*2.1. Subsection*

The original clay was extracted from the southern suburbs of Luoyang City in China. The detailed clay minerals of the clay are presented in Table 1. The density, moisture content, and solid particles of the original clay varied with the place and depth. Therefore, it was not possible to obtain the original clay samples with the same density and moisture content applied for experiments. All the samples used in this study were remolded according to the experimental needs.

**Table 1.** The mineral components of the dry original clay.


To study the influence of moisture content on the dynamical behaviors of clay, three unsaturated clay samples were prepared with a dry density of 1.70 g/cm<sup>3</sup> and initial moisture contents of 0%, 8%, and 15%, respectively. The clay samples were fabricated as follows: first, dry the dispersed clay in the oven; second, take the appropriate weight of clay with electronic scales; third, weigh the water with the measuring cylinder or injector and add it into the dried clay to the wanted moisture content; once the water diffuses in the clay uniformly, the clay sample should be put into the designed mold made of 2024 aluminum. As shown in Figure 2, the mold consists of a sample supporter, a reinforcement cylinder, and a compaction piston. The sample supporter had a circular indentation of 16 mm in diameter and 3 mm in depth. The reinforcement cylinder had a circular center hole of 16 mm in diameter and its depth was almost the same as the height of the compaction piston, which made the two parts connected closely and made the thickness of each sample almost the same. Before compressing the clay into the sample, an oil film should be evenly formed on the inner wall of the center bore of the reinforcement cylinder in order to reduce friction between the reinforcement cylinder and the piston. With the jack compressing the piston slowly, the clay in the mold can be compressed into the same size of ϕ16 × 3 mm, as shown in Figure 3. Table 2 gives the physical parameters of the final compressed clay samples of three moisture contents.

**Figure 2.** Schematic of sample mold.

**Figure 3.** Schematic of experimental sample preparation.


**Table 2.** Physical parameters of clay in plate impact test.

After the preparation of the experimental sample, the sample-holder was placed behind the target-stand to assemble a whole target, as shown in Figure 4a. After assembly, the target was installed in the terminal end of the light gas gun tube, as shown in Figure 4b.

(**a**) (**b**) 

**Figure 4.** The target for plate-impact experiment graph: (**a**) the sample-holder and the target-stand; (**b**) back view of the assembled target.

#### *2.2. Experimental Method*

To obtain a wider range of loading pressure, the flyers made of 2024 aluminum alloy, copper, and tantalum were used in the plate-impact experiments. The flyers are discs of ϕ24 × 3 mm and their Hugoniot parameters [12] are shown in Table 3. Impacting velocities of flyers were measured by a velocity magnetic-measuring device [13].

**Table 3.** The parameters [11] of flyer materials and Hugoniot.


Shock wave velocities were measured by the optical probes made from quartz fiber, which were calibrated in detonation experiments and had an uncertainty of about 1.8% [14]. When the shock wave propagated to the end of the probe, the quartz fiber would radiate optically due to shock wave arrival and the moment of shock wave arrival would be recorded by the digital oscillograph. The structure of the optical probe used in the experiments, as shown in Figure 5, consisted of one quartz fiber with a core diameter of 60 μm and an outside diameter of 175 μm, and one metal capillary with the internal diameters of 0.3 mm and 0.9 mm. The quartz fiber and metal capillary were cemented together with quick dry glue, in which the metal capillary plays an important role in enhancing the strength of the optical probe and keeping the optical probes in a fixed position perpendicularly. The surface of the end of the optical probe was coated with a 120~150 nm thick aluminum film which could prevent stray light from entering the quartz fiber.

The schematic diagram of one single-fiber probe system, as shown in Figure 6, consists of a photomultiplier tube and a digital oscillograph. The photomultiplier tube is the GDB-608 MCP and has an impulse response time of no more than 0.4 ns. The digital

oscillograph used in the experiments has a sampling rate of 5 GS/s and an analog output bandwidth of 1 GHz.

