## 2 Constitutive Model in Rate Form
Figure 2: Model parameters from odometric tests: (a) \(\mu^{*}\) determination; (b) \(\lambda^{*}\) and \(\kappa^{*}\) determination. where \(\epsilon_{v}^{vp}\) is the volumetric viscoplastic strain and \(p_{p0}^{eq}\) is a reference parameter defining the initial value of the consolidation pressure. Figure 1 shows two contour plots of \(p^{eq}\) in the \(q-p\) plane, the original one and an evolved state. The critical state line of slope \(M\) is also shown in the plot. Equation (6) shows that the evolution of the plastic potential is independent of the deviatoric component of stress and, together with (4), defines an associative evolution of the viscoplastic strains. It should be noted that these inelastic strains start to develop whenever \(p^{eq}>0\), though, due to the power law in (5), for common values of the parameters, the growth rate is very small as long as \(p^{eq}/p_{p}^{eq}<1\). In general, both \(G\) and \(K\) in (2) are taken to depend on the current pressure state \(p(t)\), and hence on time, through the following relation:  
\[K(t)=\frac{p(t)}{\kappa^{*}},\quad G(t)=\frac{3}{2}\frac{1-2v}{1+\
u}\frac{p(t) }{\kappa^{*}} \tag{7}\]  
\(\
u\) being Poisson's coefficient. For the subsequent developments, it is convenient to introduce the projector operators \(\mathbf{m}\) and \(\mathbf{n}\) defined as:  
\[\mathbf{m}=\frac{\partial p}{\partial\boldsymbol{\sigma}}=\frac{1}{3} \mathbf{I},\quad\mathbf{n}=\frac{\partial q}{\partial\boldsymbol{\sigma}}= \frac{\partial q}{\partial\mathbf{s}}=\frac{3}{2}\frac{\mathbf{s}}{q} \tag{8}\]  
such that:  
\[p=\mathbf{m}:\boldsymbol{\sigma},\quad q=\mathbf{n}:\boldsymbol{\sigma}, \quad\mathbf{m}:\mathbf{D}=3K\mathbf{m},\quad\mathbf{n}:\mathbf{D}=2G\, \mathbf{n} \tag{9}\]  
Following [20], the key idea is that, by using \(\mathbf{m}\) and \(\mathbf{n}\), the viscoplastic strain rate can be easily decomposed into its volumetric and deviatoric components:  
\[\dot{\epsilon}^{vp}=\dot{\gamma}\frac{\partial p^{eq}}{\partial\boldsymbol{ \sigma}}=\dot{\gamma}\Big{(}p_{,p}^{eq}\,\mathbf{m}+p_{,q}^{eq}\,\mathbf{n} \Big{)} \tag{10}\]  
Noting that \(\mathbf{m}:\mathbf{n}=0\), \(\mathbf{m}:\mathbf{m}=1/3\), and \(\mathbf{n}:\mathbf{n}=3/2\), the rates of the stress invariants \(p\) and \(q\) can be directly obtained from (1), (4), and (10) as:  
\[\dot{p} =\mathbf{m}:\mathbf{D}:\dot{\boldsymbol{\epsilon}}-\mathbf{m}: \mathbf{D}:\Big{(}p_{,p}^{eq}\,\mathbf{m}+p_{,q}^{eq}\,\mathbf{n}\Big{)}\dot {\gamma}=\dot{p}^{\text{trial}}-Kp_{,p}^{eq}\dot{\gamma} \tag{11}\] \[\dot{q} =\mathbf{n}:\mathbf{D}:\dot{\boldsymbol{\epsilon}}-\mathbf{n}: \mathbf{D}:\Big{(}p_{,p}^{eq}\,\mathbf{m}+p_{,q}^{eq}\,\mathbf{n}\Big{)}\dot {\gamma}=\dot{q}^{\text{trial}}-3 Gp_{,q}^{eq}\dot{\gamma} \tag{12}\]  
In (12), \(\dot{p}^{\text{trial}}=K\dot{\epsilon}_{v}/3\) and \(\dot{q}^{\text{trial}}=2G\,\mathbf{n}:\dot{\boldsymbol{\epsilon}}\) (and hence, also \(\dot{\boldsymbol{\epsilon}}^{\text{trial}}=\mathbf{D}:\dot{\boldsymbol{ \epsilon}}\)) represent tentative values of the stress rates under the assumption of a purely elastic stress variation. In view of the already mentioned absence of a stress threshold for the activation of the viscoplastic strains, a purely elastic stress variation is impossible and, therefore, the trial quantities are purely theoretical. Equation (10) implicitly defines the rate of the volumetric viscoplastic strain as:  
\[\dot{\epsilon}_{v}^{vp}=p_{,p}^{eq}\dot{\gamma} \tag{13}\]  
whose integration is required to compute the evolution of the internal variable \(p_{p}^{eq}\) in (6).