# Review of Particle Breakage Measurement Methods for Calcareous Sand

Authors: Liang-Jie Xu; Ren Wang; Dong-Sheng Xu; Xin-Zhi Wang; Qing-Shan Meng; Chang-Qi Zhu
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

In order to better understand the particle breakage mechanics and characterization methods of calcareous sand, the particle breakage characteristics of calcareous sand in one-dimensional compression tests, isotropic compression tests, triaxial shear tests, and ring shear tests are reviewed and analyzed. The results show that the mechanism of shear crushing is essentially different from compression crushing. Compared with one-dimensional compression, it is easier to break calcareous sand under triaxial shear. In the shearing process, the particle breakage of calcareous sand will not continue indefinitely. The gradation tends to be stable and controlled by confining pressure and shear strain. The characteristic particle size measurement method for particle breakage is simple and has limitations. The description method using the particle size distribution curve is more comprehensive, but it is impossible to compare the crushing degree of particle with different particle sizes, and a suitable measurement method needs to be proposed.


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## Body

## 1. Introduction

Construction on coral reefs with calcareous sand began during World War II. Then, the engineering mechanical properties of calcareous sand were widely studied. In the mid-1960s, engineering problems caused by calcareous sand were encountered for the first time in the oil exploitation projects in the Arabian Gulf of Middle East. At that time, the particularities of the physical and mechanical properties of calcareous sand were ignored. Subsequently, calcareous sand caused a series of engineering problems during the construction of offshore oil platforms in northwest continental shelf of Australia, southern waters of Brazil, and Florida of North America, resulting in heavy losses.With the development of coral reef engineering, calcareous sand, as the primary foundation filling material in the offshore reefs, is receiving more and more attention. The physical and mechanical properties of calcareous sand have gradually attracted the attention of researchers and engineers [1–3]. The sedimentary environment, origin, and material composition of calcareous sand are very special. Compared with siliceous sand, particle breakage is an important factor affecting the deformation and strength of calcareous sand. It is helpful to avoid unnecessary losses in engineering design by studying the crushing evolution law of calcareous sand and its influence on mechanical behavior.Therefore, it is necessary to study the geotechnical engineering characteristics of calcareous sand and understand its engineering mechanical effect. This article systematically summarizes particle breakage characteristics and measurement methods of calcareous sand under different conditions, which provides a scientific basis for coral reef engineering construction, and also provides reference for the in-depth study on coral reef geotechnical engineering characteristics. It has certain guiding and reference significance for the study of the particle breakage.


## 2. Physical Properties of Calcareous Sand

### 2.1. Origin and Distribution

Calcareous sand is a kind of carbonate sediment formed by the fragments of coral residues and shells in the marine environment, and the engineering mechanical properties are significantly different from those of terrestrial sediments [1, 2]. This difference is mainly due to the characteristics of calcareous sand particles, such as porosity, irregular shape, and easy crushing. The primary material sources of calcareous sand are reef-building corals, coral algae, and skeleton detritus from other marine organisms deposited in situ or transported from a nearby source [4–6].In addition, since the surface of coral reefs is composed of an un-cemented loose sediment layer (biological sediments, 0–20 m) and a cemented solid rock layer (reef limestone) below the surface. Therefore, it can be considered that the distribution of calcareous sand and coral reef is basically the same [7]. Moreover, the distribution of coral reefs or calcareous sands in the global is primarily concentrated in the tropical and subtropical regions between 30° S and 30° N, such as the north and south sides of the Pacific Ocean, Indian Ocean, and Atlantic Ocean equator (Figure 1), while 92 percent coral reefs are located in the Indian and Pacific oceans (including the Red Sea, Persian Gulf, South China Sea, and Caribbean Sea) [3].Figure 1
Distribution of coral reefs in the world [3].


### 2.2. Physical Properties of Calcareous Sand

Table1 lists the basic physical parameters of calcareous sand samples taken from Nansha Islands and Xisha Islands in the South China Sea as well as other parts of the world, such as the Persian Gulf, Ireland, Caribbean Sea, and the north coast of Egypt. According to the existing literature, the specific gravity (GS) of calcareous sand is between 2.70 and 2.86, which is higher than that of ordinary quartz sand (GS = 2.65), indicating that calcareous sand contains other carbonate substances, such as calcium carbonate and magnesium carbonate, which fully reflects the characteristics of marine sediments. Calcareous sand from the Persian Gulf of Iran, Egypt, and Western Australia are similar, with the GS ranging from 2.68 to 2.76, 2.69 to 2.79, and 2.72 to 2.76, respectively, while the GS of calcareous sand from Dog’s Bay, Sarb, and Mischief Island are all similar as well. The maximum void ratio (emax) ranges from 0.63 to 1.86, and the minimum void ratio (emin) ranges from 0.43 to 1.34. It is observed that the GS and void ratio (e) of calcareous sand vary greatly in different regions. Moreover, it also demonstrates that the compressibility of calcareous sand foundations in different regions is significantly different. The calcareous sand foundation in the South China Sea is moderately high in compressibility, while in Western Australia, Ireland (Dog’s Bay), Puerto Rico (Cabo Rojo), and Abu Dhabi (Sarb), it is highly compressible.Table 1
Physical parameters of calcareous sand.
AuthorsemaxeminCuD50 (mm)D10 (mm)GSSampling locationsZhang [4]1.4601.0502.2000.3770.2082.790Li [5]1.4230.8962.5000.500–0.6000.200–0.2602.730Yongshu Reef, Nansha, ChinaWang [6]1.4050.7668.9500.600–0.7000.100–0.2002.760Morioka and Nicholson [8]1.3000.6605.4200.8402.720Ewa Plains, HawaiiSharma and Fahey [9]1.2100.9001.8350.2402.760Ledge point, Western Australia2.3171.2664.6040.1002.720Goodwyn, Western AustraliaArango et al. [10]1.7101.3401.0500.2002.860Cabo Rojo, Puerto RicoTan [11]1.4801.0002.7802.800Yongshu Reef, Nansha, ChinaHu [12]1.4100.7705.6400.1322.760Mischief Island, Nansha, ChinaDonohue et al. [13]1.8601.1700.3302.700Dogs Bay, IrelandSalem et al. [14]1.0400.7502.4000.1502.790Dabaa, EgyptShahnazari and Rezvani [15]0.9090.6254.4700.7802.760Hormuz Island, Persian gulf1.0510.7263.2000.4302.710Bushehr Port, Persian gulfShahnazari et al. [16]0.9090.6258.3300.7802.764Hormuz Island, Persian gulf1.0510.7267.8700.4302.709Bushehr Port, Persian gulfPham et al. [17]1.3300.9033.4600.7302.787Sarb, Abu DhabiShen et al. [18]1.1900.8004.8002.700Mischief Island, Nansha, ChinaWang et al. [19]1.4500.9803.3502.0500.7902.860Jinqing Island, Xisha, ChinaMorsy et al. [20]0.6300.4302.0000.4000.2002.720Agami, EgyptNote: (a)GS: specific gravity; (b) emax: maximum void ratio; (c) emin: minimum void ratio; (d) Cu: nonuniform coefficient; (e) D10: effective particle size; (f) D50: median particle size.In addition, macroscopic mechanical properties of calcareous sand are determined by its microscopic properties, because calcareous sand retains abundant micropores during its formation. The porosity of calcareous sand containing biological skeleton is as high as 41%, while the porosity of calcareous sand formed by weathering deposition is less than 20% [21]. Cui et al. [22] pointed out that the particle shape and surface roughness are the main factors that influence pore shape, pore throat size, and global connectivity. A large number of internal pores give calcareous sand different characteristics from quartz sand, such as easy breakage, difficult saturation, and high compressibility [22–24]. Therefore, it is an important research direction in the future to deeply analyze the influence of internal pores on the macro- and micromechanical properties of calcareous sand, such as saturation and crushing.The biogenic origin of calcareous sand leads to complex and diverse particle shape. Figure2 shows four typical particle shapes of calcareous sand, which are lump-shaped (Figure 2(a)), spindle like-shaped (Figure 2(b)), flaky-shaped (Figure 2(c)), and branched-shaped (Figure 2(d)). In geotechnical engineering research and engineering practice, particle shape is an important factor affecting the mechanical properties of soil [17, 24–28]. Pham et al. [17] pointed out that the particle characteristics (shape and size) are very important parameters that affect the shear strength properties. Lv et al. [26] found that the apparent dynamic stiffness of calcareous sand is approximately 10% of that for silica sand due to the different particle shapes and mineral composition. The physical properties of calcareous sand include GS, e, and particle shape, which are also important reasons for the difference in mechanical properties of calcareous sand and siliceous sand. For a detailed comment on the particle shape and e of calcareous sand, refer to another published paper of calcareous sand review by the author [3]. Furthermore, regarding the physical and mechanical properties of calcareous sand, the marine engineering geology team led by Professor Wang Ren of the Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, has carried out a lot of experiments and achieved fruitful results [1–7, 11, 12].Figure 2
Typical shape of calcareous sand [4]: (a) lump-shaped; (b) spindle like-shaped; (c) flaky-shaped; (d) branched-shaped.
(a)(b)(c)(d)


## 2.1. Origin and Distribution

Calcareous sand is a kind of carbonate sediment formed by the fragments of coral residues and shells in the marine environment, and the engineering mechanical properties are significantly different from those of terrestrial sediments [1, 2]. This difference is mainly due to the characteristics of calcareous sand particles, such as porosity, irregular shape, and easy crushing. The primary material sources of calcareous sand are reef-building corals, coral algae, and skeleton detritus from other marine organisms deposited in situ or transported from a nearby source [4–6].In addition, since the surface of coral reefs is composed of an un-cemented loose sediment layer (biological sediments, 0–20 m) and a cemented solid rock layer (reef limestone) below the surface. Therefore, it can be considered that the distribution of calcareous sand and coral reef is basically the same [7]. Moreover, the distribution of coral reefs or calcareous sands in the global is primarily concentrated in the tropical and subtropical regions between 30° S and 30° N, such as the north and south sides of the Pacific Ocean, Indian Ocean, and Atlantic Ocean equator (Figure 1), while 92 percent coral reefs are located in the Indian and Pacific oceans (including the Red Sea, Persian Gulf, South China Sea, and Caribbean Sea) [3].Figure 1
Distribution of coral reefs in the world [3].