**Figure 5.** The schematic diagram of the optical probe of quartz fiber.

**Figure 6.** The schematic diagram of one single-fiber probe system.

In the experiments, there were a total of five fiber-optic pins (No. A-E) arranged for measuring the Hugoniot parameters of the sample. As shown in Figure 7, four fiber-optic pins (No. A-D) were arranged at the impacted surface and the outer edge of the sample to record the moment when the flyer just impacted the clay sample. Two diagonal fiber-optic pins formed a recording channel; there were a total of two channels that could amend the error arising from an oblique collision between the flyer and the clay sample. The last fiber-optic pin (No. E) was arranged in the center of the rear plane of the clay sample to record the moment of shock wave arrival.

**Figure 7.** The schematic diagram for measuring Hugoniot parameters. (**a**) Arrangement position of fiber-optic pins. (**b**) Fiber-optic pins test system.

#### *2.3. Data Processing Method*

The typical signals in the experiments, as shown in Figure 8, were obtained from two measuring channels. The signal in Figure 8a was taken from the four fiber-optic pins (No. E, A, B, and C) and the signals from fiber-optic pins E, B, and C had almost the same wave-form. It can be seen from Figure 8 that the signals from fiber-optic pins A, B, C, and D had almost the same jumping moment, which indicated that the flyer had preferable planarity and a small deflection error when the flyer impacted the target. In all the signals, the inflection points showed the moments that the flyer or shock wave arrived at the corresponding fiber-optic pin. With these moments, the shock wave velocity *U*s in the sample at this shocked state could be calculated from the following equation:

$$
\Delta l\_s = \Delta h / \Delta t \tag{1}
$$

where Δ*h* was the thickness of the sample and Δ*t* was the interval that the shock wave took for traveling in the sample.

**Figure 8.** The typical signals of fiber optical pins in shock experiments: (**a**) optical pin signals from the first channel; (**b**) optical pin signals from the second channel.

For most materials, one linear relation between the shock wave velocity *U*s and the particle velocity *<sup>u</sup>*p existed as the following:

$$\mathcal{U}\_{\rm s} - u\_{\rm 0} = \mathcal{c}\_{\rm 0} + s \left( u\_{\rm P} - u\_{\rm 0} \right) \tag{2}$$

where *c*0 and *s* were the Hugoniot parameters and *u*0 was the initial particle velocity. If a flyer impacted a sample plate with the impacting velocity *W*, according to the Rayleigh line and Equation (2), the shock pressure in the flyer should be written as:

$$p\_{\rm f} = \rho\_{0\rm f}(\mathcal{U}\_{\rm s} - \mathcal{W})(\mathcal{u}\_{\rm p} - \mathcal{W}) = \rho\_{0\rm f} \left[ -\mathfrak{c}\_{0\rm f} + s\_{\rm f}(\mathcal{u}\_{\rm p} - \mathcal{W}) \right](\mathcal{u}\_{\rm p} - \mathcal{W})\tag{3}$$

According to Equation (3), for the given impacting velocity *W* and the measured Hugoniot parameters in Table 2, the shock pressure in the flyer could be expressed as a function of particle velocity *<sup>u</sup>*p.

For the *i*th impact experiment with the velocity *Wi* and the measured shock wave velocity *U*s*i*, the particle velocity *<sup>u</sup>*p*<sup>i</sup>* can be calculated according to the impedance match method [8] as shown in Figure 9:

$$
\mu\_{\rm pi} = \frac{-B - \sqrt{B^2 - 4AC}}{2A} \tag{4}
$$

where

$$A = \rho\_{0\text{fsf}} \tag{5}$$

$$B = -\left(\rho\_{0\text{f}}c\_{0\text{f}} + 2\rho\_{0\text{f}}s\_{\text{f}}\mathcal{W}\_{\text{i}} + \rho\_{0\text{s}}lI\_{\text{si}}\right) \tag{6}$$

$$C = \rho\_{0\text{f}} \mathcal{W}\_{i} (c\_{0\text{f}} + s\_{\text{f}} \mathcal{W}\_{i}) \tag{7}$$

**Figure 9.** The impedance match method.

The subscript f and s meant the flyer and sample plates, and the subscript 0 meant the initial state.

