Mechanical Characteristics and Particle Breakage of Calcareous Sand under Quasi-One-Dimensional Impact Load

Abstract

Calcareous sand, a type of marine sediment formed from the skeletal remains of marine life, exhibits unique characteristics such as high porosity and fragility due to its biological origin. Particle breakage is a key attribute of calcareous sand. Given that foundations on calcareous sand islands encounter various types of loads, including pile driving, aircraft loading, earthquakes, and tsunamis, it is imperative to investigate its mechanical properties and particle breakage under high strain rates. This study focuses on assessing the dynamic mechanical properties of calcareous sand under quasi-one-dimensional impact loads using split Hopkinson pressure bar (SHPB) tests. Three particle sizes of calcareous sand with different water contents, strain rates, and relative densities were examined. The particle fragmentation degree of each sand sample was also analyzed quantitatively. The results indicated that stress–strain curves progress through an elastic phase with rapid elevation, followed by a plastic stage with a slower increase under various factors. Within the plastic phase, there are multiple instances of stress drops and recoveries. The stress–strain curves generally decrease as particle size increases, concurrent with an increase in particle breakage. Moisture content has minimal impact on the stress–strain curve; a higher moisture content does correspond to reduced particle breakage. Both the maximum strain and peak stress increase as the strain rate increases, resulting in a higher relative crushing rate. The difference between stress–strain curves under different relative densities diminishes as particle size increases, and greater relative density leads to reduced particle breakage. Functional relationships among peak stress and strain rate, relative fragmentation rate and water content, strain rate and relative density, as well as relative density and peak stress are also established.

1. Introduction

Calcareous sand is widely distributed throughout various regions including the South China Sea [1,2], Croatia [3], and coastal areas of Egypt [4,5]. It is formed by deposits of skeletal remains from marine life [6], with more than 50% percent CaCO₃ [7]. Calcareous sand is porous [8,9] and brittle [10,11] due to its unique biological origins. As the foundation material for reefs and islands, it presents distinct geotechnical challenges due to its specific mechanical properties and propensity for particle breakage. In-depth investigations into the mechanical properties and particle breakage characteristics of calcareous sand are essential for the sustainable exploitation of marine resources.

The split Hopkinson pressure bar (SHPB) test is a widely used method for examining the mechanical properties of construction materials subject to impact loading, including concrete [12,13], rock [14,15], and granular material cement [16]. Luo et al. [17] performed SHPB tests to investigate the dynamic compression behavior of Eglin sand with different particle sizes and moisture contents at a high strain rate. Song et al. [18] investigated the quasi-static compressive properties of dry silica sand with SHPB tests to find that its compressive response is not sensitive to strain rate, but significantly dependent on the initial density. These studies affirm that the behavior of dry silica sand is not sensitive to the strain rate but is indeed linked to relative density [19] and moisture content [20] under quasi-one-dimensional impact loading.

In contrast to silica sand, calcareous sand has numerous internal pores and distinct mineralogical properties. Zhao et al. [21] found that dry calcareous sand exhibits no significant rate effects within the strain rate range of 335 s⁻¹ to 1253 s⁻¹ in SHPB tests. Lv et al. [22,23] conducted SHPB tests to observe the effects of relative density, strain rate, and particle size on the dynamic behavior of dry calcareous sand subjected to impact loading. They found that the calcareous sand shows yielding and strain-hardening behaviors followed by particle crushing, and that the mass of crushed particles increases as both strain rate and particle size increase. Lv et al. [24] studied the effects of moisture on calcareous sand at a high strain rate using the SHPB tests, where the dynamic stiffness of the sand was not sensitive to moisture changes within a water content range of 0–35%, thus, a high level of water content is necessary to influence particle friction in porous calcareous sand.

Calcareous sand is widely used as a construction material in the context of rapid offshore development across the globe. Understanding the dynamic characteristics of calcareous sand is crucial in certain engineering situations (e.g., pile driving, aircraft wheel loading, mining activities) and during natural disasters (e.g., earthquakes, tsunamis). However, the dynamic behavior of calcareous sand under quasi-one-dimensional impact loading conditions is not yet fully understood [25,26]. In this study, a series of SHPB tests were conducted to determine the mechanical impact characteristics of calcareous sand. The breakage of calcareous sand particles influenced by water content, strain rate, and relative density was also quantitatively assessed.

