## 2 Theoretical and Computational Models
### Principle and Application of Antilons Digging Resistance Reduction
#### 2.3.1 Non-Smooth Surfaces and Vibration Resistance Reduction Mechanism  
During digging, the body surface of the antilons came in contact with the soil, and the areas of contact could be divided into five layers. The adhesion between soil and antilon surface depended on the condition of the weakest layer among the five layers [32]. The body surface of the antilon and the soil are simplified to create a contact model, as shown in Figure 4.  
Taking \(\gamma_{S}\) as the surface free energy of the touching soil component, \(\gamma_{L}\) as the surface free energy of soil water, \(\gamma_{SL}\) as the free energy of the solid-liquid interface, and \(\theta\) as the equilibrium contact angle of soil water on the solid surface. \[\gamma_{S} = \gamma_{SL}+\gamma_{L}\cos\theta \tag{1}\]  
Figure 4: Schematic diagram of five-layer interface model: (**a**) Smooth surface; (**b**) Non-smooth surfaces. Figure 3: Statistics of the up–down and left–right vibration parameters of the antilons in four conditions: (**a**) Vibration frequencies; (**b**) Amplitudes. The adhesion work \(W_{A}\) required to separate the soil-touching component from the soil-water interface per unit area is  
\[W_{A}\ =\ \gamma_{S}+\gamma_{L}-\gamma_{SL}\ =\ \gamma_{L}(1+\cos\theta) \tag{2}\]  
Considering that there is an adsorbed water vapor layer on the surface of the soil-touching component, its expansion pressure is \(\varphi\), so the free energy on the surface of the soil-touching component changes from \(\gamma_{S}\) to \(\gamma_{SV}\), that is  
\[\gamma_{SV}\ =\ \gamma_{S}-\varphi \tag{3}\]  
In this case, the adhesion work required to separate the adhesion interface into two parts, the soil-touching component with water vapor and the soil surface, is denoted as \(W^{\prime}_{A}\),  
\[W^{\prime}_{A}\ =\ \gamma_{SV}+\gamma_{L}-\gamma_{SL}\ =\ \gamma_{S}+\gamma_{L}- \gamma_{SL}-\varphi\ =\ \gamma_{L}(1+\cos\theta)-\varphi \tag{4}\]  
Obviously, \(W_{A}\ >\ W^{\prime}_{A}\), so the typical damage could only occur in the water film layer. When the contact surface between the soil and the soil-touching components is smooth, the adhesion between the water film layers is larger, and the destructive force is also bigger. When it is a non-smooth surface, the adhesion between the water film layers is destroyed by the non-smooth surface so that the adhesion force at the water film interface changes from anisotropic to isotropic tendency and reduces the resistance to movement. Vibrational resistance reduction characteristics of antilons could also be analyzed from the perspective of soil shear destruction. The essence of its penetration process is the process of soil shear destruction under the action of external force. The maximum shear stress of soil is an important index of resistance analysis. According to the general expression of the Mohr-Coulomb failure criterion [33]:  
\[\tau\ =\ c+\sigma\cdot\tan\varphi \tag{5}\]  
The relationship between the Coulomb shear line and the Mohr circle in static and dynamic shear states was made, and the Mohr circle was simplified to a semicircle, as shown in Figure 5. \(\tau\) is the shear stress, \(c\) is the cohesion, \(\sigma\) is the normal stress, and \(\varphi\) is the angle of internal friction. According to the general rule of the Mohr-Coulomb damage criterion, no shear damage occurs in the soil when the Coulomb shear line deviates from the Mohr circle. When they are tangent, it is precisely at the critical position for soil damage; when they are cut, shear damage occurs in the soil [34]. In the static shear state, a Mohr circle tangent to the shear line is made. At this state, the tangent point between the static shear line and the Mohr circle is the critical point of shear damage, the shear stress is \(\tau_{0}\), and the angle of internal friction is \(\varphi_{0}\).