#### *2.4. Experimental Results and Analysis*

According to the moisture content of the samples, the shock-impacted experiments were divided into three groups and each group had four samples with the same moisture content. Twelve effective experimental data were obtained in total. From these experimental data, the flyer plate impacting velocities, the shock wave velocities, and the particle velocities were derived with the experimental method in Section 2.2 and the experimental data processing method in Section 2.3. Moreover, the shock wave pressure in the samples was also obtained with the Hugoniot relation. All the experimental measured data and the processed data are shown in Table 4.


**Table 4.** Experimental results for unsaturated clay.

According to the shock wave velocities and the particle velocities (*Usi*, *upi*) for four samples of the same moisture content, the linear relation such as Equation (2) could be fitted with the least square method. Processed linear relations for the clay of the three different moisture contents are shown in Figure 10.

**Figure 10.** The linear relation *<sup>U</sup>*s-*u*p for the clay of three moisture contents.

With Figure 10, the material parameters in the Hugoniot linear relation for the clay of 0% moisture content, 8% moisture content, and 15% moisture content could be derived and they had the following expressions.

For the clay samples of 0% moisture content, the material parameters were:

$$c\_0 = 1.08 \pm 0.30 \text{ km/s} \quad s = 1.62 \pm 0.17 \tag{8}$$

For the clay samples of 8% moisture content, the material parameters were:

$$c\_0 = 1.29 \pm 0.24 \text{ km/s} \quad s = 1.72 \pm 0.14 \tag{9}$$

For the clay samples of 15% moisture content, the material parameters were:

$$
\omega\_0 = 1.91 \pm 0.18 \text{ km/s} \quad s = 1.71 \pm 0.10 \tag{10}
$$

#### **3. Equation of State of Unsaturated Clay**

In Section 2, shock pressures in the experiments were not more than 30 GPa and the shock temperature rise can be neglected. Therefore, the EOS of the unsaturated clay can take the form *p* = *f*(*ρ*). As shown in Figure 1, the equation of state of the unsaturated clay should include two deformation mechanisms when considering the critical pressure *p*c. Our solution is to use the *p*-*alpha* compaction model [15] when the pressure *p* < *p*c, and to use the EOS of a solid–liquid two-phase mixture when the pressure *p* > *p*c.

#### *3.1. p-alpha Compaction Model*

The *p*-*alpha* compaction model was firstly presented by Herrmann [16] for porous materials. In this model, the stiffness of the skeleton material was neglected and the porous material could be thought of as an isotopic material. Therefore, the stress state of porous material can be described with hydrostatic pressure *p*, and the EOS for *p*-*alpha* compaction model has the following forms, containing the hydrostatic pressure *p*, specific volume *v,* and internal energy *e*:

$$
\boldsymbol{v} = \boldsymbol{f}(\boldsymbol{v}, \boldsymbol{e}) \tag{11}
$$

To differentiate the specific volume change of the skeleton material from the change of pore shape in the deformation process under external force, the porosity *α* was introduced and defined as:

$$\alpha = \frac{\upsilon}{\upsilon\_{\text{s}}} = \frac{\upsilon(p, e)}{\upsilon\_{\text{s}}(p, e)} \tag{12}$$

where *v*s was the specific volume of skeleton material and *v* was the specific volume of corresponding porous material at the same state.

