Analysis of Experimental Results on the Bearing Capacity of Sand in Low-Gravity Conditions

Abstract

With advancements in space exploration, operations in low-gravity environments such as the lunar surface are expected to be conducted. Geotechnical engineering problems such as those associated with the foundation bearing capacity considerably influence the feasibility and safety of these operations. In this study, the findings of experiments conducted by Japan Aerospace Exploration Agency (JAXA) scientists on parabolic flight, aimed at measuring the ultimate bearing capacity of shallow foundations under 1/6 g to 2 g of gravity, are analysed. Specifically, the results are analysed to calculate the ultimate friction angle based on the classical Terzaghi limiting equilibrium solution in soil mechanics. The friction angle of foundation sand increases as the gravity level decreases. This finding is verified through advanced arbitrary Lagrangian–Eulerian (ALE) finite element simulations based on a simple Mohr–Coulomb model. Moreover, the underlying mechanism for this phenomenon is examined considering an ALE finite element simulation based on a newly developed rheological model known as the Tsinghua–MiDi sand model. The pressure-sensitive and rate-dependency constitutive behaviour of sand is clarified. Notably, this phenomenon increases the viscous shear stress and ultimate friction angles in low-gravity conditions. The coupled effects of the loading rate and low-gravity level on the bearing capacity of foundation sand are predicted. The findings can provide a novel theoretical prospective for geotechnical studies in space exploration engineering in low-gravity conditions.

Appendix 1

Tsinghua–Midi Sand Constitutive Model

The model assumptions are as follows:

• (1) Sand can be regarded as an incompressible granular flow in the steady shear state. In the unsteady shear state, a linear elastic bulk module $K$ is used to describe the relationship between the isotropic effective stress ${p}^{^{\prime}}$ and volume strain ${\varepsilon }_{kk}$.

• (2) The shear-stress–strain relationship of sand is developed for isotopically consolidated samples.

The total stress component can be decomposed into the quasi-static and viscous components, formulated as

$${\sigma }_{ij}\left({\varepsilon }_{ij},{\dot{\varepsilon }}_{ij}\right)={\delta }_{ij}{p}^{^{\prime}}\left({\varepsilon }_{kk}\right)+{\tau }_{ij}^{s}\left({\varepsilon }_{ij}\right)+{\tau }_{ij}^{v}\left({\varepsilon }_{ij},{\dot{\varepsilon }}_{ij}\right)$$

(4)

To calculate the isotropic stress component ${p}^{^{\prime}}$ a constant bulk module is considered according to the first assumption, as indicated in Eq. (5). For the quasi-static shear stress components ${\tau }_{ij}^{s}$, a nonlinear shear module based on a large database (Cheng et al. 2019) of sand shear tests is used, as indicated in Eqs. (6) and (7). In the formulation, ${e}_{ij}={\varepsilon }_{ij}-{\varepsilon }_{kk}{\delta }_{ij}/3$ is the deviator strain tensor, ${\gamma }_{ij}=2{e}_{ij}$ is the defined as engineering deviatoric strain tensor, and $\gamma =\sqrt{\frac{1}{2}{\gamma }_{ij}{\gamma }_{ij}}$ is the second invariant of such tensor.

$${p}^{^{\prime}}=K{\varepsilon }_{kk}$$

(5)

$${\tau }_{ij}^{s}=G\left(\gamma \right){\gamma }_{ij}$$

(6)

$$G=\frac{{A}_{0}}{1+\gamma /{\gamma }_{r2}}\frac{{p}_{a}}{{\left(1+e\right)}^{3}}{\left(\frac{{p}^{^{\prime}}}{{p}_{a}}\right)}^{m\left(\gamma \right)}$$

(7)

The quasi-static friction coefficient ${\mu }_{m}$, defined as the ratio of the quasi-static shear stress to the effective isotropic stress, can be formulated in terms of $\left|{\tau }^{s}\right|=\sqrt{\frac{1}{2}{\tau }_{ij}^{s}{\tau }_{ij}^{s}}$, which is the second invariant of the quasi-static shear stress tensor ${\tau }_{ij}^{s}$:

$${\mu }_{m}=\frac{\left|{\tau }^{s}\right|}{{p}^{^{\prime}}}$$

(8)

The viscous shear stress component is assessed by extending the MiDi rheological steady state theory to the transient shear state (MiDi, 2004). The viscous shear stress ${\tau }_{ij}^{v}$ is determined using the viscous friction coefficient ${\mu }_{v}$ as shown in Eqs. (9) and (10).

$${\tau }_{ij}^{v}={\mu }_{v}\left(I\right){p}^{^{\prime}}\frac{{\dot{\gamma }}_{ij}}{\left|\dot{\gamma }\right|}$$

(9)

$${\mu }_{v}\left(I\right)=\frac{{\mu }_{1}-{\mu }_{m}}{1+{I}_{0}/I}$$

(10)

$$I=\dot{\gamma }d\sqrt{\frac{{\rho }_{s}}{{p}^{^{\prime}}}}$$

(11)

In these expressions, the inertial number $I$ is an innovative state variable, $\dot{\gamma }$ is the engineering shear strain rate, $d$ is the average particle diameter, ${\rho }_{s}$ is the specific density of sand particle, and ${\mu }_{1}$ is the upper bound value of the dense granular steady flow friction coefficient.