## 2.2. Physical Properties of Calcareous Sand

Table1 lists the basic physical parameters of calcareous sand samples taken from Nansha Islands and Xisha Islands in the South China Sea as well as other parts of the world, such as the Persian Gulf, Ireland, Caribbean Sea, and the north coast of Egypt. According to the existing literature, the specific gravity (GS) of calcareous sand is between 2.70 and 2.86, which is higher than that of ordinary quartz sand (GS = 2.65), indicating that calcareous sand contains other carbonate substances, such as calcium carbonate and magnesium carbonate, which fully reflects the characteristics of marine sediments. Calcareous sand from the Persian Gulf of Iran, Egypt, and Western Australia are similar, with the GS ranging from 2.68 to 2.76, 2.69 to 2.79, and 2.72 to 2.76, respectively, while the GS of calcareous sand from Dog’s Bay, Sarb, and Mischief Island are all similar as well. The maximum void ratio (emax) ranges from 0.63 to 1.86, and the minimum void ratio (emin) ranges from 0.43 to 1.34. It is observed that the GS and void ratio (e) of calcareous sand vary greatly in different regions. Moreover, it also demonstrates that the compressibility of calcareous sand foundations in different regions is significantly different. The calcareous sand foundation in the South China Sea is moderately high in compressibility, while in Western Australia, Ireland (Dog’s Bay), Puerto Rico (Cabo Rojo), and Abu Dhabi (Sarb), it is highly compressible.Table 1
Physical parameters of calcareous sand.
AuthorsemaxeminCuD50 (mm)D10 (mm)GSSampling locationsZhang [4]1.4601.0502.2000.3770.2082.790Li [5]1.4230.8962.5000.500–0.6000.200–0.2602.730Yongshu Reef, Nansha, ChinaWang [6]1.4050.7668.9500.600–0.7000.100–0.2002.760Morioka and Nicholson [8]1.3000.6605.4200.8402.720Ewa Plains, HawaiiSharma and Fahey [9]1.2100.9001.8350.2402.760Ledge point, Western Australia2.3171.2664.6040.1002.720Goodwyn, Western AustraliaArango et al. [10]1.7101.3401.0500.2002.860Cabo Rojo, Puerto RicoTan [11]1.4801.0002.7802.800Yongshu Reef, Nansha, ChinaHu [12]1.4100.7705.6400.1322.760Mischief Island, Nansha, ChinaDonohue et al. [13]1.8601.1700.3302.700Dogs Bay, IrelandSalem et al. [14]1.0400.7502.4000.1502.790Dabaa, EgyptShahnazari and Rezvani [15]0.9090.6254.4700.7802.760Hormuz Island, Persian gulf1.0510.7263.2000.4302.710Bushehr Port, Persian gulfShahnazari et al. [16]0.9090.6258.3300.7802.764Hormuz Island, Persian gulf1.0510.7267.8700.4302.709Bushehr Port, Persian gulfPham et al. [17]1.3300.9033.4600.7302.787Sarb, Abu DhabiShen et al. [18]1.1900.8004.8002.700Mischief Island, Nansha, ChinaWang et al. [19]1.4500.9803.3502.0500.7902.860Jinqing Island, Xisha, ChinaMorsy et al. [20]0.6300.4302.0000.4000.2002.720Agami, EgyptNote: (a)GS: specific gravity; (b) emax: maximum void ratio; (c) emin: minimum void ratio; (d) Cu: nonuniform coefficient; (e) D10: effective particle size; (f) D50: median particle size.In addition, macroscopic mechanical properties of calcareous sand are determined by its microscopic properties, because calcareous sand retains abundant micropores during its formation. The porosity of calcareous sand containing biological skeleton is as high as 41%, while the porosity of calcareous sand formed by weathering deposition is less than 20% [21]. Cui et al. [22] pointed out that the particle shape and surface roughness are the main factors that influence pore shape, pore throat size, and global connectivity. A large number of internal pores give calcareous sand different characteristics from quartz sand, such as easy breakage, difficult saturation, and high compressibility [22–24]. Therefore, it is an important research direction in the future to deeply analyze the influence of internal pores on the macro- and micromechanical properties of calcareous sand, such as saturation and crushing.The biogenic origin of calcareous sand leads to complex and diverse particle shape. Figure2 shows four typical particle shapes of calcareous sand, which are lump-shaped (Figure 2(a)), spindle like-shaped (Figure 2(b)), flaky-shaped (Figure 2(c)), and branched-shaped (Figure 2(d)). In geotechnical engineering research and engineering practice, particle shape is an important factor affecting the mechanical properties of soil [17, 24–28]. Pham et al. [17] pointed out that the particle characteristics (shape and size) are very important parameters that affect the shear strength properties. Lv et al. [26] found that the apparent dynamic stiffness of calcareous sand is approximately 10% of that for silica sand due to the different particle shapes and mineral composition. The physical properties of calcareous sand include GS, e, and particle shape, which are also important reasons for the difference in mechanical properties of calcareous sand and siliceous sand. For a detailed comment on the particle shape and e of calcareous sand, refer to another published paper of calcareous sand review by the author [3]. Furthermore, regarding the physical and mechanical properties of calcareous sand, the marine engineering geology team led by Professor Wang Ren of the Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, has carried out a lot of experiments and achieved fruitful results [1–7, 11, 12].Figure 2
Typical shape of calcareous sand [4]: (a) lump-shaped; (b) spindle like-shaped; (c) flaky-shaped; (d) branched-shaped.
(a)(b)(c)(d)


## 3. Particle Breakage Analysis with Various Test Conditions

In recent years, research on particle breakage of calcareous sand has mainly focused on the characteristics of particle breakage under compression and shearing. In terms of laboratory tests, the primary test methods include the one-dimensional compression test, isotropic consolidation test, conventional drained and undrained triaxial test, and cyclic loading and unloading triaxial test, as well as the ring shear test. In this article, the characteristics and laws of particle breakage of calcareous sand under different experimental conditions are analyzed and summarized.


### 3.1. One-Dimensional and Isotropic Compression Tests

#### 3.1.1. One-Dimensional Compression Test

Due to the brittleness and irregular shape of calcareous sand, particle breakage occurs at a stress level of 1.6 MPa [29], and the relative particle breakage ratio (Br) has a significant functional relationship with pressure, while the median particle size (D50) has a significant linear relationship (negative correlation) and a semilogarithmic linear relationship (positive correlation) with the crushing stress and ultimate relative particle breakage ratio (Brmax), respectively [30]. Wang, Lei, and Wang et al. [31] further pointed out that the relationship between Br and load value (Pmax) is defined by an exponential function. However, some studies show that the particle breakage degree of calcareous sand under a vertical pressure of 3.2 MPa is limited (Br = 0.019) or negligible. The high porosity and high pressure contractive of calcareous sand are primarily caused by particle morphology [18]. Meanwhile, the Br of calcareous sand should have an exponential relationship with volume and a linear relationship with input power per unit volume (Br ≤ 9%). When the relative density is low, more particles are broken compared to a relatively high density [29, 32]. Cai et al. [33] found that the particle breakage degree increased with increasing relative density and positively correlated with stress level (confining pressure or normal stress). Calcareous sand is easier to break under triaxial shear than under one-dimensional compression, and the crushing law under the two stress paths is significantly different. The particle breakage of calcareous sand with a relatively low density (loose sample) is more serious than that with relatively high density (dense sample) and has different ideas. Under the condition of triaxial shear tests, the greater the relative density of calcareous sand, the more serious the particle breakage.In addition, changes in the particle size, shape, and surface roughness are the most remarkable features of particle breakage [34, 35], which leads to the more complicated mechanical properties of calcareous sand [36, 37]. For instance, Wang et al. [31] pointed out that the particle shape has an important influence on the density and compressibility and believed that the compressibility of calcareous sands with different particle shapes is obviously different. Wei et al. [38] discussed the relationship between the particle shape of calcareous sand and its compressibility as well as particle breakage and believed that the change in the total perimeter of particles can effectively quantify particle breakage. However, Ata et al. [39] demonstrated that the crushability of medium to fine uniform calcareous sand was found to be insignificant and proposed the simplified equations to express the degradation of the constrained modulus due to crushability. In addition, the existing research mainly focuses on the particle breakage characteristics of calcareous sand and the influence of stress level on the crushing mechanism and compression characteristics. The particle size distribution curves of pre and post the test were used to measure the degree of particle breakage, while the influence of particle breakage and overlapping cover up during crushing of calcareous sand was not considered. Peng et al. [40], based on the method of dyeing calibration and image particle segmentation, put forward a cumulative particle breakage ratio considering the crushing overlapping coverage and pointed out that the absolute crushing amount for calcareous sand with medium particle size is the largest. However, the factors affecting particle breakage are relatively complex, and the mechanical mechanism and influencing factors of crushing overlapping coverage are unclear.The stress level ranging from 0 to 4 MPa was the main focus of the above research, while the compression deformation mechanism of calcareous sand under other stress levels (ranging from 0 to 100 MPa) was studied. Altuhafi et al. [41] studied the influence of initial gradation and density on the deformation mechanism of calcareous sand during one-dimensional compression and found that the particle breakage of calcareous sand is significantly reduced when the initial grading is changed to that of well graded. Miao and Airey [42] determined the particle breakage law of calcareous sand under high stress level and large shear strain through one-dimensional compression and ring shear tests, and found that the mechanisms of particle breakage caused by compression and shear are essentially different. Cai et al. [33] made use of one-dimensional compression (stress level ranging from 0.4 to 3.2 MPa) and triaxial shear test (confining pressure ranging from 0.1 to 0.4 MPa) to draw the same conclusion. Zhang et al. [43], Ma et al. [44], and Lv et al. [45] carried out compression tests on quartz sand and calcareous sand under high stress level (ranging from 0 to 100 MPa) and found that with increasing stress level, the particle breakage first increased and then gradually stabilized. Furthermore, the yield stresses of calcareous sand and quartz sand were 2 MPa and 10 MPa, respectively, and the compression deformation of calcareous sand before the yield point is primarily due to particle slip and grinding. However, deformation after the yield point is primarily caused by particle breakage, which is basically consistent with the conclusions of Zhang et al. [29]. The stress level at the yield point (about 1.6 MPa) is also relatively close to 2 MPa.


