3. Formulation of the viscoplastic model VISC_DRUC_PRAG

3.1. Model equations

This model is based on a viscoplastic formulation of the Drücker-Prager type, where the loading surface \(f(\sigma ,p)\) is defined by:

\(f=\sqrt{\frac{3}{2}}{s}_{\mathrm{II}}+\alpha (p){I}_{1}-R(p)\)

\(\alpha (p)\) and \(R(p)\) are functions of the cumulative deviatoric viscoplastic deformation \(p\),

A viscoplastic flow potential \(g(\sigma ,p)\) is introduced:

\(g=\sqrt{\frac{3}{2}}{s}_{\text{II}}+\beta (p){I}_{1}\)

For the evolution of the criterion \(f\) and the potential \(g\) we distinguish three distinct thresholds corresponding to three values of the work-hardening variable: an elastic threshold, a peak threshold and an ultimate threshold. Between these thresholds, the work hardening is linear. Between the elastic threshold and the peak threshold, the work hardening is positive, after the peak the work hardening is negative and becomes constant after the ultimate threshold.

The functions related to cohesion are written in the following form:

\(\alpha (p)\mathrm{=}(\frac{{\alpha }_{\mathit{pic}}\mathrm{-}{\alpha }_{0}}{{p}_{\mathit{pic}}})p+{\alpha }_{0}\) for \(0<p<{p}_{\mathit{pic}}\)

\(\alpha (p)\mathrm{=}(\frac{{\alpha }_{\mathit{ult}}\mathrm{-}{\alpha }_{\mathit{pic}}}{{p}_{\mathit{ult}}\mathrm{-}{p}_{\mathit{pic}}})(p\mathrm{-}{p}_{\mathit{pic}})+{\alpha }_{\mathit{pic}}\) for \({p}_{\mathit{pic}}<p<{p}_{\mathit{ult}}\)

\(\alpha (p)\mathrm{=}{\alpha }_{\mathit{ult}}\) for \(p>{p}_{\mathit{ult}}\)

The functions related to dilatance are written in the following form:

\(\beta (p)\mathrm{=}(\frac{{\beta }_{\mathit{pic}}\mathrm{-}{\beta }_{0}}{{p}_{\mathit{pic}}})p+{\beta }_{0}\) for \(0<p<{p}_{\mathit{pic}}\)

\(\beta (p)\mathrm{=}(\frac{{\beta }_{\mathit{ult}}\mathrm{-}{\beta }_{\mathit{pic}}}{{p}_{\mathit{ult}}\mathrm{-}{p}_{\mathit{pic}}})(p\mathrm{-}{p}_{\mathit{pic}})+{\beta }_{\mathit{pic}}\) for \({p}_{\mathit{pic}}<p<{p}_{\mathit{ult}}\)

\(\beta (p)\mathrm{=}{\beta }_{\mathit{ult}}\) for \(p>{p}_{\mathit{ult}}\)

The work hardening functions are written as:

\(R(p)\mathrm{=}(\frac{{R}_{\mathit{pic}}\mathrm{-}{R}_{0}}{{p}_{\mathit{pic}}})p+{R}_{0}\) for \(0<p<{p}_{\mathit{pic}}\)

\(R(p)\mathrm{=}(\frac{{R}_{\mathit{ult}}\mathrm{-}{R}_{\mathit{pic}}}{{p}_{\mathit{ult}}\mathrm{-}{p}_{\mathit{pic}}})(p\mathrm{-}{p}_{\mathit{pic}})+{R}_{\mathit{pic}}\) for \({p}_{\mathit{pic}}<p<{p}_{\mathit{ult}}\)

\(R(p)\mathrm{=}{R}_{\mathit{ult}}\) for \(p>{p}_{\mathit{ult}}\)

Stresses are linked to deformations by Hooke’s law:

\(\sigma ={D}^{e}(\varepsilon -{\varepsilon }^{\text{vp}})\)

When the viscoplastic threshold is reached, irreversible viscoplastic deformations are generated and expressed according to Perzyna’s theory by:

\(d{\varepsilon }_{\text{ij}}^{\text{vp}}=A{\langle \frac{f}{{P}_{\text{ref}}}\rangle }^{n}\frac{\partial g}{\partial {\sigma }_{\text{ij}}}\text{dt}\)

\(f\) being the viscoplasticity criterion; \(A\) and \(n\) are model parameters; \({P}_{\text{ref}}\) a reference pressure.

