# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 2 Constitutive Model
### Eigendegradation Model  
Following the work of [PERSON] and [PERSON] [54], and the later implementations by [PERSON] et al. [54] (similar also to some other implementations [55, 56]), the behavior of sensitive clays can be modeled by strain-softening curves in order to reduce the strength of the material by a degradation related to the accumulation of strain. [PERSON] and [PERSON] assumed that the current shear strength depends on the accumulated absolute shear strain, \(\xi\), which is assumed as a state variable in order to calculate the isotropic strength reduction, \(\delta(\xi)\), as:  
\[\delta(\xi)=s_{u}/s_{ui}=\delta_{\text{rem}}+(1-\delta_{\text{rem}})\text{e} ^{-3\xi/\xi_{95}}, \tag{15}\]  
where  
\[\xi=\int_{t}\lvert\dot{\gamma}_{\text{max}}\rvert\text{d}t \tag{16}\]  
and \(\lvert\dot{\gamma}_{\text{max}}\rvert\) represents the cumulative absolute shear strain. \(s_{u}\) and \(s_{ui}\) are considered the softened strength and initial strength, respectively. In the above equations, \(\delta_{\text{rem}}\) is the fully remolded strength ratio, and \(\xi_{95}\) is the cumulative shear strain required to cause 95% reduction from the peak to fully remolded material. An appropriate value of \(\xi_{95}\) must be obtained from laboratory test data as well as from cyclic penetration and extraction tests with T-bar or ball. Furthermore, \(\delta_{\text{rem}}\) is assumed to be the inverse of the sensitivity of the soil.  
The calculation of the cumulative shear strain can be achieved by the eigendeformation technique, departing from Equation (11), and considering that, for the calculation of the eigendegradation, that the stress remains constant in a neighborhood \(\epsilon\). Thus, Equation (11) can be simplified as  
\[\delta W_{p}^{\epsilon}=\frac{\delta\tau_{p}}{m_{p}}\sum_{x_{q,k+1}\in\mathcal{ R}_{\epsilon}(x_{p,k+1})}m_{q}\gamma_{q}, \tag{17}\]  
where \(\delta\tau_{p}\) refers to the increment of tangential stress of the neighborhood and \(\gamma_{p}^{\epsilon}\) is the current local shear strain, which is obtained as:  
\[\gamma_{p}^{\epsilon}=\frac{1}{m_{p}}\sum_{x_{q,k+1}\in\mathcal{R}_{\epsilon}( x_{p,k+1})}m_{q}\gamma_{q} \tag{18}\]  
Similarly, in the neighborhood \(\epsilon\), the non-local cumulative strain of a material point \(p\) is calculated only when plasticity is activated, as follows:  
\[\xi_{p}^{\epsilon}=\int_{t_{p0}}^{t_{k+1}}\lvert\dot{\gamma}_{p}^{\epsilon} \rvert\text{d}t \tag{19}\]  
where \(t_{k+1}\) refers to the current step, and \(t_{p0}\) refers to the step when plasticity begins.  
Considering only shear failure, yield shear stress \(\tau\) is equivalent to the softened strength, \(s_{u}\), and the residual yield shear stress, \(\tau_{95}\), can be reached by \(\tau_{95}=\tau rem=s_{ui}\,\delta_{rem}\). Thus, in every state of degradation, the current yield shear stress, referring to the epsilon neighborhood, \(\tau^{\epsilon}\), reads:  
\[\tau^{\epsilon}=\tau_{95}+(\tau_{i}-\tau_{95})\text{e}^{-3\xi_{p}^{\epsilon}/ \xi_{95}} \tag{20}\]It is remarkable that in the laboratory, the parameter \(\xi_{95}\) is not obtained. Instead, the displacement \(\delta_{95}\) is achieved. In Figure 3A, the degradation of the strength in terms of the displacement is plotted.  
It can be seen how this law can be translated to the shear strain measurement (Figure 3B) by multiplying by \(\epsilon\), which, in this problem, is considered as the sliding length. Depending on the size of the soft layer, this parameter \(\epsilon\) is obtained as the minimum length between the neighbor radius, [38], and the size of the soft layer (see Figure 4) as follows:  
\[2\epsilon=\min(h_{s},2C_{c}h) \tag{21}\]