# Numerical Study of Cone Penetration Tests in Lunar Regolith for Strength Index
## 1 Introduction  
In recent years, large-scale lunar exploration initiatives have been undertaken worldwide, with notable projects including NASA's Artemis program, China Lunar Exploration Program (CLEP) [1], and India's Chandrayaan program [2]. The study of the in situ physical properties of lunar soil is of great significance for the construction of the lunar research base. As a practical in situ geotechnical test method, the cone penetration test (CPT) is widely used in various soil materials. With mature theoretical and technical support, it has become a possible means to be applied to the lunar surface to measure in situ soil parameters of lunar soil in the future. However, the relevant formulas must consider the influence of the special environment on the lunar surface. During the Apollo missions in the United States, [PERSON] et al. [3] used parabolic flights of airplanes toresistance and spherical stress, and these values are linearly correlated with the reciprocal gravity. These works mainly studied the experimental results of the CPT in the lunar environment, but there is no summary of the empirical formula of CPTs considering the lunar environment. The essence of lunar soil is silty sand. Based on a CPT, a variety of relevant empirical formulas have been proposed to explain soil property parameters of sandy soil in the ground environment. [PERSON] (1977) reported that the internal friction angle of sand can be calculated using Equation (1), which is based on both laboratory tests and field in situ test data from sand foundations,  
\[\varphi=\tan^{-1}(\frac{q_{\mathrm{c}}}{0.5N_{\mathrm{q}}}) \tag{1}\]  
where \(\varphi\) is the internal friction angle; \(q_{\mathrm{c}}\) represents the cone resistance; \(N_{\mathrm{q}}\) is the bearing capacity factor. [PERSON] (1977) found that the cone resistance obtained from CPTs is influenced by the effective overburden stress of soil. Consequently, the internal friction angle (\(\varphi\)) of sand is related to both cone resistance (\(q_{\mathrm{c}}\)) and effective overburden stress (\(\sigma_{\mathrm{v}}^{{}^{\prime}}\)). They subsequently introduced Equation (2) to calculate the internal friction angle of sand. \[\varphi=\tan^{-1}\left(0.1+0.38\log\left(\frac{q_{\mathrm{c}}}{\sigma_{\mathrm{ v}}^{{}^{\prime}}}\right)\right) \tag{2}\]  
[PERSON] and [PERSON] (1977), building on [PERSON]'s work and utilizing triaxial test data, proposed a logarithmic formula for calculating the internal friction angle (\(\varphi\)) of sand using cone resistance (\(q_{\mathrm{c}}\)) and effective overburden stress (\(\sigma_{\mathrm{v}}^{{}^{\prime}}\)), which is presented in Equation (3). The results of further model tests indicated that this formula can be used to effectively calculate the internal friction angle for both normally consolidated and overconsolidated sand. \[\varphi=17.6+11\log(\frac{q_{\mathrm{c}}}{\sqrt{100\sigma_{\mathrm{v}}^{{}^{ \prime}}}}) \tag{3}\]  
[PERSON] et al. (2010) developed a high-quality sand database and proposed a method for calculating the internal friction angle (\(\varphi\)) of sand, as shown in Equation (4). This method demonstrates a high degree of fit with experimental results. \[\varphi=25(\frac{q_{\mathrm{c}}}{\sqrt{100\sigma_{\mathrm{v}}^{{}^{\prime}}}} )^{0.1} \tag{4}\]  
Relative density, an important parameter of sand, is a key indicator used for evaluating the engineering properties of sand. [PERSON] (2011), utilizing data from laboratory CPT model tests, was the first to propose a formula for calculating the relative density (\(D_{\mathrm{r}}\)) of sand using cone resistance (\(q_{\mathrm{c}}\)), as shown in Equation (5),  
\[D_{\mathrm{r}}=\frac{1}{C_{2}}\ln[(\frac{q_{\mathrm{c}}\left(\sigma_{\mathrm{ v}}^{{}^{\prime}}\right)^{C_{1}}}{C_{0}})] \tag{5}\]  
where \(C_{0}\), \(C_{1}\), and \(C_{2}\) are empirical correlation coefficients not specifically provided in the literature.