# Numerical Study of Cone Penetration Tests in Lunar Regolith for Strength Index
## 2 Numerical Model  
### Constitutive Particle Contact Model of Lunar Regolith  
In terms of particle size range, lunar soil is similar to dry silt on the Earth, but the special environment of high vacuum, low gravity, and strong radiation on the lunar surface makes lunar soil show different mechanical properties from dry silt. The most significant difference is that the van der Waals force between particles of lunar soil makes lunar soil have a certain cohesion. [PERSON] et al. [5] incorporated van der Waals forces into the discrete element method (DEM) particle contact model to simulate the cohesive effects of lunar regolith in lunar surface environments. They used a rolling resistance model to explain the high inter-particle friction angle resulting from the angular and subangular shapes of lunar regolith particles. Many scholars have proposed lunar regolith particle contact models that consider van der Waals forces [15; 16; 17; 18] and have continuously revised them to more accurately reflect the true mechanical behavior of lunar regolith. However, none of these studies have addressed the hysteresis effect of van der Waals forces. The lunar regolith particle contact model used in the simulations in this study was the one modified in a previous study [19], which is shown in Figure 1.  
In Figure 1, n, s, r, and t represent the normal, tangential, bending, and torsional directions of the contact surface, respectively; l, v, and d denote the properties of the linear force, van der Waals force, and damping force, respectively; g represents the separator; \(D\) indicates the micro-contact distance between particles; \(\mu\) denotes the friction coefficient at the contact; \(k\) and \(\beta\) represent the spring stiffness and damping coefficient on the contact surface, respectively; \(F\) and \(M\) denote the force and moment (bending moment/torsion) acting on the contact surface, respectively. Research [20] shows that the completely nonhysteretic van der Waals force exists only when the distance scale is very small, and with the increase of distance, the van der Waals force will show hysteresis and gradually transform into Casimir force. [PERSON] et al. [21] studied the conversion between van der Waals forces and Casimir forces on nanostructured surfaces based on [PERSON]'s Nonlocal Elasticity Theory. The results show that the surface spacing dominated by van der Waals forces ranges from 0.5 nm to 4 nm. When the spacing gradually increases to 18 nm, the dominance of the van der Waals force decreases with the increase of the spacing, and the Casimir force gradually dominates, and when the spacing is greater than 18 nm, it is completely dominated by the Casimir force. Therefore, the hysteresis effect of van der Waals forces should be considered in the contact model of lunar soil particles, so as to reflect the variation of van der Waals forces between particles and ensure the accuracy and rationality of the contact model. This contact model accounts for the hysteresis effect of van der Waals forces as the microscopic distance (\(D\)) between particles increases, as well as the influence of particle contact proximity on the magnitude of these forces. From the model, a piecewise function is derived from the van der Waals forces between lunar regolith particles with respect to \(D\), based on Lifshitz theory, as shown in Equation (9),  
\[F_{\rm v}(D)=\left\{ \begin{array}{ll}-\frac{A_{\rm L}R_{\rm c}^{2}}{6D_{\rm min }^{3}}&D_{min}>D\\ -\frac{A_{\rm L}R_{\rm c}^{2}}{6D_{\rm max}^{3}}&D_{min}\leq D\leq D_{0}\\ -\frac{A_{\rm L}R_{\rm c}^{2}(D_{\rm 1}-D)}{6D_{\rm 0}^{3}(D_{\rm 1}-D_{\rm 0})}- \frac{\pi_{\rm J}hcR_{\rm c}^{2}(D_{\rm 1}-D_{\rm 0})}{240D_{\rm 1}^{4}(D_{ \rm 1}-D_{\rm 0})}&D_{0}\leq D\leq D_{1}\\ -\frac{\pi_{\rm J}hcR_{\rm c}^{2}}{240D^{4}}&D_{\rm 1}\leq D\leq D_{max}\\ 0&D_{max}\leq D\end{array} \right.