# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 2 Constitutive Model  
### Rate Dependent Plasticity  
Apart from the mathematical and numerical models that will be described in Section 3, it is essential to choose an appropriate constitutive model for the degradable soils. The mechanisms through which soft clays display a viscous phenomenon and delayed deformation because of the creep have been extensively studied in the past ([28]). The concept of the visco-plastic material outlined in this section and employed in the proposed simulations is based on [PERSON]'s theory [34], which is a modification of the classical plasticity, wherein viscous-like behavior is introduced by a time-rate flow rule thanks to a plastic yield function adapted to dynamic conditions. In a similar manner to the rate-independent theory, the strain rate is split into a visco-plastic strain rate and an elastic one \[\dot{\mathbf{\varepsilon}}=\dot{\mathbf{\varepsilon}}^{\mathbf{\varepsilon}}+\dot{\mathbf{\varepsilon} }^{\rm VP} \tag{1}\]  
The rate of the Cauchy's stress tensor, \(\dot{\mathbf{\sigma}}\), is linked to the elastic strain rate through the constitutive tensor \(\mathbf{D}_{\mathbf{\varepsilon}}\), which, in our case, is variable or stress-dependent due to the hyperelasticity and large strain theory. \[\dot{\mathbf{\sigma}}=\mathbf{D}_{\mathbf{\varepsilon}}(\dot{\mathbf{\varepsilon}}-\dot{ \mathbf{\varepsilon}}^{pp}) \tag{2}\]  
In the model proposed by [PERSON] (2016), and later modified by [PERSON] et al. (2016), the rate of the visco-plastic strain can be defined similarly than in the rate-independent plasticity approach:  
\[\dot{\mathbf{\varepsilon}}^{\rm VP}=\langle\lambda\rangle\frac{\partial g}{ \partial\sigma} \tag{3}\]  
where \(\langle\dot{\lambda}\rangle\) is the function of the viscous flow, which denotes the current magnitude of visco-plastic strain rate; \(g\) represents the visco-plastic potential function and \(\frac{\partial g}{\partial\sigma}\) represents the current direction of the visco-plastic strain rate. The viscous flow function is defined by:  
\[\langle\lambda\rangle=\left\{ \begin{array}{cc}\gamma\left[\left(\frac{q}{ \sigma_{y}}\right)^{\alpha}-1\right]&,\phi>0\\ 0&,\phi\leq 0\end{array} \right., \tag{4}\]  
where \(<>\) denotes Macauley brackets, \(\gamma\) is the fluidity parameter, also thought as the reciprocal of the viscosity, and \(\alpha\) is a material constant. In this work, associative flow is invoked by \(\phi=g\). \(\phi\) is the plasticity function that plays the role of the loading surface; a von [PERSON] \(\phi=g\) function, with a degradation curve for the undrained shear strength, has been adopted for the plastic criteria. Function \(\phi\) is defined as:  
\[\phi=q-\sigma_{y}, \tag{5}\]  
where \(q\) is the deviatoric stress invariant and \(\sigma_{y}\) is defined by the degradation law, which will be detailed later. Regarding algorithmic aspects, in displacement-based numerical methods, stress updates take place at the Gauss points (or material points if material-based methodologies are adopted) for a known nodal displacement. Initial conditions, which are the ones from the last converged state, depart from time \(t_{n}\):  
\[\left[\varepsilon_{n},\varepsilon_{n}^{vp},\sigma_{n},\kappa_{n}\right], \tag{6}\]  
in which the variables are total strain, visco-plastic strain, stress and a scalar internal variable, respectively. The latter characterizes the size of the yield surface as well as any other plastic aspect. The final goal is to calculate the corresponding values at time \(n+1\) through:  
\[t_{n+1}=t_{n}+\Delta t:\left[\varepsilon_{n+1},\varepsilon_{n+1}^{vp},\sigma _{n+1},\kappa_{n+1}\right]. \tag{7}\]  
Indeed, this process has been carried out in an incremental way, being calculated:  
\[\Delta\mathbf{\varepsilon} = \varepsilon^{\rm e}+\Delta\varepsilon^{\rm VP} \tag{8}\] \[\Delta\mathbf{\sigma} = \mathbf{D}_{\mathbf{\varepsilon}}(\Delta\varepsilon-\Delta \varepsilon^{\rm VP}) \tag{9}\]  
Thus, the main objective of the stress updates is to estimate the visco-plastic strain increment, \(\Delta\varepsilon^{vp}\).