, Δϵ is the total strain increment, and Δϵelas 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 rt is calculated from the relaxation time t* and the time increment Δt 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.