#### 3.1.2. Isotropic Compression Test

Based on the analysis of isotropic compression and one-dimensional compression tests, the compression characteristics of calcareous sand are similar to those of normal consolidated cohesive soil, and the yield stress of calcareous sand under isotropic compression is larger than that under one-dimensional compression, which is primarily due to the presence of shear stress in the process of one-dimensional compression, which makes its particle breakage more significant [4]. In the Hormuz Island (HI) of the Persian Gulf, calcareous sand is significantly crushed under a confining pressure of 1.5 MPa, and the crushing amount of medium dense calcareous sand samples was the largest [46]. Similarly, the particle breakage of calcareous sand in Bushehr Port (BP) and Hormuz Island (HI) of the Persian Gulf positively correlates with stress level, while particle breakage is directly related to input energy [15]. The yield stress of BP sand and HI sand is 0.65 MPa and 1.00 MPa [15], respectively, which are smaller than that of calcareous sand in the South China Sea (2.00 MPa) [44, 45]. From isotropic compression tests, Shahnazari et al. [15] found that the denser the calcareous sand sample, the smaller the particle breakage. It is believed that this is primarily due to a decrease in average contact stress with increasing relative density. This conclusion is contrary to triaxial shear test results [47–50], but whether or not this conclusion is consistent with one-dimensional compression test results remains controversial [29, 32, 33]. However, Dehnavi et al. [46] found that the particle breakage of medium dense calcareous sand samples is the largest.Isotropic compression tests were the main measurement methods to explore the effects of stress level and relative density on the particle breakage of calcareous sand. Under the one-dimensional and isotropic compression test conditions, there are differences in the research results regarding the influence of relative density of calcareous sand on particle breakage, which needs further study. In addition, although Ma and Chiu [51] explored the crushing mechanism of calcareous sand particles under varying fines content and pointed out that when the fine content is less than or equal to 30%, the soil mixture (calcareous sand and fine crushed quartz silt) exhibits a unique compression line duo to the substantial breakage of calcareous sand. There are still few studies on the compression behavior of the mixture of calcareous sand and siliceous sand. Furthermore, the mechanical mechanism of the particle breakage of calcareous sand should be accurately analyzed from a microscopic point of view, and the mathematical relationship between particle breakage, stress level, and relative density needs to be established.


### 3.2. Triaxial Shear Test

#### 3.2.1. Triaxial Monotonic Shear Test

Under triaxial shear, the particle breakage of calcareous sand will not increase indefinitely, but eventually tend to be constant. Particle breakage is affected and controlled by as well as positively correlates with confining pressure and shear strain [50, 52]. The stress-strain curve shows strain softening under low confining pressure and strain hardening under high confining pressure, while the influence of relative density on particle breakage is smaller than that of confining pressure [49]. Zhang et al. [53] proposed a hyperbolic function and an inverted S-shaped function to describe the prepeak portion and the postpeak portion of stress-strain curve of calcareous sand, respectively. However, the mathematical model for confining pressure, shear strain, and particle breakage is still unclear. Studies have shown that in the triaxial shearing process, the degree of particle breakage is closely related to the amount of absorbed plastic strain [54]. Based on monotonic triaxial shear tests, Yu [55] established a hyperbolic model that connected the relative particle breakage and fractal dimension of particles with the plastic strain per unit volume, which was believed to be the most reliable method of explaining the energy consumption characteristics of particle breakage. Although this method considered the fractal characteristics of soil, it was not suitable for cyclic shearing.Furthermore, the effect of particle breakage on shear strength is primarily manifested in the dissipation of energy and inhibition of dilatancy by particle breakage [48]. The energy consumed by particle breakage is a function of Br or crushing strength, and the influence of particle breakage on strength of calcareous sand first increases and then decreases with increasing confining pressure and finally tends to be stable. The effect of dilatancy on its strength is significantly stronger than that of particle breakage at low confining pressure and gradually weaker than that of particle breakage with increasing confining pressure [19, 56–58]. Some studies also show that the particle shape plays a key role in the shear contraction of sands, and particle breakage has a significant influence on the shear response of calcareous sand during dilatancy. Particle breakage primarily occurs in the dilatancy stage, which can inhibit the dilatancy behavior [16, 47, 59, 60]. Moreover, Zhang and Luo [61] discussed the effect of particle breakage on the dilatancy deformation and critical state of calcareous sand, put forward the dilatancy equation considering particle breakage, and found that particle breakage significantly affected the critical state line, which is consistent with the conclusion of Bandini and Coop [62], but there were still shortcomings in determining the degree of particle breakage under external force. In addition, particle breakage will not only change the shear strength of calcareous sand [55], but also change the position of the critical state line and friction angle [62, 63]. Bandini and Coop [62] did find that due to particle breakage, the critical state line of calcareous sand is not unique, but moves with the increase of particle breakage. Unfortunately, the paper does not establish a quantitative relationship between particle breakage and critical state. Meanwhile, He et al. [63] pointed out that the peak friction angle, critical friction angle, and Br follow a power function and linear function. Although the relationship between particle breakage and critical state line has been qualitatively analyzed, the influencing factors of particle breakage of calcareous sand are complex, and its influencing mechanism and constitutive relationship remain largely unknown.Although there were many research studies on the mechanical properties of calcareous sand, all focused on the strength deformation relationship, ignoring the time-dependent deformation effect of calcareous sand [50, 64–66]. Lv et al. [64] found that the creep behavior of calcareous sand under general stress conditions is mainly caused by particle interlocking. Zhang et al. [67] believed that the deformation caused by the refining of calcareous sand and the filling of pores is the main cause of creep, and pointed out that the increase in the relative particle breakage with time is not obvious. However, this research on the creep properties of calcareous sand is mainly focused on phenomenon analysis, and the creep model of calcareous sand has not been established. Therefore, Wang, Ma, and Wang et al. [65], and Chen, Chao, and Wu et al. [68] established an empirical creep model of calcareous sand and believed that the particle breakage of calcareous sand is time-dependent, and the movement and reorganization of particles, particle slip, and particle breakage are the main causes of creep. At the same time, Cao and Ye [66] proposed a new four-parameter creep model with creep strain related to time, deviator stress, and effective confining pressure, which can provide a better mathematical description of the creep process of calcareous sand. Because of the significant rheological effect of geotechnical materials, it is very important to establish the strength-time-deformation relationship and a semiempirical creep constitutive model considering the particle breakage of calcareous sand.The existing research is primarily based on a single test method, focusing on the relationship between particle breakage and dilatancy behavior, energy dissipation law, physical properties (particle size, particle shape, and porosity), test conditions (axial strain, stress ratio, or confining pressure), stress path, and drainage condition (drained or undrained shear). However, the loading mode has a significant effect on the particle breakage of calcareous sand [69]. Therefore, Yu [55] proposed a hyperbolic model for the monotonic tests to correlate the particle breakage in relative breakage with the plastic work per unit volume, but this model could not be applicable in the cyclic tests. This is mainly due to the different mechanism of particle breakage induced by loading mode for calcareous sand. Meanwhile, calcareous sand was easier to break under triaxial shear than under one-dimensional compression [33]. The primary difference between one-dimensional compression, monotonic shear, and cyclic shear is that the relative motion between calcareous sand particles is limited and the contact force is large. The distribution law of particle size range is similar under different loading modes, but the distribution of new particles produced during particle breakage is different [70]. In addition, existing studies primarily focus on the size range of fine-grained (calcareous sand) in coral reef sediments and rarely consider the related research on the mechanical properties of calcareous gravel particles [19, 71].