2. Materials and Methods

2.1. Materials

The material used in this experiment was calcareous sand, primarily composed of calcium carbonate. A high-resolution, innovative Smartlab SE X-ray diffractometer (Rigaku Co., Tokyo, Japan, The equipment used in this study was sourced from Jiangsu University of Science and Technology) was used to carry out a laboratory XRD test on calcareous sand powder. Combined with measurement and analysis software, comprehensive analysis of the material was conducted. The mineral composition of the measured calcium sand samples is presented in

Table 1. X-ray diffraction analysis report of calcareous sand.

Mineral Composition	Quartz /%	Potassium Feldspar/%	Plagioclase /%	Calcite /%	Aragonite /%	Halite /%	Gypsum /%	Magnesium Calcite/%	Clay Mineral/%
Content	5.6	0.5	0.4	10.0	23.0	0.3	/	58.9	1.3

. The calcareous sand contained abundant magnesium calcite (58.9%) with low clay content (1.3%).

To consider the influence of particle size on the SHPB results for calcareous sand, tests on 0.25–0.5, 0.5–1.0, and 1.0–2.0 mm sand samples (Figure 1) were carried out. The designated quantity of natural calcareous sand was taken, dried, and screened prior to each test.

The maximum and minimum dry densities of sand samples (three grains) were determined using the method described by Wang et al. [27]. The maximum dry densities of calcareous sand with particle sizes of 0.25–0.5, 0.5–1.0, and 1.0–2.0 mm were 1.44, 1.40, and 1.36 g/cm³, and the minimum dry densities were 1.24, 1.20, and 1.16 g/cm³, respectively.

2.2. Specimen Preparation and Test Protocol

In this study, the mechanical impact characteristics of calcareous sand were examined as they respond to water content, strain rate, and relative density. The set values of water content were 0%, 4%, 8%, 12%, and 16%. The set values of strain rate were 300 s⁻¹, 600 s⁻¹, 900 s⁻¹, 1200 s⁻¹, and 1500 s⁻¹, respectively. The relative densities were set to 40%, 70%, and 90%. To prepare samples with different moisture contents, a significant amount of dried calcareous sand with particle sizes of 0.25–0.5, 0.5–1.0, and 1.0–2.0 mm was collected. The sand of each particle size was split into five parts and placed into 15 plastic cups. The amount of water necessary to add to each calcareous sand to reach the designated moisture content was placed into a measuring cylinder and poured into the corresponding cup, which was then sealed with plastic film, and gently shaken to ensure the sand and water was thoroughly combined. After standing overnight, a portion of each sand sample was weighed from each cup to measure the actual water content. If the range between the preset and real moisture contents was reasonable, the configured sand sample was considered to have reached the set water content. A single sample of calcareous sand was taken from each cup for the formal test. The strain rate was controlled by controlling the projectile pressure in the SHPB test device. The relative densities were calculated according to the preset values of different sample masses prior to the test.

The SHPB test apparatus used in this study was sourced from Jiangsu University of Science and Technology. A photograph of the experimental setup is given in Figure 2. The loading system, drive system, and measuring and recording system comprise the majority of the SHPB test apparatus. The incident and transmission bars are each 16 mm in diameter and 1000 mm in length. The striker bar is 16 mm in diameter and 350 mm in length. The aluminum bars were used to obtain sensitive transmitted signals due to their low impedance and suitability to low-impedance soil. During the test, the sample was sandwiched between the incident bar and transmission bar while a pulse shaper incident wave was generated by the pneumatic cylinder and propagated through the incident bar. Once the incident wave reached the sample interface, a portion of it was reflected back into the incident bar as a reflected wave due to the mechanical impedance mismatch between the bars and the sample [18], while the rest of the wave transmitted through the transmission bar as a transmission wave to compress the sample. The incident wave and transmission wave were recorded by the strain gauges installed on the incident bar and transmission bar, respectively. Strain gauge data were recorded with a high-speed digital oscilloscope, operated at 12-bit A/D resolution.

