# Dynamic Characteristics of Coral Sand in the Condition of Particle Breakage

Authors: Hui Long; Ke Zhuang; Bo Deng; Jianlin Jiao; Junjie Zuo; Enlu You
Journal: Geofluids
 (2022)

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## Abstract

To study the influence of coral sand particle breakage on its dynamic characteristics, taking the South China Sea coral sand as the research object, one-dimensional consolidation test was conducted to prepare different particle breakage degree samples, and then, the GCTS resonant column test was carried out on these samples. The results show that with the degree of coral sand particle breakage increases, the maximum dynamic shear modulus of coral sand first increases and then decreases, and the minimum damping ratio also shows a similar trend. In addition, based on Hardin’s theory of the relative breakage, this study proposes a new modified relative breakageBr′ for describing particle breakage and establishes a relationship between the maximum dynamic shear modulus and the modified relative breakage Br′.


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## Body

## 1. Introduction

In offshore engineering and island reef engineering, coral sand is one of the common foundation soils and a commonly used filling material. Coral sand belongs to calcareous sand, with a calcium carbonate content of up to 90%, which is brittle and easy to break. The natural or artificial foundation formed by coral sand is often subjected to greater overburden pressure or compaction function during construction, resulting in particle breakage. In addition, the coral sand foundation will be affected by dynamic loads such as traffic loads, seismic action, and wind loads during the engineering. The impact of particle breakage on the dynamic characteristics of coral sand is very important for the structural safety of offshore engineering and island reef engineering.Generally, the breakage of sand particles is affected by factors such as consolidation confining pressure, moisture content, relative density, particle shape, and particle gradation. The results of the one-dimensional consolidation test show that the crushing of sand particles will increase with the increase of consolidation pressure [1]. Some scholars had compared different types of sand in one-dimensional compression tests and found that calcareous sand is more easily broken than terrigenous sand [2, 3]. Suescun-Florez et al. [2] conducted a series of axial loading and unloading tests on calcareous and siliceous sand and found that the fracture of silica sand starts at 10% strain, and the coral sand starts at 5%. Altuhafi and Coop [3] discussed the influence of the fractal dimension and density of the initial sand sample on the fracture of sand particles in axial compression. Peng et al. [4] conducted impact load tests on calcareous sand and siliceous sand and studied the crushing of particles by dyeing and image tracking of sand particles. Wang et al. [5] also studied the influence of relative density on nonuniformly graded coral sand based on one-dimensional compression tests. Ma et al. [6] conducted single particle crushing tests on coral sand with different large particle sizes at different loading rates and studied the effects of size and loading rate on the mechanical properties of coral sand. In the triaxial test, some scholars have also studied the evolution of particle breakage during the loading process. Chen and Ueng [7] established the stress-dilatancy relationship of sand under triaxial loading conditions based on Rowe’s principle of minimum energy ratio. The test results show that the dilatancy of sand plays an important role in the shear resistance process. Yu [8, 9] studied the influence of particle breakage on its shear, dilatancy, consolidation, and friction. Wang and Zha [10] conducted triaxial drainage and undrained cyclic shear tests on calcareous sand and studied the evolution characteristics of particle breakage during cyclic shear. After that, Wang et al. [11] also conducted triaxially drained and undrained compression tests on coral sand, setting up different stress paths, and the results show that under different conditions, the particle breakage rate shows different accumulation modes. Sadrekarimi and Olson [12] carried out scanning electron microscopy and ring shear tests on the sand in three areas. The experiment observed that the degree of particle breakage is affected by consolidation normal stress, shear displacement, particle mineralogy, particle size distribution, and drainage conditions. Wei et al. [13] studied the formation of shear bands and the evolution of particle breakage under different loading stress levels through ring shear tests.To describe the degree of breakage of sand particles, many scholars have proposed different methods: Lade et al. [14] proposed a particle crushing parameter based on D10 particle size, B10=1−D10f/D10i, (D10 is the grain diameter (mm) corresponding to 10% of the material being smaller by weight. The subscript i is the initial sample before the test, and the subscript f is the sample after the test). B10 changes from 0 to 1 as the particles break. Perfect et al. [15] use the concept of fractal dimension to describe the shape of particles and analyze the fractal dimension before and after the particle crushing to characterize the degree of particle breakage. Biarez and Hicher [16] measured the uniformity coefficient of the two particle gradation curves before and after the test, respectively Cu=D60/D10 (D10 and D60 are the grain diameter (mm) corresponding to 10% and 60% of the material being smaller by weight.), and the changes in the data before and after the test can reflect the degree of breakage of the particles. Nakata et al. define the particle breakage parameter Bf=1−R/100, in which R is the percentage of the minimum particle size of the initial particle gradation curve corresponding to the percentage on the particle gradation curve after the test [17]. Hardin [18] proposed the concept of relative breakage to characterize particle breakage Br=Bt/Bp. Bp represents the breakage potential which is the area enclosed by the initial particle size gradation curve and D=0.074 mm and finer=100%. Bt represents the total breakage which is the area of the graph surrounded by the particle size gradation curve and D=0.074 mm before and after the test, as is shown in Figures 1 and 2.Figure 1
Definition of total breakage.Figure 2
Definition of breakage potential.In terms of research on dynamic characteristics of coral sand, He et al. [19] studied the effects of initial mean effective stress and cyclic stress ratio on the resilient modulus of coral sand in the South China Sea through a series of drainage cyclic triaxial tests. Dong et al. [20] used a split Hopkinson pressure bar (SHPB) apparatus to perform impact tests on coral sand and quartz sand and obtained the compressive stress-strain curves of the two kinds of sand in a one-dimensional strain state. The results show that the internal porosity of the particles is the main factor affecting the strain rate correlation of coral sand under impact load. Jafaria et al. [21, 22] carried out the resonant column and cyclic triaxial tests for the dynamic mechanical properties of calcareous sand and siliceous sand under isotropic and anisotropic conditions. The results show that calcareous sand has a higher shear modulus and a lower damping ratio. Rasouli et al. [23] studied the influence of the cyclic shear orientation on the liquefaction performance of Hormuz calcareous sand. Different initial cyclic shear orientations and reversal of shear stress have an impact on the shear and failure mechanism of calcareous sand. Giang et al. [24] analyzed the influence of calcareous sand particle shape, stiffness, and uniformity coefficient on the dynamic shear modulus and proposed a formula for predicting the maximum dynamic shear modulus based on the uniformity coefficient. Zhao et al. [25] used a split Hopkinson pressure bar (SPHB) apparatus to study the dynamic compression test of dry calcareous sand with different densities and describe the dynamic mechanical behavior of dry calcareous sand with different models.In general, the existing research on the breakage characteristics of coral sand particles is mainly carried out under static conditions, considering the influence of various factors such as relative density, particle gradation, particle shape, and stress path on its breakage characteristics. However, there are few studies considering the impact of coral sand particle breakage on its dynamic characteristics. The breakage of coral sand particles will inevitably lead to changes in its dynamic characteristics. How to evaluate the impact of coral sand particle crushing on its dynamic characteristics is an urgent problem to be solved. In this study, the resonant column test is used to study the dynamic characteristics of coral sand samples with different particle breakage degrees, attempting to establish the relationship between coral sand particle breakage and its dynamic shear modulus and damping characteristics, proposing a method to describe the degree of coral sand particle breakage under dynamic conditions. Relevant research results can provide a reference for seismic design and safety evaluation of coral sand and similar foundation structures.


