# Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering
## 4 Illustrative Example
### Comparison between the Homogeneous Case and Heterogeneous Case
Similarly, when the coefficient of friction is 0.3, if the normal contact force of the homogeneous case (35,283.8 N) is taken as a limit, the probability of exceeding the limit is 49.8%. When the coefficient of friction is 0.3, if the frictional contact force of the homogeneous case (0 N) is taken as a limit, the probability of exceeding the limit is 51%. Figure 11: PDF **(a)** and CDF **(b)** of the frictional contact force at the middle bottom point A of the elastic block from 500 realisations when the coefficient of friction is 0.3. Figure 10: PDF **(a)** and CDF **(b)** of the normal contact force at the middle bottom point A of the elastic block from 500 realisations when the coefficient of friction is 0.3. Figure 9: Probability density function (PDF) **(a)** and cumulative distribution function (CDF) **(b)** of the normal contact force at the middle bottom point A of the elastic block from 500 realisations when the coefficient of friction is 0. Since the influence of heterogeneity on the contact force is significant, the heterogeneity certainly has an influence on the corresponding displacement of the elastic block. Figure 12 shows the displacement of the elastic block from two realisations when there is no friction. Comparing Figure 12 with Figure 5a,c, it can be seen that the displacement of the heterogeneous case is significantly different from that of the homogeneous case. For the horizontal and vertical displacements, the variation of the horizontal displacement is much more obvious. Figure 13 shows the displacement of the elastic block from two realisations when the coefficient of friction is 0.3. From the figure, the variation of the horizontal and vertical displacements is not significant compared to the scenario with no friction. The variation of the vertical displacement is not obvious; in order to better investigate the influence, the vertical displacement of the middle top point B (shown in Figure 3) has been selected as an example. The vertical displacements of the homogeneous cases are \(-0.00366\) m (no friction) and \(-0.00328\) m (the coefficient of friction is 0.3), respectively. The following Figure 14 has shown the PDF of the vertical displacement at the point B when the coefficients of friction are 0 and 0.3, respectively. From Figure 14, it can be seen that the variation of the vertical displacement follows a normal distribution. When there is no friction, the mean and the standard deviation of the vertical displacement are \(-0.00367\) m and \(-0.0000848\) m. The mean is quite close to the result of the homogeneous case (\(-0.00366\) m). When the coefficient of friction is 0.3, the mean and the standard deviation of the vertical displacement is 0.3. Figure 12: Horizontal (**a,b**) and vertical (**c,d**) displacements of two realisations when the coefficient of friction is 0. Figure 13: Horizontal (**a,b**) and vertical (**c,d**) displacements of two realisations when the coefficient of friction is 0.3.  
deviation of the vertical displacement is \(-0.00329\) m and \(-0.0000747\) m. The mean is also close to the result of the homogeneous case (\(-0.00328\) m).