Each sample had a diameter of 16 mm and length of 12.8 mm. The sample loading process is depicted in Figure 3a–d. First, the glass sheet was placed on a horizontal table and the bracket, cushion, and sleeve were loaded onto it; these components were premade for consistency with the protruding section of the cushion block. The quality of sample needed according to the set relative density was calculated, then the appropriate sample was poured into the sleeve to the specified height. The other pad block, bracket, and glass sheet were placed on the sleeve. The whole sleeve was turned 90°, then the glass sheet and pad at either end was removed and affixed to the SHPB test device, then the formal test began.

As shown in Figure 3d, a sleeve design with protruding pads was used to minimize friction between the sleeves and incident bar during impact loading. The cushion block, sleeve, and bars were all made of aluminum. A total of 33 groups of test conditions are shown in

Table 2. Tests conditions.

Number	Particle Size/mm	Water Content/%	Relative Density/%	Strain Rate/s−1
1	1.0–2.0	0	70	900
2		4
3		8
4		12
5		16
6		8	70	300
7				600
8				1200
9				1500
10			40	900
11			90	900
12	0.5–1.0	0	70	900
13		4
14		8
15		12
16		16
17		8	70	300
18				600
19				1200
20				1500
21			40	900
22			90	900
23	0.25–0.5	0	70	900
24		4
25		8
26		12
27		16
28		8	70	300
29				600
30				1200
31				1500
32			40	900
33			90	900

. Each combination of test circumstances was run at least three times to ensure accuracy.

3. Results and Discussion

3.1. Dynamic Stress Equilibrium

Prior to conducting the tests, the dynamic stress equilibrium was verified. The two-wave method was used to determine the dynamic stress equilibrium according to the assumption of a one-dimensional wave in the SHPB tests [24]. The stress history σ_z(t) at the front end can be calculated for comparison against that at the back end of the samples. The stress history σ_z(t) at the front and back ends of the samples was determined as follows [28]:

$${\mathsf{\sigma }}_z\left(\mathrm{t}\right)=\frac{A_0}{A_s}{E}_0[{\epsilon }_i\left(\mathrm{t}\right)+{\epsilon }_r\left(\mathrm{t}\right)] (\mathrm{front} \mathrm{end})$$

(1)

$${\mathsf{\sigma }}_z\left(\mathrm{t}\right)=\frac{A_0}{A_s}{E}_0{\epsilon }_{\mathrm{t}}\left(\mathrm{t}\right) (\mathrm{back} \mathrm{end})$$

(2)

where ${\epsilon }_i\left(\mathrm{t}\right)$, ${\epsilon }_r\left(\mathrm{t}\right)$, and ${\epsilon }_t\left(\mathrm{t}\right)$ are the strain history of the incident, reflection, and transmission bars, respectively. If the stress histories at the front and back ends of the samples are nearly identical, then the dynamic equilibrium condition is established. A comparison of the stress histories at the front and back ends of the calcareous sand samples with grain sizes of 1.0–2.0 mm and water content of 8%, further verifying the dynamic stress equilibrium, is shown in Figure 4. The stress history at the front end of the sample almost overlaps that at the back end, indicating that the dynamic stress equilibrium is established, thus validating the test results.

3.2. Mechanical Characteristics Evident in SHPB Tests

3.2.1. Influence of Strain Rate on Mechanical Characteristics

The stress–strain curves for calcareous sand with three distinct particle sizes at 8% water content and various strain rates were generated after processing the original stress wave signals, as illustrated in Figure 5. This visualization underscores how the stress–strain profiles of the three samples progress through an elastic phase characterized by rapid development, followed by a plastic phase marked by multiple instances of stress fluctuations and subsequent increases across various strain rates. As the strain rate increases, both the peak stress and maximum strain of the samples also rise proportionally. However, when the particle size measures 1.0–2.0 mm, the stress–strain curve tends to overlap more across different strain rates; when the particle size is 0.5–1.0 mm or 0.25–0.5 mm, the stress–strain curve rises as the strain rate increases.

This rising phenomenon becomes more pronounced when the particle size is smaller, and it can be attributed to the crushing degree of samples with different particle sizes. Once particles were broken, there was a subsequent decrease in stress levels. As mentioned above, larger particles were more prone to particle breakage in the test. Consequently, at the same strain, the stress drop in large particles occurred earlier than in small particles, and the decrease was more significant.

The curve depicting the relationship between the peak stress and strain rate for samples with three different particle sizes is shown in Figure 6.

