# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 4 Applications
### Vertical Cut  
This final application allows understanding the potentiality of the proposed methodology. A weak layer is supposed in a soil with a vertical cut on the left side (Figure 11). This weak layer is located, forming a 45\({}^{\circ}\) angle. This layer, the thickness of which is 1 m, will be considered as plastic. Von Mises yield surface is employed, and its degradation is modeled through both _eigendegradation_ and traditional softening in order to assess the performance of the former one compared with the latter. Out of the weak layer, the soil is considered elastic since its failure is far from the failure of the weak layer. Parameters of both models are presented in the right part of Figure 11. In the traditional softening model, no eigendegradation parameters are needed. Instead, a negative hardening of 200 kPa is employed. This parameter is not employed in the eigendegradation simulation. The soil can be considered infinite on the right and on the bottom of the model; thus, any movement in these directions is prevented. The 12 first meters are modeled for the sake of simplicity. Figure 10: Obtained reaction with different loading rates for the problem of the strip footing using visco-plastic von Mises law, in which \(t_{f}\) is the final time of the application of the load. The dashed line represents the pseudo-static behavior. Figure 9: Obtained reaction with different \(\mu\) values for the problem of the strip footing using visco-plastic von Mises law. The dashed line represents the pseudo-static behavior. The top left part is loaded by a surface load as shown in Figure 11. This load is composed by two different waves as sketched in Figure 12.  
Both waves follow the expression:  
\[q_{\alpha}(t)=A_{\alpha}\cdot[1-cos(\omega_{\alpha}t)]\]  
where \(\alpha\) varies for each of both loads. The bigger load is the one that provokes the triggering of the plastic mechanism. It can be considered as an abnormal scenario that may lead to catastrophic consequences in the short or in the long time. Their parameters are \(A_{1}=11\) kPa and \(\omega_{1}=0.1\pi\) rad/s. This load is applied only in the first 20 s of simulation. The secondary load is of a lower magnitude. It could be considered as a usual load that the soil suffers permanently; it is not capable of provoking the breakage of the slope by itself. The amplitude of this load is half of the first one, \(A_{2}=5.5\) kPa, and the frequency is much higher, \(\omega_{2}=3\pi\) rad/s. This load is maintained through the 80 s of the simulation. Figure 11: (Left:) Geometry of the vertical cut analyzed through _eigendegradation_ and softening models and the location of the weak layer and the loaded zone (units in meters). (Right): Parameters of the employed models. Figure 12: Scheme of the load and the yield strength along the time. Following, in Figure 13, the evolution of both shear strength and the equivalent plastic strain are depicted along the time for both eigendegradation and softening models. Both OTM and FEM are carried out. The latter is made through the commercial software Abaqus. On the left, the shear strength is plotted. The first part of the figure is loaded with the first load: the material reaches the yield stress close to 10 s and starts to decrease the strength until 15 s. Both materials, until this point, behave similarly. Observing the right figure, we can see how both models obtain plastic strain until 15 s as well. Obviously, the amount of shear strain is different, since one law is logarithmic (eigendegradation) and the other one is linear. After this point, the eigendegradation model accumulates shear strain (elastic in this case) that makes the shear strength decrease. This elastic shear strain comes from the second law (the one with small amplitude). We know that it is elastic strain since, in the right figure, no accumulated plastic strain is obtained from 15 s to 50 s. From this point on, the equivalent plastic strain increases drastically. It is translated in an increment of the descent of the yield stress.