#### 3.2.2. Triaxial Cyclic Shear Test

As the degree of particle breakage is significantly different under varying drainage conditions (drained or undrained shear), Qademi and Coop [72] conducted a triaxial cyclic shear test on undrained calcareous sand in Dog's Bay and found that the particle breakage caused by cyclic loading is much less than that under monotonic loading. However, Salem et al. [14] believed that there was no significant difference in particle breakage of calcareous sand from Dabaa on the northern coast of Egypt under monotonic and cyclic loading and pointed out that particle breakage had little effect on the effective confining pressure range and normal stress. This is mainly due to the calculated values of Br ranged from 0.01 (1.09%) to 0.02 (2.23%). Therefore, it is suggested to carry out monotonic and cyclic triaxial shear tests of calcareous sand to analyze the functional relationship between effective confining pressure, normal stress, and particle breakage. In addition, Zhou et al. [73] mentioned that particle shape played a more important role in the undrained monotonic and cyclic shear process of loose calcareous sand than particle breakage. This is mainly because the crushing potential is very low in this condition and the test results do not exhibit a clear trend between the crushing value and consolidation stress ratio. For the drained test conditions, Donohue et al. [13] found that the particle breakage and volumetric strain of calcareous sand from Dog’s Bay primarily occurs in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases gradually, and also believed that the degree of particle breakage was related to volumetric strain, stress level, cyclic stress ratio, and creep. However, no quantitative model between particle breakage and stress-strain has been proposed. Therefore, Wang et al. [74] analyzed the influence of confining pressure, cyclic stress ratio, and cycles on the development process of the particle breakage calcareous sand in Xisha, South China Sea. The research results were basically consistent with Donohue et al. [13], and the particle breakage form of calcareous sand under cyclic shear was primarily sharp angle wear, and a function model of the evolution law of particle breakage was preliminarily established. Furthermore, Yu [55] quantitatively analyzed particle breakage by using relative breakage and relative fractal dimension, and proposed a hyperbolic model to correlate relative fractal dimension and relative breakage, which can be used for both monotonic and cyclic tests. He et al. [75] proposed a new empirical prediction model of the resilient modulus considering particle breakage, which can better predict the resilient modulus of calcareous sand in the whole stress interval. It is also believed that the change in the fractal dimension can reflect the rule of particle breakage evolution.Based on the above analysis and the statistical results in Table2, it was found that (1) the Hardin model was primarily used by scholars at home and abroad to analyze the mechanism of particle breakage of calcareous sand. The Br of the isotropic compression test was less than 0.1 (the stress level is 1.5 MPa), but the Br of one-dimensional compression test (the stress level is basically in the ranges of 3.2 to 6.0 MPa) and triaxial shear test (the stress level is in the range of 0.2 to 3.2 MPa) is between 0.10 and 0.36, and the selected particle size is between 0.25 and 2.00 mm. (2) The primary factors affecting the particles breakage of calcareous sand are stress level (confining pressure and normal stress), drainage condition, loading mode, and volumetric strain, while the secondary factors were initial consolidation stress state, cycles, cyclic stress ratio, and creep. (3) Particle breakage and volumetric strain of calcareous sand mainly occur in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases. (4) Based on the analysis of the above research results, it is suggested that dynamic triaxial shear tests should be conducted under high confining pressure and complex dynamic stress. The functional relationship of calcareous sand under complex stress should be established. The influence of relative density, dynamic characteristics of large-sized (i.e., D > 2 mm), calcareous sand, the mechanism of particle breakage, and evolution should be considered.Table 2
Summary of the particle breakage ratio of calcareous sand.
Test typeLiteratureSampling locationStress level (MPa)Particle size (mm)ModelRelative particle breakageCCTZhang et al. [29]Yongshu Reef,4.001.0∼2.0Hardin0.038Nansha, ChinaZhang [30]Yongshu Reef,6.001.0∼2.0Hardin0.204Nansha, China0.5∼1.0Hardin0.0810.25∼0.5Hardin0.022Zhang et al. [43]Nansha, China45.41.0∼2.0Hardin0.359Xiao et al. [32]East coast of China3.20.8∼1.0Einav0.101Cai et al. [33]—3.20.509Hardin0.1100.530Hardin0.0600.533Hardin0.036ICTShahnazari and Rezvani [15]Hormuz Island1.50.780Hardin0.045Hormuz Island1.50.780Hardin0.033Bushehr Port1.50.430Hardin0.066Bushehr Port1.50.430Hardin0.049TSTZhang et al. [52]Yongshu Reef,13.80.377Hardin0.289Nansha, ChinaHu [12]Mischief Island, Nansha3.21.0∼2.0Hardin0.284Donohue et al. [13]Dog’s Bay, Ireland1.02.0∼3.0Hardin0.132Chen et al. [49]South China Sea1.21.0∼2.0Hardin0.128Wang et al. [19]Jinqing Island, Xisha0.60.0∼2.0Hardin0.052Yu [60]—3.0—Hardin0.319He et al. [63]Xisha, China0.81.0∼2.0Hardin0.075Wang et al. [74]Xisha, China0.20.6∼0.8Hardin0.025Note: (a) Confined compression test: CCT; (b) Isotropic compression test: ICT; (c) Triaxial shear test: TST.


### 3.3. Ring Shear Test

Luzzani et al. [76] and Coop et al. [77] studied the development process of the particle breakage of calcareous sand under large shear strain through a series of ring shear tests. It was found that volume compression was directly related to particle breakage, while particle breakage could not continue indefinitely, and it finally tended to a stable gradation. However, some scholars verified this conclusion through triaxial shear tests [12, 52] and one-dimensional compression tests [44]. [78] and got the same results. Moreover, Miao and Airey [42] compared the particle breakage laws of calcareous sand under the ring shear and one-dimensional compression conditions, and found that the mechanism of shear crushing and compression crushing was essentially different. This is mainly because, under the condition of shear test, the sample was subjected to shearing and the shearing force was very large, while under the condition of one-dimensional compression, the sample is compressed, so there was a significant difference in particle breakage under the two stress paths. Change under shear crushing was greater than that under compression crushing (Table 2). Furthermore, Huang et al. [79] believed the influence of relative density and normal stress on shearing and particle breakage evolution characteristics of calcareous sand, and found that the particle breakage primarily occurred from 0.50 to 2.00 mm, resulting in small particles ranging from 0.25 to 0.50 mm ins size. The proportion of particles smaller than 0.25 mm was almost unchanged, which is consistent with Qin et al. [69] and Ji et al. [70]. Wei et al. [80, 81] mainly studied the formation of shear bands, particle breakage evolution, and the shear mechanical behavior of fiber reinforced calcareous sand, and put forward the function model for the relative particle breakage and final shear strain, normal stress, initial sand particle size distribution, maximum particle size, and shear rate. However, they did not quantify the contribution of each parameter or the interdependencies among parameters. In addition, according to the analysis of particle size distribution of calcareous sand, most particles smaller than 0.074 mm can still be crushed under a small normal stress. It is suggested that the particle size limit for sand selected by Hardin [82] can further be reduced to consider fine particle breakage.In recent years, most scholars have used the analysis methods of particle breakage amount [83] and relative particle breakage ratio [82] to quantitatively analyze particle breakage (Table 2). However, whether these methods can be directly applied to particle breakage under large shear displacement as well as the mechanism of particle size distribution and shape evolution formed by these methods is still unclear [84, 85]. Under pile driving and impact loading, calcareous sand will undergo large shear deformation. The particle breakage characteristics and evolution mechanism of calcareous sand under large shear strain still need further study.


## 3.1. One-Dimensional and Isotropic Compression Tests

### 3.1.1. One-Dimensional Compression Test

Due to the brittleness and irregular shape of calcareous sand, particle breakage occurs at a stress level of 1.6 MPa [29], and the relative particle breakage ratio (Br) has a significant functional relationship with pressure, while the median particle size (D50) has a significant linear relationship (negative correlation) and a semilogarithmic linear relationship (positive correlation) with the crushing stress and ultimate relative particle breakage ratio (Brmax), respectively [30]. Wang, Lei, and Wang et al. [31] further pointed out that the relationship between Br and load value (Pmax) is defined by an exponential function. However, some studies show that the particle breakage degree of calcareous sand under a vertical pressure of 3.2 MPa is limited (Br = 0.019) or negligible. The high porosity and high pressure contractive of calcareous sand are primarily caused by particle morphology [18]. Meanwhile, the Br of calcareous sand should have an exponential relationship with volume and a linear relationship with input power per unit volume (Br ≤ 9%). When the relative density is low, more particles are broken compared to a relatively high density [29, 32]. Cai et al. [33] found that the particle breakage degree increased with increasing relative density and positively correlated with stress level (confining pressure or normal stress). Calcareous sand is easier to break under triaxial shear than under one-dimensional compression, and the crushing law under the two stress paths is significantly different. The particle breakage of calcareous sand with a relatively low density (loose sample) is more serious than that with relatively high density (dense sample) and has different ideas. Under the condition of triaxial shear tests, the greater the relative density of calcareous sand, the more serious the particle breakage.In addition, changes in the particle size, shape, and surface roughness are the most remarkable features of particle breakage [34, 35], which leads to the more complicated mechanical properties of calcareous sand [36, 37]. For instance, Wang et al. [31] pointed out that the particle shape has an important influence on the density and compressibility and believed that the compressibility of calcareous sands with different particle shapes is obviously different. Wei et al. [38] discussed the relationship between the particle shape of calcareous sand and its compressibility as well as particle breakage and believed that the change in the total perimeter of particles can effectively quantify particle breakage. However, Ata et al. [39] demonstrated that the crushability of medium to fine uniform calcareous sand was found to be insignificant and proposed the simplified equations to express the degradation of the constrained modulus due to crushability. In addition, the existing research mainly focuses on the particle breakage characteristics of calcareous sand and the influence of stress level on the crushing mechanism and compression characteristics. The particle size distribution curves of pre and post the test were used to measure the degree of particle breakage, while the influence of particle breakage and overlapping cover up during crushing of calcareous sand was not considered. Peng et al. [40], based on the method of dyeing calibration and image particle segmentation, put forward a cumulative particle breakage ratio considering the crushing overlapping coverage and pointed out that the absolute crushing amount for calcareous sand with medium particle size is the largest. However, the factors affecting particle breakage are relatively complex, and the mechanical mechanism and influencing factors of crushing overlapping coverage are unclear.The stress level ranging from 0 to 4 MPa was the main focus of the above research, while the compression deformation mechanism of calcareous sand under other stress levels (ranging from 0 to 100 MPa) was studied. Altuhafi et al. [41] studied the influence of initial gradation and density on the deformation mechanism of calcareous sand during one-dimensional compression and found that the particle breakage of calcareous sand is significantly reduced when the initial grading is changed to that of well graded. Miao and Airey [42] determined the particle breakage law of calcareous sand under high stress level and large shear strain through one-dimensional compression and ring shear tests, and found that the mechanisms of particle breakage caused by compression and shear are essentially different. Cai et al. [33] made use of one-dimensional compression (stress level ranging from 0.4 to 3.2 MPa) and triaxial shear test (confining pressure ranging from 0.1 to 0.4 MPa) to draw the same conclusion. Zhang et al. [43], Ma et al. [44], and Lv et al. [45] carried out compression tests on quartz sand and calcareous sand under high stress level (ranging from 0 to 100 MPa) and found that with increasing stress level, the particle breakage first increased and then gradually stabilized. Furthermore, the yield stresses of calcareous sand and quartz sand were 2 MPa and 10 MPa, respectively, and the compression deformation of calcareous sand before the yield point is primarily due to particle slip and grinding. However, deformation after the yield point is primarily caused by particle breakage, which is basically consistent with the conclusions of Zhang et al. [29]. The stress level at the yield point (about 1.6 MPa) is also relatively close to 2 MPa.


