# Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering
## 5 Conclusions  
In this paper, a simple elastic problem has been studied to investigate the influence of heterogeneity on the contact forces and deformation. It is the first time that the linking the penalty method has been used with the LAS. In the example, the parameter Young's modulus has been considered as a spatial variable. The penalty method and the LAS have been linked together to solve the contact problem and model the heterogeneity of Young's modulus. First, in the homogeneous case in which Young's modulus is taken to be uniform across the whole research domain, it can be found that the normal/frictional contact force and the deformation is symmetrical. Compared to the homogeneous case, the results of the case where Young's modulus is considered to be spatially variable (following a normal distribution) shows that the influence of soil heterogeneity on the elastic contact problems is significant. Due to heterogeneity, the contact force and deformation shows evident variation. The variation of the normal contact force at any specific point follows a normal distribution, and the COV changes with the \(x\) coordinate. When the soils are considered to be heterogenous, it can be found that there is a certain probability when the contact force exceeds a certain value (result calculated in the homogeneous case). This implies that the conventional analysis of the contact problems in which the soils are taken as homogeneous could induce risk. In addition, the variation of the frictional contact force is less than that of the normal contact force. Later, the influences of different COVs and horizontal scales of fluctuation have been investigated. The results show that with the increase of the COV of Young's modulus, the standard deviation of the normal contact force increases. However, with the increase of the horizontal scale of fluctuation of Young's modulus, the standard deviation of the normal contact force decreases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Horizontal Scale of** & **Mean of the Normal Contact** & **Standard Deviation of the** \\
**Fluctuation (m)** & **Force (N)** & **Normal Contact Force (N)** \\ \hline
1.0 & 35,279 & 677.2 \\ \hline
4.0 & 35,292 & 310.8 \\ \hline
10.0 & 35,308 & 166.8 \\ \hline
20.0 & 35,304 & 147.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean and standard deviation of the normal contact force for different horizontal scales of fluctuation. Figure 17: PDFs (**a**) and CDFs (**b**) of the normal contact force at the point A of the elastic block when the horizontal scales of fluctuation are 1.0 m, 4.0 m, 10.0 m and 20.0 m.  
From the analysed results, it suggests that, in order to better predict the deformation/stress in the contact bodies (such as soil-concrete contact), the spatial variability needs to be considered. To achieve this, the proposed approach of linking the penalty method with the LAS can be applied. Conceptualization, H.G. and K.L. ; methodology, H.G. ; software, H.G. and K.L. ; validation, K.L. ; formal analysis, H.G. and K.L. ; investigation, H.G. and K.L. ; resources, K.L. ; data curation, H.G. and K.L. ; writing--original draft preparation, H.G. ; writing--review and editing, K.L. ; visualization, H.G. and K.L. ; supervision, K.L. ; project administration, K.L. ; funding acquisition, K.L. All authors have read and agreed to the published version of the manuscript. The research was funded by the Natural Science Foundation of China (NSFC), grant number 51908175 and 41807223; the Natural Science Foundation of Anhui Province, grant number 1908085 QE242; and the Fundamental Research Funds for the Central Universities, grant number JZ2018 HGBZ0119. The financial support is gratefully acknowledged. Not applicable. The data presented in this study are openly available in [Dryad] at [[https://doi.org/10.5061/dryad.5x69p8d1m](https://doi.org/10.5061/dryad.5x69p8d1m)]([https://doi.org/10.5061/dryad.5x69p8d1m](https://doi.org/10.5061/dryad.5x69p8d1m)) (accessed on 5 May 2021). The authors declare no conflict of interest. * Weyler et al.