**1. Introduction**

Unsaturated clay is generally considered a three-constituent mixture composed of soil skeleton, water, and air, where the soil skeleton is formed by many solid mineral particles. Under external load, unsaturated clay will be compressed, and the three constituents inside will also deform correspondingly. During the initial loading period, the soil skeleton of unsaturated clay is generally the main bearing component and will deform from elastic to plastic with the increase of the external load. At the same time, most of the water and air inside can flow freely in the connected space among the solid particles due to the existence of the voids. When the external load increases to a threshold, the solid particles will be closely compacted and form many discrete enclosed rooms, in which nearly all the water and air are locked. At this stage, the soil skeleton, water, and air bear the same load. However, since the air has a greater compressibility than the water and the soil skeleton, it can be compressed to such an extent that its volume can be nearly ignored. Consequently, the force that the air bears is very small, compared with the force that the water and the soil skeleton bear, and it can also be neglected.

According to the Henrych theory [1], the deformation of an unsaturated clay under external load can be divided into two stages by the compacted pressure *p*c: the low-pressure stage and the high-pressure stage. As shown in Figure 1a, the low-pressure stage can be further separated into an elastic-deformed part and a plastic-deformed part by the elastic limit *p*e. In the elastic-deformed part, the soil skeleton deforms elastically, resulting in the elastic deformation of both the solid particles and the void content. When the external load is higher than the elastic limit *p*e, the shear force among some solid particles will exceed the bond strength, which leads to the fracture of some solid-particle combinations and the displacement among some solid particles. This deformation cannot be recovered once the external load has been released, indicating that the plastic deformation of the unsaturated clay has happened. During the external load increases from the elastic limit *p*e to the compacted pressure *p*c, numerous solid particles will be sheared to slip and rearrange to a new position to accommodate the external load, and form more and more enclosed

**Citation:** Ran, X.; Zou, X.; Zhou, J.; Tang, W. Shock Properties of One Unsaturated Clay and Its Equation of State Up to 30 GPa. *Crystals* **2022**, *12*, 119. https://doi.org/10.3390/ cryst12010119

Academic Editors: Shuhai Zhang, Yong He, Wenhui Tang, Yuanfeng Zheng and Chuanting Wang

Received: 4 December 2021 Accepted: 27 December 2021 Published: 17 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

rooms in which air and water are locked. At the compacted pressure *p*c, nearly all the solid particles are closely compacted, all the air and water are locked in, and there will be no void existing in the unsaturated clay. Therefore, in the low-pressure stage, the *p*-*alpha* equation of state (EOS) can be used to describe the state changes of the unsaturated clay. In the highpressure stage, all of the soil skeleton, water, and air have been greatly compressed and the elastic–plastic effect is no longer the dominant factor in the deformation, as shown in Figure 1b. With the increase of the external load, the volume ratio of air in the unsaturated clay decreases because of the pressure balance among the solid particles, water, and air, and the relatively grea<sup>t</sup> compressibility of the air, which indicates that the effect of air on the state of the unsaturated clay in the high-pressure stage can be neglected. Hence, in the high-pressure state, it is adequate to consider only the contributions of the water and the solid particles in EOS of the unsaturated clay. In addition, mineral components and water usually have a relatively high thermal capacity and there will be a relatively small temperature-rise in the dynamic shock process for the unsaturated clay with low porosity. Therefore, when the shock pressure is not high enough, the thermal contribution on shock pressure is small compared with cold contribution, and the EOS of unsaturated clay can be simplified to the form *p* = *f*(*ρ*) from the form *p* = *f* (*ρ*, *T*).

**Figure 1.** Schematic graph of the relation of *p* and *ρ*: (**a**) low pressure part; (**b**) high pressure part.

To study the mechanical and physical behaviors of the soil, many works have been conducted. Schofield [2] studied the mechanical behavior of saturated remolded soil, based on the critical state concept. Thiel [3], Kalashnikov [4], and Trunin [5] studied the shock wave data of the porous or hydrated earth materials and the sand–water mixture at several saturation levels. Tsembelis [6], Chapman [7], and Brown [8] completed a series of shock compressed experiments using a one-stage light gas gun to obtain the shock compressed behaviors of the dry sand. Resnyansky [9] theoretically put forward a two-substance EOS by regarding dry soil as the mixture of solid particles and gas. Using a three-substance EOS, Wang [10,11] carried out several numerical analyses on soil to obtain the dynamic response. These research results were important references for predicting the dynamic response of geotechnical material, but they could not be directly applied to unsaturated clay because moisture content was a significant factor affecting dynamic behavior. The shock loadings are the impact loads in terms of the wave dynamics.

To assess the effects of the underground explosion that happened in the southern suburbs of Luoyang city, China, in this study we conducted many experiments on the unsaturated clay with moisture contents of 0%, 8%, and 15%, respectively. Their Hugoniot parameters are obtained using a one- or two-stage light gas gun, based on which the high-pressure EOS up to 30 GPa is built. The developed EOS in this study can be applied for the numerical simulation of an underground explosion.

#### **2. Experimental Measurement of the Hugoniot Parameters**