### 3.1.2. Isotropic Compression Test

Based on the analysis of isotropic compression and one-dimensional compression tests, the compression characteristics of calcareous sand are similar to those of normal consolidated cohesive soil, and the yield stress of calcareous sand under isotropic compression is larger than that under one-dimensional compression, which is primarily due to the presence of shear stress in the process of one-dimensional compression, which makes its particle breakage more significant [4]. In the Hormuz Island (HI) of the Persian Gulf, calcareous sand is significantly crushed under a confining pressure of 1.5 MPa, and the crushing amount of medium dense calcareous sand samples was the largest [46]. Similarly, the particle breakage of calcareous sand in Bushehr Port (BP) and Hormuz Island (HI) of the Persian Gulf positively correlates with stress level, while particle breakage is directly related to input energy [15]. The yield stress of BP sand and HI sand is 0.65 MPa and 1.00 MPa [15], respectively, which are smaller than that of calcareous sand in the South China Sea (2.00 MPa) [44, 45]. From isotropic compression tests, Shahnazari et al. [15] found that the denser the calcareous sand sample, the smaller the particle breakage. It is believed that this is primarily due to a decrease in average contact stress with increasing relative density. This conclusion is contrary to triaxial shear test results [47–50], but whether or not this conclusion is consistent with one-dimensional compression test results remains controversial [29, 32, 33]. However, Dehnavi et al. [46] found that the particle breakage of medium dense calcareous sand samples is the largest.Isotropic compression tests were the main measurement methods to explore the effects of stress level and relative density on the particle breakage of calcareous sand. Under the one-dimensional and isotropic compression test conditions, there are differences in the research results regarding the influence of relative density of calcareous sand on particle breakage, which needs further study. In addition, although Ma and Chiu [51] explored the crushing mechanism of calcareous sand particles under varying fines content and pointed out that when the fine content is less than or equal to 30%, the soil mixture (calcareous sand and fine crushed quartz silt) exhibits a unique compression line duo to the substantial breakage of calcareous sand. There are still few studies on the compression behavior of the mixture of calcareous sand and siliceous sand. Furthermore, the mechanical mechanism of the particle breakage of calcareous sand should be accurately analyzed from a microscopic point of view, and the mathematical relationship between particle breakage, stress level, and relative density needs to be established.


## 3.1.1. One-Dimensional Compression Test

Due to the brittleness and irregular shape of calcareous sand, particle breakage occurs at a stress level of 1.6 MPa [29], and the relative particle breakage ratio (Br) has a significant functional relationship with pressure, while the median particle size (D50) has a significant linear relationship (negative correlation) and a semilogarithmic linear relationship (positive correlation) with the crushing stress and ultimate relative particle breakage ratio (Brmax), respectively [30]. Wang, Lei, and Wang et al. [31] further pointed out that the relationship between Br and load value (Pmax) is defined by an exponential function. However, some studies show that the particle breakage degree of calcareous sand under a vertical pressure of 3.2 MPa is limited (Br = 0.019) or negligible. The high porosity and high pressure contractive of calcareous sand are primarily caused by particle morphology [18]. Meanwhile, the Br of calcareous sand should have an exponential relationship with volume and a linear relationship with input power per unit volume (Br ≤ 9%). When the relative density is low, more particles are broken compared to a relatively high density [29, 32]. Cai et al. [33] found that the particle breakage degree increased with increasing relative density and positively correlated with stress level (confining pressure or normal stress). Calcareous sand is easier to break under triaxial shear than under one-dimensional compression, and the crushing law under the two stress paths is significantly different. The particle breakage of calcareous sand with a relatively low density (loose sample) is more serious than that with relatively high density (dense sample) and has different ideas. Under the condition of triaxial shear tests, the greater the relative density of calcareous sand, the more serious the particle breakage.In addition, changes in the particle size, shape, and surface roughness are the most remarkable features of particle breakage [34, 35], which leads to the more complicated mechanical properties of calcareous sand [36, 37]. For instance, Wang et al. [31] pointed out that the particle shape has an important influence on the density and compressibility and believed that the compressibility of calcareous sands with different particle shapes is obviously different. Wei et al. [38] discussed the relationship between the particle shape of calcareous sand and its compressibility as well as particle breakage and believed that the change in the total perimeter of particles can effectively quantify particle breakage. However, Ata et al. [39] demonstrated that the crushability of medium to fine uniform calcareous sand was found to be insignificant and proposed the simplified equations to express the degradation of the constrained modulus due to crushability. In addition, the existing research mainly focuses on the particle breakage characteristics of calcareous sand and the influence of stress level on the crushing mechanism and compression characteristics. The particle size distribution curves of pre and post the test were used to measure the degree of particle breakage, while the influence of particle breakage and overlapping cover up during crushing of calcareous sand was not considered. Peng et al. [40], based on the method of dyeing calibration and image particle segmentation, put forward a cumulative particle breakage ratio considering the crushing overlapping coverage and pointed out that the absolute crushing amount for calcareous sand with medium particle size is the largest. However, the factors affecting particle breakage are relatively complex, and the mechanical mechanism and influencing factors of crushing overlapping coverage are unclear.The stress level ranging from 0 to 4 MPa was the main focus of the above research, while the compression deformation mechanism of calcareous sand under other stress levels (ranging from 0 to 100 MPa) was studied. Altuhafi et al. [41] studied the influence of initial gradation and density on the deformation mechanism of calcareous sand during one-dimensional compression and found that the particle breakage of calcareous sand is significantly reduced when the initial grading is changed to that of well graded. Miao and Airey [42] determined the particle breakage law of calcareous sand under high stress level and large shear strain through one-dimensional compression and ring shear tests, and found that the mechanisms of particle breakage caused by compression and shear are essentially different. Cai et al. [33] made use of one-dimensional compression (stress level ranging from 0.4 to 3.2 MPa) and triaxial shear test (confining pressure ranging from 0.1 to 0.4 MPa) to draw the same conclusion. Zhang et al. [43], Ma et al. [44], and Lv et al. [45] carried out compression tests on quartz sand and calcareous sand under high stress level (ranging from 0 to 100 MPa) and found that with increasing stress level, the particle breakage first increased and then gradually stabilized. Furthermore, the yield stresses of calcareous sand and quartz sand were 2 MPa and 10 MPa, respectively, and the compression deformation of calcareous sand before the yield point is primarily due to particle slip and grinding. However, deformation after the yield point is primarily caused by particle breakage, which is basically consistent with the conclusions of Zhang et al. [29]. The stress level at the yield point (about 1.6 MPa) is also relatively close to 2 MPa.


## 3.1.2. Isotropic Compression Test

Based on the analysis of isotropic compression and one-dimensional compression tests, the compression characteristics of calcareous sand are similar to those of normal consolidated cohesive soil, and the yield stress of calcareous sand under isotropic compression is larger than that under one-dimensional compression, which is primarily due to the presence of shear stress in the process of one-dimensional compression, which makes its particle breakage more significant [4]. In the Hormuz Island (HI) of the Persian Gulf, calcareous sand is significantly crushed under a confining pressure of 1.5 MPa, and the crushing amount of medium dense calcareous sand samples was the largest [46]. Similarly, the particle breakage of calcareous sand in Bushehr Port (BP) and Hormuz Island (HI) of the Persian Gulf positively correlates with stress level, while particle breakage is directly related to input energy [15]. The yield stress of BP sand and HI sand is 0.65 MPa and 1.00 MPa [15], respectively, which are smaller than that of calcareous sand in the South China Sea (2.00 MPa) [44, 45]. From isotropic compression tests, Shahnazari et al. [15] found that the denser the calcareous sand sample, the smaller the particle breakage. It is believed that this is primarily due to a decrease in average contact stress with increasing relative density. This conclusion is contrary to triaxial shear test results [47–50], but whether or not this conclusion is consistent with one-dimensional compression test results remains controversial [29, 32, 33]. However, Dehnavi et al. [46] found that the particle breakage of medium dense calcareous sand samples is the largest.Isotropic compression tests were the main measurement methods to explore the effects of stress level and relative density on the particle breakage of calcareous sand. Under the one-dimensional and isotropic compression test conditions, there are differences in the research results regarding the influence of relative density of calcareous sand on particle breakage, which needs further study. In addition, although Ma and Chiu [51] explored the crushing mechanism of calcareous sand particles under varying fines content and pointed out that when the fine content is less than or equal to 30%, the soil mixture (calcareous sand and fine crushed quartz silt) exhibits a unique compression line duo to the substantial breakage of calcareous sand. There are still few studies on the compression behavior of the mixture of calcareous sand and siliceous sand. Furthermore, the mechanical mechanism of the particle breakage of calcareous sand should be accurately analyzed from a microscopic point of view, and the mathematical relationship between particle breakage, stress level, and relative density needs to be established.


