# Influence of Soil Heterogeneity on the Contact Problems in Geotechnical Engineering
## 1 Introduction  
In geotechnical engineering, contact problems are commonly seen, such as the contact between soils and concrete used in earth and rockfill dams [1, 2], tunnels [3] and coastal levees. In order to accurately predict the performance of these engineering projects, the contact between soils and concrete needs to be analysed. For example, the contact force applied on the concrete lining of tunnels is important for checking whether the strength of the concrete is qualified or not. If it is not qualified, the breakage of the concrete could result in water leakage and even subsequent failure of these projects. However, due to the unknown contact region and contact forces, the contact problems have strong boundary nonlinearity, which makes the problem difficult. Currently, the frequently used method in solving contact problems contains the Goodman element method [4], the thin-layer element method [5], the Lagrange multipliermethod [6; 7; 8; 9] and the penalty method [10; 11]. The Goodman element method is the most widely used method to simulate the interface contact between two materials with different modulus. In this method, the thickness of the Goodman element is zero [2; 12]. In contrast, in the thin-layer element method, the interface is replaced by an equivalent solid element with a small thickness. In addition, the element is given a relatively low modulus and can experience large deformation [13]. The Lagrange multiplier method introduces additional variables (the Lagrange multiplier), e.g., the contact force, to enforce directly and exactly the contact constraints. The penalty method enforces the contact constraints by applying virtual springs with zero rest lengths to surfaces in contact. A point is penalized for penetrating another project. Qian et al. [14] compared the Goodman element method, the thin-layer method and the penalty method in a numerical simulation of a concrete-faced rockfill dam. The comparison results indicated that the penalty method can reflect the contact stress along the interface between the soil and the concrete slab well, especially when there is a gap which emerges between the soil and the concrete slab. In addition, compared to the Lagrange multiplier, the penalty method is easy to implement and there are no additional efforts required to solve the multiplier [15]. Therefore, it is worth investigating the contact behavior by using the penalty method. In the previous research of contact problems, parameters are usually considered to be homogeneous. However, heterogeneity has proven to be influential in many fields, such as Zeng et al. [16], Amato et al. [17], etc. In geotechnical engineering, soils have already been recognized as heterogeneous materials. The effect of soil heterogeneity has been investigated in many geotechnical problems, such as bearing capacity [18; 19], seepage [20; 21; 22] and slope stability/displacement [23; 24]. In Fenton and Griffiths [18] and Cho and Park [19], the cohesion and friction angle are considered as random variables. The results showed that the bearing capacity of heterogeneous foundations presented significant variation compared to the homogeneous foundations. The variation was strongly related to the mean, standard deviation and the correlation length of the soil properties. Griffiths and Fenton [20] investigated the seepage beneath a water retaining structure. In the study, the hydraulic conductivity was taken to be a random variable. The heterogeneity of hydraulic conductivity influenced the exit gradient, the flow rate and the uplift force significantly. Liu et al. [21] demonstrated the influence of the heterogeneity in hydraulic conductivity on the pore pressure distribution and the subsequent slope stability of an embankment. Yetbarek et al. [22] numerically examined the effects of the heterogeneity in hydraulic parameters on subsurface water movement which accounts for root water uptake.