The peak stress and strain rate have a primary functional relationship that can be expressed by Formula (3):

$${\mathsf{\sigma }}_{\mathrm{M}}={\mathsf{\sigma }}_0+\mathsf{\lambda }\epsilon$$

(3)

where σ₀ and λ are related to the properties of the material. In this test, the value of λ gradually increased as the particle size decreased, i.e., the slope of the fitting line gradually trended upward. In effect, the peak stress increased with the increase in strain rate more noticeably as the particle size decreased.

3.2.2. Effect of Water Content on Mechanical Characteristics

The stress–strain curves for calcareous sand with three distinct particle sizes under conditions of varying water content with a constant strain rate of 900 s⁻¹ are shown in Figure 7. Each sample’s stress–strain curve exhibits a distinct quick increase in the elastic phase regardless of particle size and water content, followed by a steady climb into the plastic phase. The increase in the plastic-phase curve is not linear, but rather shows a recurring pattern of multiple stress fluctuations. This behavior is closely tied to particle breakage and the subsequent reconstruction of particles within the sample.

The stress–strain curves for various water content situations generally show similar patterns when the particle sizes are 1.0–2.0 mm and 0.25–0.5 mm, as illustrated in Figure 7. When the particle size was 0.5–1.0 mm, the stress–strain curves with water contents of 0% and 16% differed significantly. The stress–strain curves with water contents of 4%, 8%, and 12% were quite similar at the end of the elastic phase and plastic phase. The moisture content of wet sand appears to have no impact on the stress–strain relationship of the samples subjected to SHPB testing. The initial deviation observed in the first half of the curve (Figure 7b) may be attributable to non-uniform moisture distribution within the pores of the initial sample. After the test, particle breakage and reconstruction led to a more uniform dispersion of internal pore water, resulting in the almost-coinciding second half of the curve.

However, it is important to note that the stress–strain relationship differed between dry and wet sand due to the different transmission modes of stress waves in pore gas and pore water. In addition, when comparing Figure 7a–c, it is clear that the varying degrees of particle breakage experienced by calcareous sand particles of various particle sizes throughout the test caused the stress–strain curve to decrease as the particle size increased.

3.2.3. Effect of Relative Density on Mechanical Properties

The stress–strain curves reflecting three samples of calcareous sand with various relative densities are shown in Figure 8. The curves show similar elastic phases, which feature a swift increase, as well as plastic phases, which show a sluggish rise and many stress fluctuations. The stress–strain curves for the three relative densities are nearly identical for particle sizes of 1.0–2.0 mm. For particle sizes of 0.5–1.0 mm, the curves generally coincide but are not quite as close. The three relative density graphs all show an increase when the particle size falls between 0.25 and 0.5 mm. The difference between the stress–strain curves under different relative densities thus appear to increase gradually as the particle size decreases.

3.3. Particle Breakage Evident in SHPB Tests

3.3.1. Establishment of Breakage Index

Particle breakage is an essential characteristic that distinguishes calcareous sand from terrigenous silica sand. Based on the relative breakage rate $B_r$ proposed by Hardin [29], Einav [30] established the relative fragmentation rate $B_r^{\mathit{*}}$.

$$B_r^{\mathit{*}}=\frac{{\int }_{d_m}^{d_M}(F\left(d\right) -F_0\left(d\left)\right)d\right(\mathit{lgd})}{{\int }_{d_m}^{d_M}(F_u\left(d\right) - F_0(d\left)\right)d\left(\mathit{lgd}\right)}$$

(4)

where $d_m$ indicates the minimum particle size; $d_M$ indicates the maximum particle size; $F\left(d\right)$ is the loading proper function of the current grading curve; $F_u\left(d\right)$ is the proper function of the limit state grading curve; and $F_0\left(d\right)$ is the proper function of the initial state grading curve.

Einav [30] also provided a fitting function expression for different grading curves based on the classification model.

$$F(d) =\frac{M\left(\mathit{∆} < d\right)}{M_{\mathrm{t}}}=\frac{d^{\mathit{3}-\alpha } - d_m^{\mathit{3}-\alpha }}{d_M^{\mathit{3}-\alpha } -{ d}_m^{\mathit{3}-\alpha }}$$

(5)

where $\mathit{∆}$ and $d$ denote the particle diameter; $M(\mathit{∆}<d)$ is the mass of particles smaller than $d$ in size; $M_T$ is the total mass of particles; and α is the fractal dimension.