## 3.2. Triaxial Shear Test

### 3.2.1. Triaxial Monotonic Shear Test

Under triaxial shear, the particle breakage of calcareous sand will not increase indefinitely, but eventually tend to be constant. Particle breakage is affected and controlled by as well as positively correlates with confining pressure and shear strain [50, 52]. The stress-strain curve shows strain softening under low confining pressure and strain hardening under high confining pressure, while the influence of relative density on particle breakage is smaller than that of confining pressure [49]. Zhang et al. [53] proposed a hyperbolic function and an inverted S-shaped function to describe the prepeak portion and the postpeak portion of stress-strain curve of calcareous sand, respectively. However, the mathematical model for confining pressure, shear strain, and particle breakage is still unclear. Studies have shown that in the triaxial shearing process, the degree of particle breakage is closely related to the amount of absorbed plastic strain [54]. Based on monotonic triaxial shear tests, Yu [55] established a hyperbolic model that connected the relative particle breakage and fractal dimension of particles with the plastic strain per unit volume, which was believed to be the most reliable method of explaining the energy consumption characteristics of particle breakage. Although this method considered the fractal characteristics of soil, it was not suitable for cyclic shearing.Furthermore, the effect of particle breakage on shear strength is primarily manifested in the dissipation of energy and inhibition of dilatancy by particle breakage [48]. The energy consumed by particle breakage is a function of Br or crushing strength, and the influence of particle breakage on strength of calcareous sand first increases and then decreases with increasing confining pressure and finally tends to be stable. The effect of dilatancy on its strength is significantly stronger than that of particle breakage at low confining pressure and gradually weaker than that of particle breakage with increasing confining pressure [19, 56–58]. Some studies also show that the particle shape plays a key role in the shear contraction of sands, and particle breakage has a significant influence on the shear response of calcareous sand during dilatancy. Particle breakage primarily occurs in the dilatancy stage, which can inhibit the dilatancy behavior [16, 47, 59, 60]. Moreover, Zhang and Luo [61] discussed the effect of particle breakage on the dilatancy deformation and critical state of calcareous sand, put forward the dilatancy equation considering particle breakage, and found that particle breakage significantly affected the critical state line, which is consistent with the conclusion of Bandini and Coop [62], but there were still shortcomings in determining the degree of particle breakage under external force. In addition, particle breakage will not only change the shear strength of calcareous sand [55], but also change the position of the critical state line and friction angle [62, 63]. Bandini and Coop [62] did find that due to particle breakage, the critical state line of calcareous sand is not unique, but moves with the increase of particle breakage. Unfortunately, the paper does not establish a quantitative relationship between particle breakage and critical state. Meanwhile, He et al. [63] pointed out that the peak friction angle, critical friction angle, and Br follow a power function and linear function. Although the relationship between particle breakage and critical state line has been qualitatively analyzed, the influencing factors of particle breakage of calcareous sand are complex, and its influencing mechanism and constitutive relationship remain largely unknown.Although there were many research studies on the mechanical properties of calcareous sand, all focused on the strength deformation relationship, ignoring the time-dependent deformation effect of calcareous sand [50, 64–66]. Lv et al. [64] found that the creep behavior of calcareous sand under general stress conditions is mainly caused by particle interlocking. Zhang et al. [67] believed that the deformation caused by the refining of calcareous sand and the filling of pores is the main cause of creep, and pointed out that the increase in the relative particle breakage with time is not obvious. However, this research on the creep properties of calcareous sand is mainly focused on phenomenon analysis, and the creep model of calcareous sand has not been established. Therefore, Wang, Ma, and Wang et al. [65], and Chen, Chao, and Wu et al. [68] established an empirical creep model of calcareous sand and believed that the particle breakage of calcareous sand is time-dependent, and the movement and reorganization of particles, particle slip, and particle breakage are the main causes of creep. At the same time, Cao and Ye [66] proposed a new four-parameter creep model with creep strain related to time, deviator stress, and effective confining pressure, which can provide a better mathematical description of the creep process of calcareous sand. Because of the significant rheological effect of geotechnical materials, it is very important to establish the strength-time-deformation relationship and a semiempirical creep constitutive model considering the particle breakage of calcareous sand.The existing research is primarily based on a single test method, focusing on the relationship between particle breakage and dilatancy behavior, energy dissipation law, physical properties (particle size, particle shape, and porosity), test conditions (axial strain, stress ratio, or confining pressure), stress path, and drainage condition (drained or undrained shear). However, the loading mode has a significant effect on the particle breakage of calcareous sand [69]. Therefore, Yu [55] proposed a hyperbolic model for the monotonic tests to correlate the particle breakage in relative breakage with the plastic work per unit volume, but this model could not be applicable in the cyclic tests. This is mainly due to the different mechanism of particle breakage induced by loading mode for calcareous sand. Meanwhile, calcareous sand was easier to break under triaxial shear than under one-dimensional compression [33]. The primary difference between one-dimensional compression, monotonic shear, and cyclic shear is that the relative motion between calcareous sand particles is limited and the contact force is large. The distribution law of particle size range is similar under different loading modes, but the distribution of new particles produced during particle breakage is different [70]. In addition, existing studies primarily focus on the size range of fine-grained (calcareous sand) in coral reef sediments and rarely consider the related research on the mechanical properties of calcareous gravel particles [19, 71].


### 3.2.2. Triaxial Cyclic Shear Test

As the degree of particle breakage is significantly different under varying drainage conditions (drained or undrained shear), Qademi and Coop [72] conducted a triaxial cyclic shear test on undrained calcareous sand in Dog's Bay and found that the particle breakage caused by cyclic loading is much less than that under monotonic loading. However, Salem et al. [14] believed that there was no significant difference in particle breakage of calcareous sand from Dabaa on the northern coast of Egypt under monotonic and cyclic loading and pointed out that particle breakage had little effect on the effective confining pressure range and normal stress. This is mainly due to the calculated values of Br ranged from 0.01 (1.09%) to 0.02 (2.23%). Therefore, it is suggested to carry out monotonic and cyclic triaxial shear tests of calcareous sand to analyze the functional relationship between effective confining pressure, normal stress, and particle breakage. In addition, Zhou et al. [73] mentioned that particle shape played a more important role in the undrained monotonic and cyclic shear process of loose calcareous sand than particle breakage. This is mainly because the crushing potential is very low in this condition and the test results do not exhibit a clear trend between the crushing value and consolidation stress ratio. For the drained test conditions, Donohue et al. [13] found that the particle breakage and volumetric strain of calcareous sand from Dog’s Bay primarily occurs in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases gradually, and also believed that the degree of particle breakage was related to volumetric strain, stress level, cyclic stress ratio, and creep. However, no quantitative model between particle breakage and stress-strain has been proposed. Therefore, Wang et al. [74] analyzed the influence of confining pressure, cyclic stress ratio, and cycles on the development process of the particle breakage calcareous sand in Xisha, South China Sea. The research results were basically consistent with Donohue et al. [13], and the particle breakage form of calcareous sand under cyclic shear was primarily sharp angle wear, and a function model of the evolution law of particle breakage was preliminarily established. Furthermore, Yu [55] quantitatively analyzed particle breakage by using relative breakage and relative fractal dimension, and proposed a hyperbolic model to correlate relative fractal dimension and relative breakage, which can be used for both monotonic and cyclic tests. He et al. [75] proposed a new empirical prediction model of the resilient modulus considering particle breakage, which can better predict the resilient modulus of calcareous sand in the whole stress interval. It is also believed that the change in the fractal dimension can reflect the rule of particle breakage evolution.Based on the above analysis and the statistical results in Table2, it was found that (1) the Hardin model was primarily used by scholars at home and abroad to analyze the mechanism of particle breakage of calcareous sand. The Br of the isotropic compression test was less than 0.1 (the stress level is 1.5 MPa), but the Br of one-dimensional compression test (the stress level is basically in the ranges of 3.2 to 6.0 MPa) and triaxial shear test (the stress level is in the range of 0.2 to 3.2 MPa) is between 0.10 and 0.36, and the selected particle size is between 0.25 and 2.00 mm. (2) The primary factors affecting the particles breakage of calcareous sand are stress level (confining pressure and normal stress), drainage condition, loading mode, and volumetric strain, while the secondary factors were initial consolidation stress state, cycles, cyclic stress ratio, and creep. (3) Particle breakage and volumetric strain of calcareous sand mainly occur in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases. (4) Based on the analysis of the above research results, it is suggested that dynamic triaxial shear tests should be conducted under high confining pressure and complex dynamic stress. The functional relationship of calcareous sand under complex stress should be established. The influence of relative density, dynamic characteristics of large-sized (i.e., D > 2 mm), calcareous sand, the mechanism of particle breakage, and evolution should be considered.Table 2
Summary of the particle breakage ratio of calcareous sand.
Test typeLiteratureSampling locationStress level (MPa)Particle size (mm)ModelRelative particle breakageCCTZhang et al. [29]Yongshu Reef,4.001.0∼2.0Hardin0.038Nansha, ChinaZhang [30]Yongshu Reef,6.001.0∼2.0Hardin0.204Nansha, China0.5∼1.0Hardin0.0810.25∼0.5Hardin0.022Zhang et al. [43]Nansha, China45.41.0∼2.0Hardin0.359Xiao et al. [32]East coast of China3.20.8∼1.0Einav0.101Cai et al. [33]—3.20.509Hardin0.1100.530Hardin0.0600.533Hardin0.036ICTShahnazari and Rezvani [15]Hormuz Island1.50.780Hardin0.045Hormuz Island1.50.780Hardin0.033Bushehr Port1.50.430Hardin0.066Bushehr Port1.50.430Hardin0.049TSTZhang et al. [52]Yongshu Reef,13.80.377Hardin0.289Nansha, ChinaHu [12]Mischief Island, Nansha3.21.0∼2.0Hardin0.284Donohue et al. [13]Dog’s Bay, Ireland1.02.0∼3.0Hardin0.132Chen et al. [49]South China Sea1.21.0∼2.0Hardin0.128Wang et al. [19]Jinqing Island, Xisha0.60.0∼2.0Hardin0.052Yu [60]—3.0—Hardin0.319He et al. [63]Xisha, China0.81.0∼2.0Hardin0.075Wang et al. [74]Xisha, China0.20.6∼0.8Hardin0.025Note: (a) Confined compression test: CCT; (b) Isotropic compression test: ICT; (c) Triaxial shear test: TST.