## 2. Materials and Methods

The coral sand material was shipped from an island reef in the South China Sea. To eliminate the influence of large pieces of coral sand on the test, the test blocks above 2 mm were removed. To facilitate the comparison of results, all the original samples are controlled with a dry density of 1.56 g/cm3 of and a relative density of 50%. All original samples are prepared with the same batch of materials.


### 2.1. The One-Dimensional Consolidation Test

The one-dimensional consolidation test adopts the GZQ-1 automatic one-dimensional consolidation apparatus of the University of South China. The load range of GZQ-1 is 0 ~ 4.8 kN, and the maximum pressure can reach 3200 kPa. The one-dimensional consolidation test is carried out on the original sample under four consolidation pressures of 200, 400, 800, and 1200 kPa, and five sets of samples are set under each consolidation pressure. The test mold has a height of 20 mm and a diameter of 61.8 mm. The test plan is shown in Table1. Before the start of the consolidation test, the sample is saturated with water, and the saturation time is more than 8 hours so that the pores between the samples are filled with water.Table 1
Test scheme.
Test no.Consolidation pressure/kPaTest no.Consolidation pressure/kPaCS-201200 kPaCS-801800 kPaCS-202200 kPaCS-802800 kPaCS-203200 kPaCS-803800 kPaCS-204200 kPaCS-804800 kPaCS-205200 kPaCS-805800 kPaCS-401400 kPaCS-12011200 kPaCS-402400 kPaCS-12021200 kPaCS-403400 kPaCS-12031200 kPaCS-404400 kPaCS-12041200 kPaCS-405400 kPaCS-12051200 kPa


### 2.2. The Resonant Column Test

The test adopts the GCTS resonant column apparatus TSH-100. The axial displacement sensor used in the test has a range of ±7.5 mm, and the built-in axial force sensor has a range of ±4 kN. The main purpose of the test is to study the dynamic shear modulus and damping ratio of saturated coral sand under isobaric consolidation. According to the breakage of coral sand under different confining pressures obtained by the one-dimensional consolidation test, prepare the resonant column loading sample. The sample diameter is 50 mm, and the height is 100 mm.Saturate the sample, and consolidate isobaric after the sample is saturated. The axial strain amplitude increases step by step from1×10−4% to 1×10−2%. After each level of loading, the sample is subjected to a recovery time of not less than 15 minutes to the initial consolidation state, and then, the next level of loading is performed. Load 5 cycles per level, and the loading frequency is 0.1 Hz. The consolidation pressure is 100 and 200 kPa, respectively.