In Formula (4), when $d_m=0$, the fitting function expression of the limit state grading curve can be obtained as follows:

$$F_u\left(d\right)={(\frac{d}{d_M})}^{\mathit{3}-D}$$

(6)

where D indicates the fractal dimension corresponding to the loading final grading curve. According to the empirical values, the calcareous sand tested in this study had fractal dimensions of 2.6 [31].

3.3.2. Influence of Strain Rate on Particle Breakage

The grain size distribution curves of calcareous sand with three particle sizes, prior to and following the tests at various strain rates, are shown in Figure 9. The grain size distribution curve expanded as a result of the experiment and significantly increased with higher strain rates, implying a corresponding escalation in the degree of particle breaking. Factors such as impact energy, shock wave propagation speed, and particle breakage degree all appear to be influenced by the strain rate, intensifying as the strain rate increased.

The proper function of the grading curve before and after the test, and the fractal dimension α after the test, were calculated using Formula (5). After the test, the particle size screening curve could be calculated using Tyler’s mass particle size classification formula [32].

Table 3. Fractal dimension α of calcareous sand with three particle sizes under different strain rates.

Particle Size/mm	Strain Rates/s−1	α
1.0–2.0	300	1.21
	600	1.75
	900	2.06
	1200	2.30
	1500	2.42
0.5–1.0	300	0.70
	600	1.44
	900	1.83
	1200	2.05
	1500	2.23
0.25–0.5	300	0.80
	600	1.28
	900	1.81
	1200	1.92
	1500	2.05

displays the fractal dimensions α of samples with three particle sizes following the test at various strain rates, estimated in accordance with Figure 9.

Because the sample selected in this study was calcareous sand with a single particle size, Tyler’s mass particle size classification formula [32] was inapplicable for determining the fractal dimension α. To ascertain the sample’s starting state’s fractal dimension α, the strain rate $\dot{\epsilon }$ and 3-α were fitted as shown in Figure 10.

The fitting curve is expressed by Equation (7):

$$\mathit{3} - \alpha =\mathrm{m}{\mathrm{e}}^{-\frac{\epsilon }{\mathrm{n}}}+{\alpha }_0$$

(7)

where $\mathrm{m}$, $\mathrm{n}$ and α₀ denote parameters related to the particle composition of calcareous sand and $\epsilon$ is the strain rate. In this test, $\mathrm{m}$ increased, while $\mathrm{n}$ decreased as particle size increased.

When $\dot{\epsilon }$ was close to 0, the sample was in its initial form free from exterior disturbance. At this point, the initial fractal dimension of the sample could be calculated using Formula (7). The sand sample with three different particle sizes had initial fractal dimensions of 2.26, 2.42, and 2.58. When $\epsilon$ tends to infinity, the sample could be considered to have reached the limit-breaking state after the test. The fractal dimensions of the sand samples at this time were 2.62, 2.57, and 2.57, respectively. These results align with previous research where the fractal dimension of the limit breaking of coarse particles was found to be approximately 2.6 [31].

Substituting the calculated initial fractal dimension and the fractal dimension of the tested material (Table 3) into Formula (5), combined with Formulas (4) and (6), revealed the relative fragmentation rate $B_r^{\mathit{*}}$ of calcareous sand with three particle sizes under various strain rates, as illustrated in

Table 4. Relative fragmentation rate $B_r^{\mathit{*}}$ of calcareous sand with three particle sizes under different strain rates.

Particle Size/mm	Strain Rate/s−1	$B_r^{\mathit{*}}$
1.0–2.0	300	0.06
	600	0.18
	900	0.30
	1200	0.48
	1500	0.62
0.5–1.0	300	0.06
	600	0.13
	900	0.24
	1200	0.35
	1500	0.49
0.25–0.5	300	0.03
	600	0.09
	900	0.20
	1200	0.25
	1500	0.32

. To further explore the relationship between the strain rate and relative fragmentation rate $B_r^{\mathit{*}}$, a connection curve between the strain rate and the relative fragmentation rate $B_r^{\mathit{*}}$ was obtained as shown in Figure 11.

The fitting curve can be expressed by Equation (8):

$$B_r^{\mathit{*}}= a\frac{\epsilon }{100}+B_1$$

(8)

where a and B₁ are parameters related to the materials.