In the study on the state change of porous material, the surface energy of voids was usually neglected, thus the porous material had the same specific internal energy as the skeleton material. If the hydrostatic pressures of the porous and skeleton material were thought to be identical in any condition, it meant that only the specific volumes were different when the porous and skeleton material were in the same state. If the skeleton material was dense, the EOS of solid as the following could be used:

$$
\sigma = f(v\_{\mathfrak{k}}, \mathfrak{e}) \tag{13}
$$

 After introducing porosity *α*, the following equation could be obtained:

$$p = f\left(\frac{v}{\alpha}, c\right) \tag{14}$$

In the low-pressure range *p* < *p*c, the porosity *α* depended on the hydrostatic pressure *p* and the specific internal energy *e*. It was a key problem to determine the porosity *α* in the EOS of porous material under low-pressure. Herrmann thought that the porosity *<sup>α</sup>*(*p*, *e*) could be approximated by *α*(*p*) along the Hugoniot curve as shown in Figure 11 under the condition that the compressibility of the porous material was insensitive to the temperature.

**Figure 11.** Schematic of porous material *α*(*p*).

The porosity *α* has little influence on the compaction during elastic loading, so the initial porosity *α*0 can be approximated by *α*e and then the porosity *α* can be specified as a function of pressure:

$$\alpha(p) = 1 + (\alpha\_0 - 1) \left( \frac{p\_\text{s} - p}{p\_\text{s} - p\_\text{c}} \right)^N \tag{15}$$

where *p*e was the elastic limit of porous material, when the pressure was larger than *pe*, the voids began to collapse; *p*s was the pressure at which the porous material was compacted into a completely solid state. *N* was one parameter and equals 2 in Herrmann's article, but later studies [17,18] showed that the parameter *N* could be determined according to the experimental results to ge<sup>t</sup> a better description of the compaction process of porous material.

In this study, the unsaturated clay was considered as a kind of porous material and *p*c = *p*s was assumed. Therefore, *p*c could be given by the Hugoniot equation:

$$p\_{\mathbb{C}} = \frac{\rho\_{\mathbb{C}} c\_0^2 \eta\_{\mathbb{C}}^2}{\left(1 - s \eta\_{\mathbb{C}}\right)^2} \tag{16}$$

where *c*0 and *s* were Hugoniots and had been determined in experiments as shown in Section 2 and *η*c = 1 − *ρ*0/*ρ*c, *ρ*c was the density of the fully compacted clay.

After being fully compacted, the clay contained only water and solid particles, and the fully compacted density *ρ*c was given by:

$$\rho\_{\text{c}} = \frac{m}{v\_{\text{c}}} = \frac{m\_{\text{W}} + m\_{\text{s}}}{v\_{\text{WC}} + v\_{\text{sc}}} \tag{17}$$

where *m* was the total mass of clay; *v*c was the volume of the fully compacted clay, corresponding to the pressure *p*c; the subscript *w* and *s* represented water and solid particles, respectively.

#### *3.2. p-alpha Compaction Model*

Unsaturated clay was a three-phase media comprised of solid particles, water, and air. When the unsaturated clay had been fully compacted, the fully compacted pressure *p*c (about 1 GPa) was so high that not much air was left in the clay and the compressibility of the skeleton could also be neglected. It meant that the high-pressure EOS of the unsaturated clay mainly came from the contribution of water and solid particles.

The relative volumes *β*wp and *β*sp for the water and the solid phase in the clay could be introduced as:

$$
\beta\_{\rm WP} = \frac{v\_{\rm WP}}{v\_{\rm}}, \quad \beta\_{\rm SP} = \frac{v\_{\rm NP}}{v\_{\rm}} \tag{18}
$$

where *<sup>v</sup>*wp (or *v*sp) was the volume that the water (or the solid) phase had if the hydrostatic pressure in the clay was *p*. When the unsaturated clay was fully compacted, the relative volumes *β*wc and *β*sc were as follows:

$$
\beta\_{\rm wc} = \frac{v\_{\rm wc}}{v\_{\rm c}}, \quad \beta\_{\rm sc} = \frac{v\_{\rm wc}}{v\_{\rm c}} \tag{19}
$$