## 2.1. The One-Dimensional Consolidation Test

The one-dimensional consolidation test adopts the GZQ-1 automatic one-dimensional consolidation apparatus of the University of South China. The load range of GZQ-1 is 0 ~ 4.8 kN, and the maximum pressure can reach 3200 kPa. The one-dimensional consolidation test is carried out on the original sample under four consolidation pressures of 200, 400, 800, and 1200 kPa, and five sets of samples are set under each consolidation pressure. The test mold has a height of 20 mm and a diameter of 61.8 mm. The test plan is shown in Table1. Before the start of the consolidation test, the sample is saturated with water, and the saturation time is more than 8 hours so that the pores between the samples are filled with water.Table 1
Test scheme.
Test no.Consolidation pressure/kPaTest no.Consolidation pressure/kPaCS-201200 kPaCS-801800 kPaCS-202200 kPaCS-802800 kPaCS-203200 kPaCS-803800 kPaCS-204200 kPaCS-804800 kPaCS-205200 kPaCS-805800 kPaCS-401400 kPaCS-12011200 kPaCS-402400 kPaCS-12021200 kPaCS-403400 kPaCS-12031200 kPaCS-404400 kPaCS-12041200 kPaCS-405400 kPaCS-12051200 kPa


## 2.2. The Resonant Column Test

The test adopts the GCTS resonant column apparatus TSH-100. The axial displacement sensor used in the test has a range of ±7.5 mm, and the built-in axial force sensor has a range of ±4 kN. The main purpose of the test is to study the dynamic shear modulus and damping ratio of saturated coral sand under isobaric consolidation. According to the breakage of coral sand under different confining pressures obtained by the one-dimensional consolidation test, prepare the resonant column loading sample. The sample diameter is 50 mm, and the height is 100 mm.Saturate the sample, and consolidate isobaric after the sample is saturated. The axial strain amplitude increases step by step from1×10−4% to 1×10−2%. After each level of loading, the sample is subjected to a recovery time of not less than 15 minutes to the initial consolidation state, and then, the next level of loading is performed. Load 5 cycles per level, and the loading frequency is 0.1 Hz. The consolidation pressure is 100 and 200 kPa, respectively.


## 3. Results and Discussion

### 3.1. The Results of One-Dimensional Consolidation Test

Figure3 shows the average gradation curve of coral sand samples under different consolidation pressures. CS-0 is the gradation curve of the original sample, and CS-2 ~ CS-12 is, respectively, the average gradation curve of the sample after the particles are crushed under the consolidation pressure of 200 kPa~1200 kPa. The particle gradation is shown in Table 2.Figure 3
The average particle gradation after one-dimensional consolidation test.Table 2
The average particle gradation under different consolidation test.
Grain sizeCS-0CS-2CS-4CS-8CS-121 ~ 2 mm11.00%7.84%7.10%6.78%6.20%0.5 ~1 mm25.00%25.72%27.78%27.00%28.04%0.25~0.5 mm33.00%33.31%31.71%32.80%32.50%0.1 ~ 0.25 mm29.00%30.95%31.20%31.33%31.05%0.075~0.1 mm1.60%1.78%1.73%1.64%1.65%<0.075 mm0.40%0.40%0.48%0.45%0.56%


### 3.2. The Results of Resonant Column Test

#### 3.2.1. The Dynamic Shear Modulus and the Damping Ratio

The decay curve of the dynamic shear modulus and the growth curve of the damping ratio with the increase of shear strain in a small strain range were obtained through the resonant column test. The research results of this study, Carraro and Bortolotto [26], Seed and Idriss [27], Rollins et al. [28], and Kokusho [29], are compared as shown in Figures 4 and 5.Figure 4
Shear modulus ratio curve.Figure 5
Damping ratio curve.The figures show that the overall trend of dynamic shear modulus ratio and damping ratio is similar to the previous research results, and the distribution range of the test results is also within the results given by scholars. The test dynamic shear modulus ratio is the closest to the lower envelope curve of the results given by Carraro and Rollins; the damping ratio is also close to the upper envelope curve given by Carraro, and the result of the damping ratio is within the range given by Rollins.The hyperbolic model of Darendeli is used to establish the relationship between coral sand dynamic shear modulus ratio and shear strain [30]. The expression is as formula (1), where γr is the shear strain when G/Gmax=0.5, and the range of the parameter a is 0.88~1.03. For the damping ratio, formula (2) is used to establish the relationship between the damping ratio and the shear strain, where s and b are fitting parameters, the range of s is 5.93~7.89, and the range of b is 0.91~1.26. Model parameters are shown in Table 3.
(1)GGmax=11+γ/γra,(2)λ=λmin+sγ/γrb1+γ/γrb.Table 3
Model parameters.
Test no.Consolidation pressure/kPaG/Gmax−γR2λ−γR2aλminsbCS-01000.88640.99030.855.98421.36510.98632000.94380.99980.767.35610.74130.9851CS-21000.93140.98530.987.84590.98430.98512000.98530.99160.897.51941.10660.9909CS-41000.97520.99060.948.81291.10010.99692000.95710.99120.97.34811.56230.9847CS-81001.06290.98270.876.10531.71610.99622000.96980.99641.027.86890.85390.9885CS-121000.96960.97480.97.94371.08460.99332000.935730.99490.977.24310.92520.9914