As shown in Figure 11, when the strain rate was set to 300 s⁻¹, the relative fragmentation rate $B_r^{\mathit{*}}$ for all samples with initial particle sizes of 1.0–2.0 mm was lower than that of the samples with initial particle sizes of 0.5–1.0 mm. However, when the strain rate was 600–1500 s⁻¹, the change law of the relative fragmentation rate $B_r^{\mathit{*}}$ with particle size conformed to the positive correlation law described above. The departure from this pattern at a strain rate of 300 s⁻¹ may be attributable to measurement errors. On the whole, under varying strain rates, the relative fragmentation rate $B_r^{\mathit{*}}$ consistently tended to increase as particle size increased.

The results also revealed that as particle size increased from 0.25–0.5 mm to 1.0–2.0 mm, the slope of the corresponding curve increased from 0.02 to 0.05. This suggests that a larger particle size caused the relative fragmentation rate $B_r^{\mathit{*}}$ to become more susceptible to the effects of different strain rates.

3.3.3. Influence of Water Content on Particle Breakage

The three samples’ distribution curves before and after the test, with various water contents, are shown in Figure 12. The particle size distribution curve trended upward as the experiment progressed, however, the curve significantly declined as the water content increased.

The fractal dimensions α for three sand particle sizes after various water content tests were determined by taking the grading distribution curve before and after loading, substituting it into Formula (4), combining Formulas (3) and (5), and calculating the relative fragmentation rate $B_r^{\mathit{*}}$ for different water contents. The results are shown in

Table 5. Fractal dimension α and relative fragmentation rate $B_r^{\mathit{*}}$ of calcareous sand with three particle sizes under different water contents.

Particle Size/mm	Water Contents/%	α	$B_r^{*}$
1.0–2.0	0	2.15	0.36
	4	2.08	0.32
	8	2.07	0.31
	12	2.04	0.29
	16	2.01	0.28
0.5–1.0	0	1.90	0.28
	4	1.86	0.26
	8	1.83	0.25
	12	1.78	0.23
	16	1.75	0.21
0.25–0.5	0	1.82	0.22
	4	1.80	0.20
	8	1.79	0.19
	12	1.77	0.18
	16	1.72	0.17

.

To further explore the relationship between the water content and relative fragmentation rate $B_r^{\mathit{*}}$, a curve connecting the water content and relative fragmentation rate $B_r^{\mathit{*}}$ was drawn as shown in Figure 13. All three curves were approximated as straight lines, which can be expressed as follows:

$$B_r^{\mathit{*}}=\mathsf{\beta }\mathsf{\omega }+B_0$$

(9)

where β and $B_0$ are related to the materials and ω indicates the water content. The fitted curve coincides well with the measured results for all three samples with initial particle sizes of 1.0–2.0 mm, 0.5–1.0 mm, and 0.25–0.5 mm, showing regression parameters (R²) of 0.93, 0.99, and 0.97, respectively.

The relative crushing rate consistently decreased as water content increased (Figure 13), in alignment with findings by Lv et al. [33]. This phenomenon can be primarily attributed to water acting as a particle buffer. As the water content increased, the buffer medium’s volume increased and the buffering effect became more pronounced, subsequently reducing particle breakage. For calcareous sand with an initial particle size of 1.0–2.0 mm, the relative fragmentation rate $B_r^{\mathit{*}}$ decreased from 0.37 to 0.28 as the water content increased from 0% to 16%—a decrease of 24.3%. For calcareous sand with an initial particle size of 0.5–1.0 mm, when the water content increased from 0% to 16%, the relative fragmentation rate $B_r^{\mathit{*}}$ decreased from 0.27 to 0.21—a 22.2% decrease. For calcareous sand with an initial particle size of 0.25–0.5 mm, the relative fragmentation rate $B_r^{\mathit{*}}$ decreased from 0.22 to 0.17 by 22.7% as the water content increased from 0% to 16%. This trend, combined with the β value outcomes, suggests that larger particle sizes rendered the relative fragmentation rate $B_r^{\mathit{*}}$ more susceptible to the influence of water content.

Moreover, Figure 13 also demonstrates that the relative fragmentation rate $B_r^{\mathit{*}}$ increased as particle size increased; in other words, as the particle size grew, so did the degree of particle breakage. There are two likely reasons for this. First, that large-particle calcareous sand is more random than small-particle calcareous sand; and second, that the spatial distribution of large-particle calcareous sand is more complex. Together, these factors have caused the giant-particle sand to be more prone to particle breakage in the tests.