\(\frac{\partial g}{\partial {\sigma }_{\text{ij}}}=\sqrt{\frac{3}{2}}\frac{\partial {s}_{\text{II}}}{\partial {\sigma }_{\text{ij}}}+\beta (p)\frac{\partial {I}_{1}}{\partial {\sigma }_{\text{ij}}}\) and \(\dot{p}\mathrm{=}\sqrt{\frac{2}{3}{\dot{\tilde{\varepsilon }}}_{\text{ij}}^{\text{vp}}{\dot{\tilde{\varepsilon }}}_{\text{ij}}^{\text{vp}}}\)

with, \({\tilde{\varepsilon }}_{\text{ij}}^{\text{vp}}\) the viscoplastic deformation tensor deviator,

\(\frac{\mathrm{\partial }{s}_{\text{II}}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\frac{\mathrm{\partial }{s}_{\text{II}}}{\mathrm{\partial }{s}_{\text{kl}}}\frac{\mathrm{\partial }{s}_{\text{kl}}}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}\frac{{s}_{\text{kl}}}{{s}_{\text{II}}}({\delta }_{\text{ik}}{\delta }_{\text{jl}}\mathrm{-}\frac{1}{3}{\delta }_{\text{ij}}{\delta }_{\text{kl}})\mathrm{=}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}\)

and \(\frac{\partial {I}_{1}}{\partial {\sigma }_{\text{ij}}}=\frac{\partial \text{tr}({\sigma }_{\text{ij}})}{\partial {\sigma }_{\text{ij}}}={\delta }_{\text{ij}}\)

Hence \(\frac{\partial g}{\partial {\sigma }_{\text{ij}}}=\sqrt{\frac{3}{2}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}+\beta (p){\delta }_{\text{ij}}\)

Summary of equations: The criterion: \(f\mathrm{=}\sqrt{\frac{3}{2}}{s}_{\text{II}}+\left[(\frac{{\alpha }_{\text{pic}}\mathrm{-}{\alpha }_{0}}{{p}_{\text{pic}}})p+{\alpha }_{0}\right]{I}_{1}\mathrm{-}\left[(\frac{{R}_{\text{pic}}\mathrm{-}{R}_{0}}{{p}_{\text{pic}}})p+{R}_{0}\right]\) for \(0<p<{p}_{\text{pic}}\) \(f\mathrm{=}\sqrt{\frac{3}{2}}{s}_{\text{II}}+\left[(\frac{{\alpha }_{\text{ult}}\mathrm{-}{\alpha }_{\text{pic}}}{{p}_{\text{ult}}\mathrm{-}{p}_{\text{pic}}})(p\mathrm{-}{p}_{\text{pic}})+{\alpha }_{\text{pic}}\right]{I}_{1}\mathrm{-}\left[(\frac{{R}_{\text{ult}}\mathrm{-}{R}_{\text{pic}}}{{p}_{\text{ult}}\mathrm{-}{p}_{\text{pic}}})(p\mathrm{-}{p}_{\text{pic}})+{R}_{\text{pic}}\right]\) for \({p}_{\text{pic}}<p<{p}_{\text{ult}}\) \(f\mathrm{=}\sqrt{\frac{3}{2}}{s}_{\text{II}}+{\alpha }_{\text{ult}}{I}_{1}\mathrm{-}{R}_{\text{ult}}\) for \(p\mathrm{\ge }{p}_{\text{ult}}\): Flow potential: \(g=\sqrt{\frac{3}{2}}{s}_{\text{II}}+\left[(\frac{{\beta }_{\text{pic}}-{\beta }_{0}}{{p}_{\text{pic}}})p+{\beta }_{0}\right]{I}_{1}\) for \(0<p<{p}_{\text{pic}}\) \(g=\sqrt{\frac{3}{2}}{s}_{\text{II}}+\left[(\frac{{\beta }_{\text{ult}}-{\beta }_{\text{pic}}}{{p}_{\text{ult}}-{p}_{\text{pic}}})(p-{p}_{\text{pic}})+{\beta }_{\text{pic}}\right]{I}_{1}\) for \({p}_{\text{pic}}<p<{p}_{\text{ult}}\) \(g=\sqrt{\frac{3}{2}}{s}_{\text{II}}+{\beta }_{\text{ult}}{I}_{1}\) for \(p\mathrm{\ge }{p}_{\text{ult}}\) \({\alpha }_{0}\), \({R}_{0}\) and \({\beta }_{0}\): work-hardening parameters corresponding to the elasticity threshold (\(p\mathrm{=}0\)) \({\alpha }_{\text{pic}}\), \({R}_{\text{pic}}\) and \({\beta }_{\text{pic}}\): work-hardening parameters corresponding to the parameter \({p}_{\text{pic}}\) \({\alpha }_{\text{ult}}\), \({R}_{\text{ult}}\) and \({\beta }_{\text{ult}}\): work-hardening parameters corresponding to the parameter \({p}_{\text{ult}}\) Hooke’s law: \(\sigma ={D}^{e}(\varepsilon -{\varepsilon }^{\text{vp}})\) \(f(\sigma ,p)\le 0\) elasticity domain; \({\dot{\varepsilon }}_{\text{ij}}^{\text{vp}}\mathrm{=}0\) \(f(\sigma ,p)>0\) viscoplasticity; \({\dot{\varepsilon }}_{\text{ij}}^{\text{vp}}=A{\langle \frac{f}{{P}_{\text{ref}}}\rangle }^{n}\frac{\partial g}{\partial {\sigma }_{\text{ij}}}\); \(\dot{p}\mathrm{=}\sqrt{\frac{2}{3}{\dot{\tilde{\varepsilon }}}_{\text{ij}}^{\text{vp}}{\dot{\tilde{\varepsilon }}}_{\text{ij}}^{\text{vp}}}\)