#### 3.2.2. The Maximum Dynamic Shear Modulus and the Modified Relative Breakage

The maximum dynamic shear modulusGmax can be obtained by fitting the G-γ curve [31], and Gmax can be obtained by using formula (1),
(3)Gmax=1a+bγγ⟶0=1a,wherea and b are the slope and intercept of the 1/G−γ curve.The relationship between Hardin’s relative breakage and the maximum dynamic shear modulus is shown in Table4. Under the same consolidation pressure, Hardin’s relative breakage and the maximum dynamic shear modulus first increase and then decrease. From the overall trend, particle breakage improves the dynamic strength performance of coral sand. This is because the particle breakage will make the fine particles of the broken coral sand fill its coarse particles, so that the coral sand is in a relatively good dense state, improving its dynamic bearing performance. In the process of particle breakage, we refine the role of each single particle group in particle breakage. Therefore, based on Hardin’s relative breakage, propose the concept of modified relative breakage and discuss another method to characterize coral sand particle breakage, which is used to establish the relationship between particle breakage and its maximum dynamic shear modulus. As shown in Figure 6, in the total breakage, we separately consider the role of a single particle group in particle breakage. Divide the total breakage of a single particle group by the initial breakage potential to obtain the relative breakage of each particle group, namely, Brn=Btn/Bp (Btn is the total breakage of the corresponding particle group, such as Bt1 for the 1~2 mm particle group total breakage, Bp is the initial breakage potential). The total breakage Btn of the single particle group under the corresponding particle breakage degree of each consolidation pressure is shown in Table 5. Taking into account that the change in the content of the relatively large particle group in the sand has a greater impact on its maximum dynamic shear modulus [31], the modified relative breakage potential Br′ is defined as Br′=aBt1+bBt2+cBt3+Bt4+Bt5/Bp and assigns a weight coefficient greater than 1 to the total breakage of a single particle group with relatively large particle size.Table 4
Hardin’s relative breakage and the maximum shear modulus.
BrConsolidation pressure/kPaGmax/MPa0100126.4200178.30.082434100137.9200193.70.084986100140.2200196.80.098658100136.4200188.30.100987100131.5200185.8Figure 6
Definition of modified relative breakage.Table 5
The particle breakage degree and total breakage of single particle group.
Particle breakage degreeTotal breakage of single particle groupBtnThe breakage potentialBpThe modified relative breakageBr՛Bt1Bt2Bt3Bt4Bt5CS-21.5791.3990.5710.1730.00245.370.134CS-41.9491.2540.4410.1960.00345.370.143CS-82.1341.6340.5860.1880.00145.370.167CS-122.3981.6390.5020.1850.00445.370.177The modified relative breakageBr′ and the maximum dynamic shear modulus are shown in Table 6. Fitting the modified relative breakage and the maximum dynamic shear modulus under different consolidation pressures, it is found that the modified relative breakage and the maximum dynamic shear modulus in this paper have a good fitting relationship when a=2, b=1.5, and c=1.2, which is in line with the Gauss function relationship. The fitting results are shown in Figures 7 and 8. The Gauss function formula is shown in
(4)Gmax=G0,max+Ae−2×Br′−m/n2,Table 6
The modified relative breakage and the maximum shear modulus.
Br′Consolidation pressure/kPaGmax/MPa0100126.4200178.30.134100137.9200193.70.143100140.2200196.80.167100136.4200188.30.177100131.5200185.8Figure 7
Fitting results (σc=100 kPa).Figure 8
Fitting results (σc=200 kPa).whereG0,max is the initial maximum dynamic shear modulus of the original coral sand sample and A, m, and n are fitting parameters.The changing trend of the fitted maximum dynamic shear modulus under the modified relative breakage is consistent with the change of Hardin’s relative breakage, which also first increases and then decreases. Under a certain degree of breakage, the maximum dynamic shear modulus of coral sand will reach the maximum, and the dynamic strength performance of coral sand will be the best at this time. It can be known from the functional relationship that when the maximum dynamic shear modulus takes the maximum value, the modified relative breakage is between 0.145 and 0.149. At this time, the corresponding relative breakage is the breakage degree near 400 kPa in the one-dimensional consolidation test. The filling effect of the fine particles generated by particles breakage is optimal near this consolidation pressure, so that the dynamic bearing performance of the coral sand reaches the best state. Due to the problem of the number of test samples, what is the specific optimal breakage degree is still a question that needs to be discussed. In the project, the coral sand foundation is often compacted, and the coral sand particles will also be crushed to a certain extent. From this, we can search for the optimal degree of compaction in the project based on the test results and apply a specific compaction work to the coral sand foundation to achieve the optimal breakage of the coral sand foundation, to obtain the best maximum dynamic shear modulus and adopt certain foundation reinforcement measures [32] to improve the seismic performance of the coral sand foundation.