and satisfied the equation:

$$
\beta\_{\rm wc} + \beta\_{\rm sc} = 1 \tag{20}
$$

According to the mass conservation, the density *ρ* of the clay under the pressure *p* was given as:

$$\rho = \frac{m}{v} = \frac{\rho\_\text{c} v\_\text{c}}{v} = \frac{\rho\_\text{c}}{\beta\_\text{wp} + \beta\_\text{sp}} \tag{21}$$

where *v* = *<sup>v</sup>*wp + *<sup>v</sup>*sp was the volume of the clay under the pressure *p*. Here, the relative volume of water *β*wp could be determined using the EOS of water [1]:

$$p = p\_0 + \frac{\rho\_{\rm w0} \cdot c\_{\rm w0}^2}{k\_{\rm w}} \left[ \left( \frac{\rho\_{\rm w}}{\rho\_{\rm w0}} \right)^{k\_{\rm w}} - 1 \right] \tag{22}$$

where *p*0 = 10<sup>5</sup> Pa, *ρ*w0 = 1.0 × 10<sup>3</sup> kg/m3, *c*w0 = 1415 m/s, *k*w = 3.

In the same way, the relative volume of solid particles *β*sp could also be determined by the following EOS [1]:

$$p = p\_{\rm c} + \frac{\rho\_{\rm s0} \cdot c\_{\rm s0}^2}{k\_{\rm s}} \left[ \left( \frac{\rho\_{\rm s}}{\rho\_{\rm s0}} \right)^{k\_{\rm s}} - 1 \right] \tag{23}$$

where *ρ*s0 = 2.73 × 10<sup>3</sup> kg/m3, *c*s0 = 4500 m/s, *k*s = 3, *p*c was the fully compacted pressure of clay determined by experiment results.

Substituting Equations (22) and (23) into Equation (21), the following equation could be obtained, by which the density of the clay under any pressure could be determined:

$$\rho = \rho(p) = \rho\_{\rm c} \left[ \beta\_{\rm wc} \left( \frac{p - p\_0}{\rho\_{\rm w0} c\_{\rm w0}^2} k\_{\rm w} + 1 \right)^{-1/k\_{\rm w}} + \beta\_{\rm sc} \left( \frac{p - p\_{\rm c}}{\rho\_{\rm s0} c\_{\rm s0}^2} k\_{\rm s} + 1 \right)^{-1/k\_{\rm s}} \right]^{-1} \tag{24}$$

Based on the aforementioned information and the experimental results, the physical parameters in the EOS of the unsaturated clay with the three moisture contents could be determined, and they are shown in Table 5.

Figure 12 shows the theoretical results from the EOS and the experimental results of the density variation of the clay samples with pressure. It is evident that the compressive strength of the clay sample increased with the increase of the moisture content. The water and the air trapped in the voids of the clay could not be removed under high strain rate and high loading; therefore, the shock compressible behavior of the unsaturated clay was dominated by the compressibility of the trapped water and air and the solid particles. Because water was a relatively incompressible material, the increase of water content led to an increase in the compressive strength of water-bearing clay. Al'tshuler and Pavlovskii [19] carried out shock compressed experiments on unsaturated clay samples with 4% and 20% moisture contents; the mineral components of their clay samples were mainly quartz and kaolin, which were similar to the samples used in this study. They also concluded the same results that the compressive strength of clay increases with the increase of the moisture content.


**Table 5.** The parameters of equation of state of unsaturated clay.

**Figure 12.** Comparison of theoretical and experimental data for the equation of state in terms of pressure and density.

As indicated by the agreemen<sup>t</sup> between the theoretical results and the experimental ones in Figure 12, the EOS of the unsaturated clay proposed in this study could give a better description of the relation between the pressure and the density of the clay, and it could also reflect the different shock compressed behaviors of the unsaturated clay resulting from the moisture content variation.