## 3.2.1. Triaxial Monotonic Shear Test

Under triaxial shear, the particle breakage of calcareous sand will not increase indefinitely, but eventually tend to be constant. Particle breakage is affected and controlled by as well as positively correlates with confining pressure and shear strain [50, 52]. The stress-strain curve shows strain softening under low confining pressure and strain hardening under high confining pressure, while the influence of relative density on particle breakage is smaller than that of confining pressure [49]. Zhang et al. [53] proposed a hyperbolic function and an inverted S-shaped function to describe the prepeak portion and the postpeak portion of stress-strain curve of calcareous sand, respectively. However, the mathematical model for confining pressure, shear strain, and particle breakage is still unclear. Studies have shown that in the triaxial shearing process, the degree of particle breakage is closely related to the amount of absorbed plastic strain [54]. Based on monotonic triaxial shear tests, Yu [55] established a hyperbolic model that connected the relative particle breakage and fractal dimension of particles with the plastic strain per unit volume, which was believed to be the most reliable method of explaining the energy consumption characteristics of particle breakage. Although this method considered the fractal characteristics of soil, it was not suitable for cyclic shearing.Furthermore, the effect of particle breakage on shear strength is primarily manifested in the dissipation of energy and inhibition of dilatancy by particle breakage [48]. The energy consumed by particle breakage is a function of Br or crushing strength, and the influence of particle breakage on strength of calcareous sand first increases and then decreases with increasing confining pressure and finally tends to be stable. The effect of dilatancy on its strength is significantly stronger than that of particle breakage at low confining pressure and gradually weaker than that of particle breakage with increasing confining pressure [19, 56–58]. Some studies also show that the particle shape plays a key role in the shear contraction of sands, and particle breakage has a significant influence on the shear response of calcareous sand during dilatancy. Particle breakage primarily occurs in the dilatancy stage, which can inhibit the dilatancy behavior [16, 47, 59, 60]. Moreover, Zhang and Luo [61] discussed the effect of particle breakage on the dilatancy deformation and critical state of calcareous sand, put forward the dilatancy equation considering particle breakage, and found that particle breakage significantly affected the critical state line, which is consistent with the conclusion of Bandini and Coop [62], but there were still shortcomings in determining the degree of particle breakage under external force. In addition, particle breakage will not only change the shear strength of calcareous sand [55], but also change the position of the critical state line and friction angle [62, 63]. Bandini and Coop [62] did find that due to particle breakage, the critical state line of calcareous sand is not unique, but moves with the increase of particle breakage. Unfortunately, the paper does not establish a quantitative relationship between particle breakage and critical state. Meanwhile, He et al. [63] pointed out that the peak friction angle, critical friction angle, and Br follow a power function and linear function. Although the relationship between particle breakage and critical state line has been qualitatively analyzed, the influencing factors of particle breakage of calcareous sand are complex, and its influencing mechanism and constitutive relationship remain largely unknown.Although there were many research studies on the mechanical properties of calcareous sand, all focused on the strength deformation relationship, ignoring the time-dependent deformation effect of calcareous sand [50, 64–66]. Lv et al. [64] found that the creep behavior of calcareous sand under general stress conditions is mainly caused by particle interlocking. Zhang et al. [67] believed that the deformation caused by the refining of calcareous sand and the filling of pores is the main cause of creep, and pointed out that the increase in the relative particle breakage with time is not obvious. However, this research on the creep properties of calcareous sand is mainly focused on phenomenon analysis, and the creep model of calcareous sand has not been established. Therefore, Wang, Ma, and Wang et al. [65], and Chen, Chao, and Wu et al. [68] established an empirical creep model of calcareous sand and believed that the particle breakage of calcareous sand is time-dependent, and the movement and reorganization of particles, particle slip, and particle breakage are the main causes of creep. At the same time, Cao and Ye [66] proposed a new four-parameter creep model with creep strain related to time, deviator stress, and effective confining pressure, which can provide a better mathematical description of the creep process of calcareous sand. Because of the significant rheological effect of geotechnical materials, it is very important to establish the strength-time-deformation relationship and a semiempirical creep constitutive model considering the particle breakage of calcareous sand.The existing research is primarily based on a single test method, focusing on the relationship between particle breakage and dilatancy behavior, energy dissipation law, physical properties (particle size, particle shape, and porosity), test conditions (axial strain, stress ratio, or confining pressure), stress path, and drainage condition (drained or undrained shear). However, the loading mode has a significant effect on the particle breakage of calcareous sand [69]. Therefore, Yu [55] proposed a hyperbolic model for the monotonic tests to correlate the particle breakage in relative breakage with the plastic work per unit volume, but this model could not be applicable in the cyclic tests. This is mainly due to the different mechanism of particle breakage induced by loading mode for calcareous sand. Meanwhile, calcareous sand was easier to break under triaxial shear than under one-dimensional compression [33]. The primary difference between one-dimensional compression, monotonic shear, and cyclic shear is that the relative motion between calcareous sand particles is limited and the contact force is large. The distribution law of particle size range is similar under different loading modes, but the distribution of new particles produced during particle breakage is different [70]. In addition, existing studies primarily focus on the size range of fine-grained (calcareous sand) in coral reef sediments and rarely consider the related research on the mechanical properties of calcareous gravel particles [19, 71].


## 3.2.2. Triaxial Cyclic Shear Test

As the degree of particle breakage is significantly different under varying drainage conditions (drained or undrained shear), Qademi and Coop [72] conducted a triaxial cyclic shear test on undrained calcareous sand in Dog's Bay and found that the particle breakage caused by cyclic loading is much less than that under monotonic loading. However, Salem et al. [14] believed that there was no significant difference in particle breakage of calcareous sand from Dabaa on the northern coast of Egypt under monotonic and cyclic loading and pointed out that particle breakage had little effect on the effective confining pressure range and normal stress. This is mainly due to the calculated values of Br ranged from 0.01 (1.09%) to 0.02 (2.23%). Therefore, it is suggested to carry out monotonic and cyclic triaxial shear tests of calcareous sand to analyze the functional relationship between effective confining pressure, normal stress, and particle breakage. In addition, Zhou et al. [73] mentioned that particle shape played a more important role in the undrained monotonic and cyclic shear process of loose calcareous sand than particle breakage. This is mainly because the crushing potential is very low in this condition and the test results do not exhibit a clear trend between the crushing value and consolidation stress ratio. For the drained test conditions, Donohue et al. [13] found that the particle breakage and volumetric strain of calcareous sand from Dog’s Bay primarily occurs in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases gradually, and also believed that the degree of particle breakage was related to volumetric strain, stress level, cyclic stress ratio, and creep. However, no quantitative model between particle breakage and stress-strain has been proposed. Therefore, Wang et al. [74] analyzed the influence of confining pressure, cyclic stress ratio, and cycles on the development process of the particle breakage calcareous sand in Xisha, South China Sea. The research results were basically consistent with Donohue et al. [13], and the particle breakage form of calcareous sand under cyclic shear was primarily sharp angle wear, and a function model of the evolution law of particle breakage was preliminarily established. Furthermore, Yu [55] quantitatively analyzed particle breakage by using relative breakage and relative fractal dimension, and proposed a hyperbolic model to correlate relative fractal dimension and relative breakage, which can be used for both monotonic and cyclic tests. He et al. [75] proposed a new empirical prediction model of the resilient modulus considering particle breakage, which can better predict the resilient modulus of calcareous sand in the whole stress interval. It is also believed that the change in the fractal dimension can reflect the rule of particle breakage evolution.Based on the above analysis and the statistical results in Table2, it was found that (1) the Hardin model was primarily used by scholars at home and abroad to analyze the mechanism of particle breakage of calcareous sand. The Br of the isotropic compression test was less than 0.1 (the stress level is 1.5 MPa), but the Br of one-dimensional compression test (the stress level is basically in the ranges of 3.2 to 6.0 MPa) and triaxial shear test (the stress level is in the range of 0.2 to 3.2 MPa) is between 0.10 and 0.36, and the selected particle size is between 0.25 and 2.00 mm. (2) The primary factors affecting the particles breakage of calcareous sand are stress level (confining pressure and normal stress), drainage condition, loading mode, and volumetric strain, while the secondary factors were initial consolidation stress state, cycles, cyclic stress ratio, and creep. (3) Particle breakage and volumetric strain of calcareous sand mainly occur in the initial stage, and the degree of particle breakage increases with increasing cycles and then decreases. (4) Based on the analysis of the above research results, it is suggested that dynamic triaxial shear tests should be conducted under high confining pressure and complex dynamic stress. The functional relationship of calcareous sand under complex stress should be established. The influence of relative density, dynamic characteristics of large-sized (i.e., D > 2 mm), calcareous sand, the mechanism of particle breakage, and evolution should be considered.Table 2
Summary of the particle breakage ratio of calcareous sand.
Test typeLiteratureSampling locationStress level (MPa)Particle size (mm)ModelRelative particle breakageCCTZhang et al. [29]Yongshu Reef,4.001.0∼2.0Hardin0.038Nansha, ChinaZhang [30]Yongshu Reef,6.001.0∼2.0Hardin0.204Nansha, China0.5∼1.0Hardin0.0810.25∼0.5Hardin0.022Zhang et al. [43]Nansha, China45.41.0∼2.0Hardin0.359Xiao et al. [32]East coast of China3.20.8∼1.0Einav0.101Cai et al. [33]—3.20.509Hardin0.1100.530Hardin0.0600.533Hardin0.036ICTShahnazari and Rezvani [15]Hormuz Island1.50.780Hardin0.045Hormuz Island1.50.780Hardin0.033Bushehr Port1.50.430Hardin0.066Bushehr Port1.50.430Hardin0.049TSTZhang et al. [52]Yongshu Reef,13.80.377Hardin0.289Nansha, ChinaHu [12]Mischief Island, Nansha3.21.0∼2.0Hardin0.284Donohue et al. [13]Dog’s Bay, Ireland1.02.0∼3.0Hardin0.132Chen et al. [49]South China Sea1.21.0∼2.0Hardin0.128Wang et al. [19]Jinqing Island, Xisha0.60.0∼2.0Hardin0.052Yu [60]—3.0—Hardin0.319He et al. [63]Xisha, China0.81.0∼2.0Hardin0.075Wang et al. [74]Xisha, China0.20.6∼0.8Hardin0.025Note: (a) Confined compression test: CCT; (b) Isotropic compression test: ICT; (c) Triaxial shear test: TST.