#### 3.2.3. The Minimum Damping Ratio and the Modified Relative Breakage

Based on the modified relative breakage proposed above, the minimum damping ratio of coral sand with different modified relative breakage is shown in Figure9. In general, with the increase of the modified relative breakage, the minimum damping ratio of coral sands first increases and then decreases. Within the range of modified relative breakage Br′ from 0 to 0.134, the minimum damping ratios of coral sand under the two confining pressures both show an increasing trend. When the modified relative breakage is greater than 0.134, the changing trend of the minimum damping ratio of the coral sand under the consolidation pressure of the two groups is significantly different. Neither Hardin’s relative breakage nor the modified relative breakage can establish an effective fitting relationship with the minimum damping ratio obtained by the experiment in this paper. The minimum damping ratio of coral sand in the state of particle breakage has greater dispersion, and subsequent studies can still explore a more effective method to establish the relationship between the two.Figure 9
The relationship betweenBr′ and λmin.


## 3.1. The Results of One-Dimensional Consolidation Test

Figure3 shows the average gradation curve of coral sand samples under different consolidation pressures. CS-0 is the gradation curve of the original sample, and CS-2 ~ CS-12 is, respectively, the average gradation curve of the sample after the particles are crushed under the consolidation pressure of 200 kPa~1200 kPa. The particle gradation is shown in Table 2.Figure 3
The average particle gradation after one-dimensional consolidation test.Table 2
The average particle gradation under different consolidation test.
Grain sizeCS-0CS-2CS-4CS-8CS-121 ~ 2 mm11.00%7.84%7.10%6.78%6.20%0.5 ~1 mm25.00%25.72%27.78%27.00%28.04%0.25~0.5 mm33.00%33.31%31.71%32.80%32.50%0.1 ~ 0.25 mm29.00%30.95%31.20%31.33%31.05%0.075~0.1 mm1.60%1.78%1.73%1.64%1.65%<0.075 mm0.40%0.40%0.48%0.45%0.56%


## 3.2. The Results of Resonant Column Test

### 3.2.1. The Dynamic Shear Modulus and the Damping Ratio

The decay curve of the dynamic shear modulus and the growth curve of the damping ratio with the increase of shear strain in a small strain range were obtained through the resonant column test. The research results of this study, Carraro and Bortolotto [26], Seed and Idriss [27], Rollins et al. [28], and Kokusho [29], are compared as shown in Figures 4 and 5.Figure 4
Shear modulus ratio curve.Figure 5
Damping ratio curve.The figures show that the overall trend of dynamic shear modulus ratio and damping ratio is similar to the previous research results, and the distribution range of the test results is also within the results given by scholars. The test dynamic shear modulus ratio is the closest to the lower envelope curve of the results given by Carraro and Rollins; the damping ratio is also close to the upper envelope curve given by Carraro, and the result of the damping ratio is within the range given by Rollins.The hyperbolic model of Darendeli is used to establish the relationship between coral sand dynamic shear modulus ratio and shear strain [30]. The expression is as formula (1), where γr is the shear strain when G/Gmax=0.5, and the range of the parameter a is 0.88~1.03. For the damping ratio, formula (2) is used to establish the relationship between the damping ratio and the shear strain, where s and b are fitting parameters, the range of s is 5.93~7.89, and the range of b is 0.91~1.26. Model parameters are shown in Table 3.
(1)GGmax=11+γ/γra,(2)λ=λmin+sγ/γrb1+γ/γrb.Table 3
Model parameters.
Test no.Consolidation pressure/kPaG/Gmax−γR2λ−γR2aλminsbCS-01000.88640.99030.855.98421.36510.98632000.94380.99980.767.35610.74130.9851CS-21000.93140.98530.987.84590.98430.98512000.98530.99160.897.51941.10660.9909CS-41000.97520.99060.948.81291.10010.99692000.95710.99120.97.34811.56230.9847CS-81001.06290.98270.876.10531.71610.99622000.96980.99641.027.86890.85390.9885CS-121000.96960.97480.97.94371.08460.99332000.935730.99490.977.24310.92520.9914


