# Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering
## 4 Illustrative Example
### Comparison between the Homogeneous Case and Heterogeneous Case  
In order to investigate the influence of heterogeneity on the contact problems, a homogeneous case is analysed first. As in the previous subsection, Figure 4 presents the results of the homogeneous case, which are the computed contact force applied on the bottom of the elastic block along the interface when the coefficient of friction is 0.0 and 0.3. From Figure 4, it can be seen the normal contact force is symmetrical when the Young's modulus is uniform. In addition, Figure 5 shows the deformation of the elastic block under loading for both scenarios. Figure 4: Validation of the contact results computed by the program. Figure 5: Horizontal (**a**,**b**) and vertical (**c**,**d**) displacements of the elastic block when the coefficient of friction is 0.0 and 0.3. Figure 5a,b show the horizontal displacement when the coefficients of friction are 0.0 and 0.3, respectively. Figure 5c,d show the vertical displacement. It can be seen from Figure 5 that the horizontal/vertical displacement is symmetrical. However, due to the emergence of friction, the distribution of the displacement across the whole domain is quite different. From Figure 5a,b, it can be found that because of the friction, the shape of the zone with large horizontal displacement becomes significantly smaller. The horizontal displacement reduces much near the interface. Moreover, Figure 5c,d indicate that the shape of the zone with large vertical displacement changes as well. In addition, the maximum horizontal and vertical displacements of the case with no friction are larger than those of the case when the coefficient of friction is 0.3. Then, Young's modulus of the elastic block is assumed to be spatially variable. The mean and coefficient of variation (COV) (defined as the standard deviation divided by the mean) of Young's modulus are 100 MPa and 5%, respectively. Additionally, the scale of fluctuation of Young's modulus is chosen to be 1.0 m. Figure 6 shows one realisation of the random field. From Figure 6, each finite element mesh has been assigned with a different value. In this paper, the total number of realisations used in the computation is 500. In each realisation, the contact forces can be calculated. Figures 7 and 8 show the results of contact forces from 500 realisations when there is no friction and the coefficient of friction is 0.3, respectively. From Figures 7 and 8, it can be found that the heterogeneity of Young's modulus has a significant influence on the results of contact force. First, due to the spatial variability of Young's modulus, the distribution of normal contact force along the contact surface is not symmetrical. Moreover, Figures 6 and 7 show that the COV of the contact force at a specific point changes with the \(x\) coordinate. Furthermore, it can be seen from Figure 8 that the variation of the frictional contact force is less than that of the normal contact force, which suggests that the influence of heterogeneity has more of an impact on the normal contact force. Figure 6: One realisation of the random field of Young’s modulus. and 835.6 N. When the coefficient of friction is 0.3, the mean and the standard deviation of the normal contact force are 35,279 N and 677.2 N. Meanwhile, the mean and the standard deviation of the frictional contact force are \(-\)2.0746 N and 182.1 N. The contact force of the homogeneous case is quite close to the mean of the heterogeneous case. By comparing Figure 9a and 10a, it can be seen that the contact with no friction generally has larger normal contact forces, which is consistent with the homogeneous case. In addition, from Figures 9b and 10b, the probability of the contact force exceeding a certain value can be calculated. For example, in the scenario without friction, if the normal contact force of the homogeneous case (40,099.7 N) is taken as a limit, the probability of exceeding the limit is \(1-P\) (normal contact force < 40,099.7 N), i.e., 48.2% in this paper.