## 2 Constitutive Model in Rate Form  
Under the assumption of infinitesimal strains, the elasto-viscoplastic constitutive model of [11] is based on the following definition of the effective stress rate, positive if compressive (note that, to simplify the notation and in contrast to what is usually done in soil mechanics, the symbol \(\sigma\) is used to denote the effective stress and not the total stress):  
\[\dot{\sigma}(t)=\mathbf{D}(t):[\dot{\mathbf{e}}(t)-\dot{\mathbf{e}}^{\mathit{vp}}(t)] \tag{1}\]  
where \(t\) is the physical time, \(\sigma\) denotes the effective stress, \(\mathbf{e}\) the total strain, and \(\mathbf{D}\) the isotropic elastic tensor:  
\[\mathbf{D}(t)=2G(t)\mathbf{I}\mathbf{I}+\left[K(t)-\frac{2}{3}G(t)\right] \mathbf{I}\otimes\mathbf{I} \tag{2}\]  
where \(\mathbf{I}\) is the second-order identity tensor and \(\mathbf{I}\) is the fourth-order tensor of Cartesian components:  
\[[II]_{ijhk}=\delta_{ih}\,\delta_{jk}+\delta_{ik}\,\delta_{ji}\qquad i,j,h,k=1,2,3 \tag{3}\]  
\(G\) and \(K\) are the shear and bulk modulus, respectively. Finally, \(\dot{\mathbf{e}}^{\mathit{vp}}\) in (1) is the viscoplastic strain rate:  
\[\dot{\mathbf{e}}^{\mathit{vp}}=\gamma\frac{\partial p^{\mathrm{eq}}}{\partial \sigma} \tag{4}\]  
\(\dot{\gamma}\) is a scalar viscoplastic multiplier, introduced for convenience of notation, and \(p^{\mathrm{eq}}\) is the plastic potential, defined as:  
\[ \begin{split}\dot{\gamma}=\frac{\mu^{*}}{\tau}\,\frac{1}{p_{p}^{ \mathrm{eq}}}\left(\frac{p^{\mathrm{eq}}}{p_{p}^{\mathrm{eq}}}\right)^{\frac{ \lambda^{*}-\kappa^{*}}{\mu^{*}}}\\ p^{\mathrm{eq}}(p,q)=p+\frac{q^{2}}{M^{2}\,p}\end{split}  \tag{5}\]  
In (5), a comma denotes the partial derivative with respect to the variable after the comma, \(\tau\) and the non-dimensional parameters \(\mu^{*}\), \(\lambda^{*}\), and \(\kappa^{*}\) are material parameters, which will be defined below, \(p=\sigma:\mathbf{I}/3\) is the effective hydrostatic pressure, \(q=\sqrt{3/2\,\mathbf{s}:\mathbf{s}}\) is the effective deviatoric stress, \(\mathbf{s}\) being the deviatoric stress tensor, and \(M\) is the slope of the critical state line in the \(q-p\) plane (see Figure 1). Finally, \(p_{p}^{\mathrm{eq}}\) (note that in this case, there is no comma before the subscript \(p\)) is an internal variable, to be interpreted as a generalized consolidation pressure, governing the hardening behavior of the viscoplastic response. \(\tau\), \(\mu^{*}\), \(\lambda^{*}\), and \(\kappa^{*}\) are material parameters with a clear physical meaning and easily identifiable through standard laboratory tests, such as odometric tests. \(\tau\) is a characteristic time, usually set equal to 24 h. \(\mu^{*}\) is the modified creep index (modified because it is expressed in terms of the volumetric strain rather than the void ratio), which can be obtained in the long term from: (i) the slope of volumetric strain \(e_{v}\) versus the logarithm of time [11] (see Figure 2a), (ii) the slope of the inverse of volumetric strain rate versus time (see [28]), and (iii) the slope of logarithm of strain rate versus strain (see [29]), while \(\lambda^{*}\) is the modified compression index, measuring the slope of the normal consolidation line in the \(e_{v}-\log p\) plane (see Figure 2b), and \(\kappa^{*}\) is the modified swelling index, measuring the slope of the unloading or swelling curve in the same plot. The parameters \(\lambda^{*}\) and \(\kappa^{*}\) should not be confused with the more commonly used \(\lambda\) and \(\kappa\) (without the \({}^{*}\) at the apex): Equation (31) in [11] states their relationships (it is \(\lambda^{*}=\lambda/(1+e)\) and \(\kappa^{*}=\kappa/(1+e)\), where \(e\) is the void ratio). The evolution of the consolidation pressure, \(p_{p}^{eq}\) in (5), is governed by the following law:  
\[p_{p}^{\text{eq}}=p_{p0}^{\text{eq}}\exp\left(\frac{e_{v}^{vp}}{\lambda^{*}- \kappa^{*}}\right) \tag{6}\]  
Figure 1: Original and evolved domain for the plastic potential \(p^{\text{eq}}\) in the deviatoric-hydrostatic plane.