### 3.2.2. The Maximum Dynamic Shear Modulus and the Modified Relative Breakage

The maximum dynamic shear modulusGmax can be obtained by fitting the G-γ curve [31], and Gmax can be obtained by using formula (1),
(3)Gmax=1a+bγγ⟶0=1a,wherea and b are the slope and intercept of the 1/G−γ curve.The relationship between Hardin’s relative breakage and the maximum dynamic shear modulus is shown in Table4. Under the same consolidation pressure, Hardin’s relative breakage and the maximum dynamic shear modulus first increase and then decrease. From the overall trend, particle breakage improves the dynamic strength performance of coral sand. This is because the particle breakage will make the fine particles of the broken coral sand fill its coarse particles, so that the coral sand is in a relatively good dense state, improving its dynamic bearing performance. In the process of particle breakage, we refine the role of each single particle group in particle breakage. Therefore, based on Hardin’s relative breakage, propose the concept of modified relative breakage and discuss another method to characterize coral sand particle breakage, which is used to establish the relationship between particle breakage and its maximum dynamic shear modulus. As shown in Figure 6, in the total breakage, we separately consider the role of a single particle group in particle breakage. Divide the total breakage of a single particle group by the initial breakage potential to obtain the relative breakage of each particle group, namely, Brn=Btn/Bp (Btn is the total breakage of the corresponding particle group, such as Bt1 for the 1~2 mm particle group total breakage, Bp is the initial breakage potential). The total breakage Btn of the single particle group under the corresponding particle breakage degree of each consolidation pressure is shown in Table 5. Taking into account that the change in the content of the relatively large particle group in the sand has a greater impact on its maximum dynamic shear modulus [31], the modified relative breakage potential Br′ is defined as Br′=aBt1+bBt2+cBt3+Bt4+Bt5/Bp and assigns a weight coefficient greater than 1 to the total breakage of a single particle group with relatively large particle size.Table 4
Hardin’s relative breakage and the maximum shear modulus.
BrConsolidation pressure/kPaGmax/MPa0100126.4200178.30.082434100137.9200193.70.084986100140.2200196.80.098658100136.4200188.30.100987100131.5200185.8Figure 6
Definition of modified relative breakage.Table 5
The particle breakage degree and total breakage of single particle group.
Particle breakage degreeTotal breakage of single particle groupBtnThe breakage potentialBpThe modified relative breakageBr՛Bt1Bt2Bt3Bt4Bt5CS-21.5791.3990.5710.1730.00245.370.134CS-41.9491.2540.4410.1960.00345.370.143CS-82.1341.6340.5860.1880.00145.370.167CS-122.3981.6390.5020.1850.00445.370.177The modified relative breakageBr′ and the maximum dynamic shear modulus are shown in Table 6. Fitting the modified relative breakage and the maximum dynamic shear modulus under different consolidation pressures, it is found that the modified relative breakage and the maximum dynamic shear modulus in this paper have a good fitting relationship when a=2, b=1.5, and c=1.2, which is in line with the Gauss function relationship. The fitting results are shown in Figures 7 and 8. The Gauss function formula is shown in
(4)Gmax=G0,max+Ae−2×Br′−m/n2,Table 6
The modified relative breakage and the maximum shear modulus.
Br′Consolidation pressure/kPaGmax/MPa0100126.4200178.30.134100137.9200193.70.143100140.2200196.80.167100136.4200188.30.177100131.5200185.8Figure 7
Fitting results (σc=100 kPa).Figure 8
Fitting results (σc=200 kPa).whereG0,max is the initial maximum dynamic shear modulus of the original coral sand sample and A, m, and n are fitting parameters.The changing trend of the fitted maximum dynamic shear modulus under the modified relative breakage is consistent with the change of Hardin’s relative breakage, which also first increases and then decreases. Under a certain degree of breakage, the maximum dynamic shear modulus of coral sand will reach the maximum, and the dynamic strength performance of coral sand will be the best at this time. It can be known from the functional relationship that when the maximum dynamic shear modulus takes the maximum value, the modified relative breakage is between 0.145 and 0.149. At this time, the corresponding relative breakage is the breakage degree near 400 kPa in the one-dimensional consolidation test. The filling effect of the fine particles generated by particles breakage is optimal near this consolidation pressure, so that the dynamic bearing performance of the coral sand reaches the best state. Due to the problem of the number of test samples, what is the specific optimal breakage degree is still a question that needs to be discussed. In the project, the coral sand foundation is often compacted, and the coral sand particles will also be crushed to a certain extent. From this, we can search for the optimal degree of compaction in the project based on the test results and apply a specific compaction work to the coral sand foundation to achieve the optimal breakage of the coral sand foundation, to obtain the best maximum dynamic shear modulus and adopt certain foundation reinforcement measures [32] to improve the seismic performance of the coral sand foundation.


### 3.2.3. The Minimum Damping Ratio and the Modified Relative Breakage

Based on the modified relative breakage proposed above, the minimum damping ratio of coral sand with different modified relative breakage is shown in Figure9. In general, with the increase of the modified relative breakage, the minimum damping ratio of coral sands first increases and then decreases. Within the range of modified relative breakage Br′ from 0 to 0.134, the minimum damping ratios of coral sand under the two confining pressures both show an increasing trend. When the modified relative breakage is greater than 0.134, the changing trend of the minimum damping ratio of the coral sand under the consolidation pressure of the two groups is significantly different. Neither Hardin’s relative breakage nor the modified relative breakage can establish an effective fitting relationship with the minimum damping ratio obtained by the experiment in this paper. The minimum damping ratio of coral sand in the state of particle breakage has greater dispersion, and subsequent studies can still explore a more effective method to establish the relationship between the two.Figure 9
The relationship betweenBr′ and λmin.