3.3.4. Influence of Relative Density on Particle Breakage

The particle size distribution curves for calcareous sand with three particle sizes prior to and following the test at various relative densities are shown in Figure 14. The initial particle size within the range of 1.0 to 2.0 mm is readily discernible; the curve appears to flatten as the relative density increases from 40% to 90%. However, when the relative density was set to 70% and 90%, there was no significant difference in the resulting curve. When the initial particle size was 0.5–1.0 mm, relative densities of 40% and 70% produced nearly identical particle size distribution curves. When the initial particle size was 0.25–0.5 mm, the curve decreased with increasing relative density.

The fractal dimensions of the sand samples with three particle sizes tested under different relative densities D_r and relative fragmentation rates $B_r^{\mathit{*}}$ are shown in

Table 6. Fractal dimension α and relative fragmentation rate ${\mathit{B}}_{\mathit{r}}^{\mathit{*}}$ of calcareous sand with three particle sizes under different relative densities.

Particle Size/mm	Relative Densities/%	α	${\mathit{B}}_{\mathit{r}}^{\mathit{*}}$
1.0–2.0	40	1.21	0.38
	70	1.75	0.30
	90	2.06	0.26
0.5–1.0	40	0.70	0.26
	70	1.44	0.24
	90	1.87	0.20
0.25–0.5	40	0.80	0.24
	70	1.28	0.20
	90	1.82	0.18

. To further explore the relationship between relative density and relative fragmentation rate $B_r^{\mathit{*}}$, a curve between the strain rate and $B_r^{\mathit{*}}$ was drawn as shown in Figure 15.

As illustrated in Figure 15, the relative fragmentation rate $B_r^{\mathit{*}}$ decreased as the relative density increased. Lade et al. [34] reached a similar conclusion, but Lv et al. [33] made disparate observations. The former asserted that high-density sand is surrounded by more grains, leading to a reduction in mean stress and a decrease in the amount of breakage as the relative density increased. On the other hand, the latter considered that the energy applied in their test was too high for stable calcareous sand crushing.

The relationship between the near fragmentation rate and relative density can be expressed as follows:

$$B_r^{\mathit{*}}=B_2+b{D}_{\mathrm{r}}$$

(10)

where B₂ and b are parameters related to the materials.

4. Conclusions

In this study, a comprehensive series of SHPB tests were conducted to investigate the dynamic behavior of calcareous sand as influenced by water content, strain rate, and relative density. Three particle size ranges, 1.0–2.0 mm, 0.5–1.0 mm, and 0.25–0.5 mm, were examined by analyzing their quasi-one-dimensional mechanical characteristics and particle breakage. The results of this work may not only enrich the theoretical framework of calcareous soil mechanics, but also provide data support and a sound theoretical basis for the construction of marine infrastructure. The key findings can be summarized as follows.

(1)

The stress–strain curves exhibited a consistent pattern across various influence factors, beginning with a sharp rise in the elastic phase, followed by a plastic phase marked by multiple stress fluctuations. Peak stress increased with strain rate. This effect was particularly pronounced with smaller particle sizes, attributable to increased particle breakage.

(2)

The stress–strain curves also trended downward as particle size increased, alongside a heightened degree of particle breakage. This phenomenon is linked to the intricate spatial distribution and irregular appearance of larger particles, which were more prone to breakage during testing, consequently leading to a decrease in stress levels.

(3)

The relative fragmentation rate $B_r^{\mathit{*}}$ increased as strain rate increased. For coarser particles, $B_r^{\mathit{*}}$ values were more susceptible to strain rate compared to fine particles.

(4)

The water content of calcareous sand did not significantly impact the stress–strain curves resulting from SHPB tests. However, the degree of particle breakage decreased as water content increased. A linear relationship was established between the relative fragmentation rate $B_r^{\mathit{*}}$ and water content $\omega$.

(5)

The differences between stress–strain curves increased gradually as particle size decreased. As the relative density increased, the extent of particle breakage decreased. The relationship between the relative fragmentation rate $B_r^{\mathit{*}}$ and relative density D_r was effectively characterized by a linear curve.