# Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering
## 4 Illustrative Example
### Influence of the Statistics of Young's Modulus on the Contact Force  
The previous subsection has demonstrated the influence of the heterogeneity on the results of contact force and displacement. The random field generation is controlled by the statistics of Young's modulus, i.e., the mean, the COV and the scale of fluctuation. Therefore, in this subsection, the purpose is to investigate the influence of different standard deviation and different scale of fluctuation.  
First, the influence of different standard deviation is investigated. The COV of Young's modulus is chosen to be 5% (which has been used in the previous subsection), 10% and 20%, respectively. In the investigation, the coefficient of friction remains 0.3 and the scale of fluctuation is assumed to be 1.0 m. Figure 15 shows the PDFs and CDFs of the normal contact force at the point A when the COV is 5%, 10% and 20%. From Figure 15, it proves that with the increase of the COV of Young's modulus, the mean of the normal contact force does not have significant variation, whereas the standard deviation of the normal contact force increases. It is because that with the increase of the COV, the range of the value of Young's modulus becomes wider. When the COV is 5%, the mean and the standard deviation of the normal contact force are 35,279 N and 677.2 N. When the COV is 10%, the mean and the standard deviation of the normal contact force are 35,266 N and 1371.3 N. When the COV is 20%, the mean and the standard deviation of the normal contact force are 35,255 N and 2878.4 N.  
Figure 14: PDFs of the vertical displacement at the middle top point B of the elastic block from 500 realisations when the coefficient of friction is 0 (**a**) and 0.3 (**b**).  
Figure 15: PDFs (**a**) and CDFs (**b**) of the normal contact force at the point A of the elastic block when the COVs are 5%, 10% and 20%.  
Second, the influence of the scale of fluctuation is investigated. The horizontal scale of fluctuation is chosen to be 1.0 m (which has been used in the previous subsection), 4.0 m 10.0 m and 20 m, respectively. The vertical scale of fluctuation remains 1.0 m. In the comparison, the coefficient of friction remains as 0.3 and the COV is assumed to be 5%. Figure 16 shows the random field with different horizontal scales of fluctuation, i.e., 4.0 m, 10.0 m and 20.0 m. It can be seen that the distribution of Young's modulus across the whole domain is different from each other. With the increase of the horizontal scale of fluctuation, the Young's modulus of any two distant points becomes more correlated.  
In Figure 17, it shows the PDFs and CDFs of the normal contact force at the point A when the horizontal scales of fluctuation are 1.0 m, 4.0 m, 10.0 m and 20.0 m. It can be found from the figure that the standard deviation of the normal contact force decreases with the increase of the horizontal scale of fluctuation, whereas the increase of the horizontal scale of fluctuation has no significant influence on the mean. Table 1 has shown the mean and standard deviation of the normal contact force for different horizontal scale of fluctuation. The reason could be that with the increase of the horizontal scale of fluctuation, the value of Young's modulus along the horizontal direction varies less, which can be seen in Figure 16c, so less variation causes decreases in the standard deviation.  
Figure 16: Realisations of the random field for different horizontal scales of fluctuation (**a**) 4.0 m, (**b**) 10.0 m and (**c**) 20.0 m.