## 3.2.1. The Dynamic Shear Modulus and the Damping Ratio

The decay curve of the dynamic shear modulus and the growth curve of the damping ratio with the increase of shear strain in a small strain range were obtained through the resonant column test. The research results of this study, Carraro and Bortolotto [26], Seed and Idriss [27], Rollins et al. [28], and Kokusho [29], are compared as shown in Figures 4 and 5.Figure 4
Shear modulus ratio curve.Figure 5
Damping ratio curve.The figures show that the overall trend of dynamic shear modulus ratio and damping ratio is similar to the previous research results, and the distribution range of the test results is also within the results given by scholars. The test dynamic shear modulus ratio is the closest to the lower envelope curve of the results given by Carraro and Rollins; the damping ratio is also close to the upper envelope curve given by Carraro, and the result of the damping ratio is within the range given by Rollins.The hyperbolic model of Darendeli is used to establish the relationship between coral sand dynamic shear modulus ratio and shear strain [30]. The expression is as formula (1), where γr is the shear strain when G/Gmax=0.5, and the range of the parameter a is 0.88~1.03. For the damping ratio, formula (2) is used to establish the relationship between the damping ratio and the shear strain, where s and b are fitting parameters, the range of s is 5.93~7.89, and the range of b is 0.91~1.26. Model parameters are shown in Table 3.
(1)GGmax=11+γ/γra,(2)λ=λmin+sγ/γrb1+γ/γrb.Table 3
Model parameters.
Test no.Consolidation pressure/kPaG/Gmax−γR2λ−γR2aλminsbCS-01000.88640.99030.855.98421.36510.98632000.94380.99980.767.35610.74130.9851CS-21000.93140.98530.987.84590.98430.98512000.98530.99160.897.51941.10660.9909CS-41000.97520.99060.948.81291.10010.99692000.95710.99120.97.34811.56230.9847CS-81001.06290.98270.876.10531.71610.99622000.96980.99641.027.86890.85390.9885CS-121000.96960.97480.97.94371.08460.99332000.935730.99490.977.24310.92520.9914


## 3.2.2. The Maximum Dynamic Shear Modulus and the Modified Relative Breakage

The maximum dynamic shear modulusGmax can be obtained by fitting the G-γ curve [31], and Gmax can be obtained by using formula (1),
(3)Gmax=1a+bγγ⟶0=1a,wherea and b are the slope and intercept of the 1/G−γ curve.The relationship between Hardin’s relative breakage and the maximum dynamic shear modulus is shown in Table4. Under the same consolidation pressure, Hardin’s relative breakage and the maximum dynamic shear modulus first increase and then decrease. From the overall trend, particle breakage improves the dynamic strength performance of coral sand. This is because the particle breakage will make the fine particles of the broken coral sand fill its coarse particles, so that the coral sand is in a relatively good dense state, improving its dynamic bearing performance. In the process of particle breakage, we refine the role of each single particle group in particle breakage. Therefore, based on Hardin’s relative breakage, propose the concept of modified relative breakage and discuss another method to characterize coral sand particle breakage, which is used to establish the relationship between particle breakage and its maximum dynamic shear modulus. As shown in Figure 6, in the total breakage, we separately consider the role of a single particle group in particle breakage. Divide the total breakage of a single particle group by the initial breakage potential to obtain the relative breakage of each particle group, namely, Brn=Btn/Bp (Btn is the total breakage of the corresponding particle group, such as Bt1 for the 1~2 mm particle group total breakage, Bp is the initial breakage potential). The total breakage Btn of the single particle group under the corresponding particle breakage degree of each consolidation pressure is shown in Table 5. Taking into account that the change in the content of the relatively large particle group in the sand has a greater impact on its maximum dynamic shear modulus [31], the modified relative breakage potential Br′ is defined as Br′=aBt1+bBt2+cBt3+Bt4+Bt5/Bp and assigns a weight coefficient greater than 1 to the total breakage of a single particle group with relatively large particle size.Table 4
Hardin’s relative breakage and the maximum shear modulus.
BrConsolidation pressure/kPaGmax/MPa0100126.4200178.30.082434100137.9200193.70.084986100140.2200196.80.098658100136.4200188.30.100987100131.5200185.8Figure 6
Definition of modified relative breakage.Table 5
The particle breakage degree and total breakage of single particle group.
Particle breakage degreeTotal breakage of single particle groupBtnThe breakage potentialBpThe modified relative breakageBr՛Bt1Bt2Bt3Bt4Bt5CS-21.5791.3990.5710.1730.00245.370.134CS-41.9491.2540.4410.1960.00345.370.143CS-82.1341.6340.5860.1880.00145.370.167CS-122.3981.6390.5020.1850.00445.370.177The modified relative breakageBr′ and the maximum dynamic shear modulus are shown in Table 6. Fitting the modified relative breakage and the maximum dynamic shear modulus under different consolidation pressures, it is found that the modified relative breakage and the maximum dynamic shear modulus in this paper have a good fitting relationship when a=2, b=1.5, and c=1.2, which is in line with the Gauss function relationship. The fitting results are shown in Figures 7 and 8. The Gauss function formula is shown in
(4)Gmax=G0,max+Ae−2×Br′−m/n2,Table 6
The modified relative breakage and the maximum shear modulus.
Br′Consolidation pressure/kPaGmax/MPa0100126.4200178.30.134100137.9200193.70.143100140.2200196.80.167100136.4200188.30.177100131.5200185.8Figure 7
Fitting results (σc=100 kPa).Figure 8
Fitting results (σc=200 kPa).whereG0,max is the initial maximum dynamic shear modulus of the original coral sand sample and A, m, and n are fitting parameters.The changing trend of the fitted maximum dynamic shear modulus under the modified relative breakage is consistent with the change of Hardin’s relative breakage, which also first increases and then decreases. Under a certain degree of breakage, the maximum dynamic shear modulus of coral sand will reach the maximum, and the dynamic strength performance of coral sand will be the best at this time. It can be known from the functional relationship that when the maximum dynamic shear modulus takes the maximum value, the modified relative breakage is between 0.145 and 0.149. At this time, the corresponding relative breakage is the breakage degree near 400 kPa in the one-dimensional consolidation test. The filling effect of the fine particles generated by particles breakage is optimal near this consolidation pressure, so that the dynamic bearing performance of the coral sand reaches the best state. Due to the problem of the number of test samples, what is the specific optimal breakage degree is still a question that needs to be discussed. In the project, the coral sand foundation is often compacted, and the coral sand particles will also be crushed to a certain extent. From this, we can search for the optimal degree of compaction in the project based on the test results and apply a specific compaction work to the coral sand foundation to achieve the optimal breakage of the coral sand foundation, to obtain the best maximum dynamic shear modulus and adopt certain foundation reinforcement measures [32] to improve the seismic performance of the coral sand foundation.