## 3.3. Ring Shear Test

Luzzani et al. [76] and Coop et al. [77] studied the development process of the particle breakage of calcareous sand under large shear strain through a series of ring shear tests. It was found that volume compression was directly related to particle breakage, while particle breakage could not continue indefinitely, and it finally tended to a stable gradation. However, some scholars verified this conclusion through triaxial shear tests [12, 52] and one-dimensional compression tests [44]. [78] and got the same results. Moreover, Miao and Airey [42] compared the particle breakage laws of calcareous sand under the ring shear and one-dimensional compression conditions, and found that the mechanism of shear crushing and compression crushing was essentially different. This is mainly because, under the condition of shear test, the sample was subjected to shearing and the shearing force was very large, while under the condition of one-dimensional compression, the sample is compressed, so there was a significant difference in particle breakage under the two stress paths. Change under shear crushing was greater than that under compression crushing (Table 2). Furthermore, Huang et al. [79] believed the influence of relative density and normal stress on shearing and particle breakage evolution characteristics of calcareous sand, and found that the particle breakage primarily occurred from 0.50 to 2.00 mm, resulting in small particles ranging from 0.25 to 0.50 mm ins size. The proportion of particles smaller than 0.25 mm was almost unchanged, which is consistent with Qin et al. [69] and Ji et al. [70]. Wei et al. [80, 81] mainly studied the formation of shear bands, particle breakage evolution, and the shear mechanical behavior of fiber reinforced calcareous sand, and put forward the function model for the relative particle breakage and final shear strain, normal stress, initial sand particle size distribution, maximum particle size, and shear rate. However, they did not quantify the contribution of each parameter or the interdependencies among parameters. In addition, according to the analysis of particle size distribution of calcareous sand, most particles smaller than 0.074 mm can still be crushed under a small normal stress. It is suggested that the particle size limit for sand selected by Hardin [82] can further be reduced to consider fine particle breakage.In recent years, most scholars have used the analysis methods of particle breakage amount [83] and relative particle breakage ratio [82] to quantitatively analyze particle breakage (Table 2). However, whether these methods can be directly applied to particle breakage under large shear displacement as well as the mechanism of particle size distribution and shape evolution formed by these methods is still unclear [84, 85]. Under pile driving and impact loading, calcareous sand will undergo large shear deformation. The particle breakage characteristics and evolution mechanism of calcareous sand under large shear strain still need further study.


## 4. Discussions

The shearing and compression characteristics of calcareous sand are affected by particle breakage. TheBr mean value of calcareous sand under triaxial shear is larger than that under one-dimensional compression, while the Br mean value under isotropic compression is the smallest (as shown in Table 2). It easier to crush under triaxial shear than under one-dimensional compression, and there are significant differences in the crushing laws of calcareous sand under the shearing (i.e., triaxial shear and ring shear) and compression (i.e., one-dimensional and isotropic compression) experimental conditions. Furthermore, in order to quantify the degree of particle breakage, many scholars proposed a method to measure the degree of particle breakage through a comparison of pre- and post-crushing characteristic particle size or the particle size distribution curve as the standard (as shown in Table 3 and Figure 3).Table 3
Quantitative description models of particle breakage.
LiteratureFragmentation factorCalculation formulaMeasure standardCharacteristicLee and Farhoomand [86]B15B15 = D15i/D15fCharacteristic particle sizeReflect the overall change is poorLade et al. [87]B10B10 = 1 − D10f/D10iCharacteristic particle sizeIt can be related to the input energy in the experimentBai and Cui [88]BB = d60i − d60fCharacteristic particle sizeReflect the overall change is poorBiarez and Hicher [89]CuCu = D60/D10Characteristic particle sizeReflect the overall change is poorAta et al. [39]B50B50 = 1 − D50,a/D50,bCharacteristic particle sizeReflect the overall change is poorMarsal [83]BgBg = ∑∆WKParticle size distribution curveThis method cannot describe the condition of uniform initial particle size, and it is not normalized.Hardin [82]BrBr = Bt/BpParticle size distribution curveThe change in the particle fraction curve pre and post the test is comprehensively consideredNakata et al. [90]BfBf = 1 − R/100Particle size distribution curveIt can reflect the change in the particle size distribution curve pre and post the experimentEinav [91]Br∗Br∗=Bt∗/Bp∗Particle size distribution curveThe definition is clear, which can basically evaluate and compare the crushing degree of materials, and can reflect the evolution of the overall particle size distribution and the stress-strain relationshipMuir Wood and Maeda [92]IGIG = Bt’/Bp’Particle size distribution curveIt can reflect the evolution of the overall particle size distribution, but it cannot uniformly evaluate the degree of particle breakageFigure 3
Sketch of definitions of different particle breakage description methods: (a) Lee et al.; (b) Lade et al.; (c) Bai et al.; (d) Biarez et al.; (e) Marsal; (f) Hardin; (g) Nakata et al.; (h) Einav; (i) Wood et al.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Lee et al. [86] (Figure 3(a)), Lade et al. [87] (Figure 3(b)), Bai et al. [88] (Figure 3(c)), Biarez et al. [89] (Figure 3(d)), and Ata et al. [39] all selected the characteristic particle size as the measurement standard, and the difference or ratio of the content of the single characteristic particle size before and after the test was used to measure the degree of particle breakage, but failed to reflect the evolution law of overall particle breakage. This defect can be overcome by measuring the particle size distribution curve. Marsal [83] put forward the crushing index (Bg) (Figure 3(e)) by using the absolute value of the difference between the mass percentage of each particle group pre- and postcompression. Although this method can describe the complete particle breakage distribution, it fails to analyze the evolution degree of particle breakage via a normalized comparison. Hardin [82] defined the concept of relative particle breakage ratio, pointed out that the potential for breakage increased with increasing particle size, and proposed that the critical particle size for particle breakage was 0.074 mm (Figure 3(f)); Nakata et al. [90] selected the percentage of particles smaller than the minimum particle size in the original sand (i.e., R%) and proposed a simple particle breakage coefficient (Bf) to describe the degree of particle breakage (Figure 3(g)); Einav [91] removed the restriction on the limiting particle size based on the Hardin model, corrected the particle breakage ratio, and proposed a gradation curve under the limit state (Figure 3(h)); Wood et al. [92] proposed the grading state index (Ig) (Figure 3(i)) based on the particle size distribution curve, limit particle size curve, and maximum particle size line pre- and posttesting.According to the analysis results in Table3 and Figure 3, the method of measuring the degree of particle breakage evolution using the characteristic particle size was simple and practical, but it cannot describe the breakage of other particle sizes. Although the integral particle size distribution curve was considered in the description of the particle size distribution curve, it cannot be normalized, and the difference in crushing degree for different particle sizes cannot be evaluated. That is, it is impossible to distinguish the crushing amount of a certain particle size. Hardin's and Einav's particle breakage ratio was the primary reference source for scholars to study the quantitative description model of sand particle breakage; however, both of them have shortcomings. The critical particle size in the Hardin model was 0.074 mm, and all particles were assumed to be broken into powder particles under any initial gradation without considering particles smaller than 0.074 mm. The Einav model was modified on the basis of the Hardin model, which was more in line with reality. However, because the modified crushing potential depends on the particle termination gradation, the existing experimental technical level has not solved the problem of particle termination gradation. Therefore, it is necessary to propose a suitable particle breakage measurement method.


## 5. Conclusions

This literature review clearly points out the degree of particle breakage, measurement methods, and its influence on the mechanical properties of calcareous sand under different test conditions; summarizes the primary conclusions of particle breakage of calcareous sand; and reveals shortcomings of current measurement methods of particle breakage, which is helpful for better understanding particle breakage characteristics and encourages new research on particle breakage of calcareous sand. According to the results of the review, several conclusions can be drawn as follows.(1)
Research on the particle breakage mechanism of calcareous sand is primarily based on one-dimensional compression or isotropic compression tests, triaxial (monotonic or cyclic) shear tests, and ring shear tests, and the mechanism of shear crushing is essentially different from compression crushing. Compared with one-dimensional compression, calcareous sand is more easily to break under triaxial shear.(2)
Quantification of particle breakage is primarily based on measuring the characteristic particle size and the integral particle size distribution curve. Although the characteristic particle size is simple, it cannot describe the breakage of other particle sizes and lacks the overall understanding of particle breakage. The integrity of the grading curve is taken into account in the particle size distribution curve, so it is impossible to evaluate the difference in crushing evolution degree of different crushed materials. Hardin and Einav’s models are the primary reference models for quantitatively description of sand particle breakage, but both of them have shortcomings.(3)
The yield point stresses of calcareous sands in the South China Sea range from 1.6 MPa to 2 MPa, while that of calcareous sands in the Persian Gulf range from 0.65 MPa to 1.00 MPa. Before the yield point, calcareous sand deformation includes slip and wear between particles, and strain softening is caused by slip dilatancy deformation. After the yield point, with increasing stress, the degree of particle breakage first increases and then tends to be stable, and the deformation is controlled by particle breakage.(4)
Under the shear strain, particle breakage will not continue indefinitely, and the sample will eventually show a stable gradation, which is affected and controlled by both confining pressure and shear strain. Furthermore, the particle breakage of calcareous sand is time-dependent, and the movement and reorganization of particles, particle slip, and particle breakages are the main causes of creep.Due to the complexity of the material itself and its marine environment, there are still many essential problems to be further studied. The author suggests carrying out further research from the following aspects. Firstly, research on the particle breakage characteristics of calcareous sand under a large shear displacement and the evolution mechanism of particle size distribution and particle shape should be carried out. Secondly, the dynamic characteristics of the particle breakage of calcareous sand under high confining pressure and cyclic loading need further study. Finally, it is one of the future research directions to put forward a suitable method to measure the fragmentation evolution degree of calcareous sand particles.



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*Source: 6477197-2022-05-17.xml*