# Dynamic Shear Degradation of Geosynthetic-Soil Interface in Waste Landfill Sites
## 5 Numerical Back-Prediction of Test Results  
### Methodology  
The numerical back-prediction for the reproduction of the cyclic shear stress-strain relationship was developed according to numerical interpolation procedures. The reproduction of the curves \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Chemical Conditions**} & \multicolumn{3}{c}{**Max. Decreases of Disturbance (\(\Delta\) Dmax)**} \\  & **(between the Results of 30 and 850 Days of Submersion)** \\ \cline{2-3}  & **Basic** & **Neutral** & **Acid** \\ \hline
0.3 MPa of normal stress & 31 \% & 23 \% & 22 \% \\
0.6 MPa of normal stress & 26 \% & 10 \% & 8 \% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum variations of the disturbance curves. Figure 11: Maximum decreases of the disturbance (\(\Delta\)Dmax) under 0.3 MPa: (**a**) natural scale for strain axis; (**b**) log scale for strain axis. Figure 10: New disturbance function curves: (**a**) 30 days submersion, 0.3 MPa; (**b**) 30 days submersion, 0.6 MPa; (**c**) 850 days submersion, 0.3 MPa; and (**d**) 850 days submersion, 0.6 MPa. is important in order to significantly reduce the number of laboratory tests, and to predict shear stress degradation of the interface. The methodology of numerical back-prediction is based on the idea that the hysteretic curve at a certain cycle can be divided into 4 phases and each curve can be equally divided into 10 segments again for numerical interpolation. The tangential slopes of the initial and final stress points in a curve can be calculated by linear tangential equations in sequence. Numerical interpolation enables one to obtain the cyclic shear stress-strain values at the midpoints in each segment and to reproduce the cyclic shear stress-strain curves. Shear stress degradation was automatically calculated from the test results and the shear strain increment, and was updated at the target cycle. The RI state was considered as the first cycle, and the FA was considered as the condition after 50 th cycle of loading. The procedures for the numerical back-prediction are as follows:  
1. Obtain the DSC parameters \(\tau^{i}\) and \(\tau^{c}\) from the experimental test results. 2. Define each of the phases in a curve; one cyclic shear stress-strain loop can be divided into four phases as described in Figure 12a. The tangential gradients for the definition of each phase are shown in Figure 12b. 3. Numerical interpolation: Equally divide each phase by \"\(n\)\" segments, and obtain the average tangential gradient by dividing the difference between the initial tangential gradient (\(G^{VL}\)) and the final tangential gradient (\(G^{EVL}\)), as demonstrated in Figure 13. Then, the linear equations at each dividing point (\(k/n\), \(2k/n\),... ) can be generated, as shown in Figure 13. The total number of the linear equations is (\(n+1\)), and \(n=10\) was applied in the present study. Therefore, the equation of the initial tangential line can be expressed using Equation (5), and the general equations are expressed using Equations (6)-(8) in the virgin loading phase. \[y^{0}=\tau^{0}=G^{VL}x+b^{0}=G^{VL}\varepsilon+b^{0}\] (5) \[y^{1}=\tau^{1}=\left[G^{VL}-\left\{\frac{G^{VL}-G^{EVL}}{n}\right\}\right] \varepsilon+b^{1}\] (6) \[y^{2}=\tau^{2}=\left[G^{VL}-2\left\{\frac{G^{VL}-G^{EVL}}{n}\right\}\right] \varepsilon+b^{2}\] (7) \[y^{m}=\tau^{m}=\left[G^{VL}-m\left\{\frac{G^{VL}-G^{EVL}}{n}\right\}\right] \varepsilon+b^{m},0\leq m\leq n,\;\varepsilon=\frac{mk}{n}\] (8)  
It is assumed that two consecutive linear equations intersect at the midpoint between the two consecutive dividing points. Practically speaking, \(G^{VL}\) (the tangential gradient of the starting point of virgin loading) and \(G^{BRL}\) (the tangential gradient of the starting point of reloading) are almost identical.