## 3.2.3. The Minimum Damping Ratio and the Modified Relative Breakage

Based on the modified relative breakage proposed above, the minimum damping ratio of coral sand with different modified relative breakage is shown in Figure9. In general, with the increase of the modified relative breakage, the minimum damping ratio of coral sands first increases and then decreases. Within the range of modified relative breakage Br′ from 0 to 0.134, the minimum damping ratios of coral sand under the two confining pressures both show an increasing trend. When the modified relative breakage is greater than 0.134, the changing trend of the minimum damping ratio of the coral sand under the consolidation pressure of the two groups is significantly different. Neither Hardin’s relative breakage nor the modified relative breakage can establish an effective fitting relationship with the minimum damping ratio obtained by the experiment in this paper. The minimum damping ratio of coral sand in the state of particle breakage has greater dispersion, and subsequent studies can still explore a more effective method to establish the relationship between the two.Figure 9
The relationship betweenBr′ and λmin.


## 4. Conclusion

This study discussed the impact of particle breakage on the dynamic characteristics of coral sand. First, five coral sand samples with different breakage degrees were obtained through one-dimensional consolidation tests. On this basis, the resonant column test was performed to obtain the dynamic parameters of coral sand under different breakage degrees. Based on Hardin’s relative breakageBr, propose the modified relative breakage Br′ to characterize the particle breakage of coral sand and fit the relationship between the modified relative breakage and the maximum dynamic shear modulus and damping ratio. It is found that the modified relative breakage and the maximum dynamic shear modulus accord with the Gauss function relationship, while there is no consistency between the modified relative breakage and the minimum damping ratio. The main conclusions are as follows:
(1)
The maximum dynamic shear modulus of coral sand first increases and then decreases with the increase of the breakage degree. The maximum dynamic shear modulus of coral sand reaches the maximum value at the breakage degree of 400 kPa consolidation pressure in one-dimension consolidation test. This is because, under the breakage degree of 400 kPa consolidation pressure, the fine particles produced by the crushing can better fill the coarse particles, resulting in the relative density of the coral sand at a better level, thereby improving the dynamic strength of the coral sand. This means that during construction, proper compaction or overburden pressure can make the coral sand foundation have a better dynamic performance(2)
The modified relative breakageBr′ is proposed to characterize coral sand particle breakage based on Hardin’s relative breakage. The total breakage of a single particle group is divided by the initial breakage potential to obtain the relative breakage of each particle group. Considering that particles with a relatively large particle size have a greater impact on their dynamic strength, the relative breakage of a single particle group with a larger particle size is given a weight coefficient greater than 1, and the modified relative breakage Br′ is obtained after summation. The index and the maximum dynamic shear modulus are highly according the Gauss function relationship, which can reflect the relationship between the degree of coral sand particle breakage and its dynamic shear modulus(3)
In general, with the increase of the modified relative breakage, the minimum damping ratio of coral sands first increases and then decreases. However, the dispersion of the minimum damping ratio is large, and there is no consistent corresponding relationship with the modified relative breakageBr′



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*Source: 5304179-2022-04-01.xml*