Laboratory Investigation on Particle Breakage Characteristics of Calcareous Sands

Many studies have demonstrated the fragility of calcareous sands even under small stresses. This bears an adverse inﬂuence on their engineering properties. A series of laboratory tests were carried out on poor-graded calcareous sands to investigate the crushability mechanism. Einav’s relative breakage and fractal dimension were used as the particle breakage indices. The results show that the particles broke into smaller fragments at the low-stress level by attrition which was caused by friction and slip between particles. In contrast, particles broke in the form of crushing at the relatively higher stresses. The evolution of the particle size was reﬂected by the variation in Einav’s relative breakage and fractal dimension. As testing commenced, the breakage index rapidly increased. When the stress was increased to 400kPa, the rate of increase in the breakage index was retarded. As the stress was further increased beyond 800kPa, the rate of increase in the fractal index became much smaller. This elucidated that the well-graded calcareous sands could resist crushing depending on the range of applied stresses. Based on the test ﬁndings, a new breakage law is proposed.


Introduction
Calcareous sands are rich in calcium carbonate or other insoluble carbonate products. ese sands are mainly distributed in the continental shelf, the coastline of the tropical/ subtropical climate between latitude 30°north and 30°south, the Southern China Sea, and the Red Sea. Calcareous sands differ from those sourced from land in terms of the particle shapes, internal porous structure, and the mechanical properties [1,2]. Owing to such variations, the particles are easy to crumble even under the small stresses [3][4][5][6]. In the field, the engineering structures constructed on the calcareous sand are prone to undergo large differential settlement. erefore, it is of paramount importance to study the breakage mechanism of the calcareous sand particles.
Particle breakage can significantly affect the strength [7][8][9][10][11], permeability, and deformation [12][13][14][15] of coarse-grained soils and can increase the uncertainty on the efficacy of engineering applications. Many studies dealing with particle breakage assessment have been proposed in the past [16,17]. Yu [18] explained the effect of particle fragmentation in calcareous sand by its effect on consolidation and shear properties, friction, and shear-induced volumetric expansion of coral sand. Wang et al. [19] found that irregularity in particle shape is a prerequisite for the generation of particle breakage in calcareous soils. Although the relative crushing rate defined by Hardin [20] is widely used [21][22][23][24][25][26], the concept of relative breakage is yet to be fully explored. Einav [27] assumed that soil particles would undergo complete fragmentation in the shear process and defined the relative breakage in terms of the relative distance of the current grain size distribution from the initial and ultimate distributions. He modified Hardin's crushing index to put forward the relative fragmentation rate suitable for granular soil particles. Some scholars determined the particle size distribution before and after the particle breakage of soil based on the fractal theory [26][27][28]. Hence, it is feasible to quantitatively describe the particle breakage based on the relative breaking rate and the fractal dimension.
Internationally, scholars have adopted a wide range of methodologies/tools to study the crushing mechanism of calcified sand particles, including cyclic triaxial shearing [29], under drained and undrained triaxial compression [30], dynamic image analysis [31], and uniaxial compression [32], among others. In this paper, the uniaxial compression tests were carried out on the poor-graded calcareous sands collected from the Southern China Sea. e particles before and after the testing were screened to obtain the change in particle size distributions. e influence of the consolidation stress on the crushing of the calcareous sands particles was analyzed. e micromorphology of calcareous sands with a single particle size was studied by scanning electron microscope (SEM) tests, and the mode of particle breakage was described. By using the fractal dimension and Einav's breakage rate, the grain breakage under the confined compression testing was quantified. e breakage characteristics of the calcareous sand under different vertical stresses were assessed. e present study has the potential to serve as a useful design guide for the structures founded on calcareous sands.

Test Materials and Method
e calcareous sand used in this study was collected from the reef in Nansha Island, China. e calcareous sand consisted of coral broken branches and biological debris which had the calcium carbonate content as high as 90%. Figure 1 shows the images of characteristic particles of calcareous sands. Before the test, the calcareous sand was washed and airdried. e particles larger than 4 mm in size were removed to minimize the effects of boundary conditions on test results. e particle size distribution curve of calcareous sand is shown in Figure 2, which represents poor gradation. e relative density of the sand sample was obtained by the pycnometer test. e physical characteristics are summarized in Table 1.
e compression test was carried out by using consolidation apparatus as shown in Figure 3. e samples were prepared in-ring knife samples of 61.8 mm × 20 mm. e rate of deformation was controlled as 0.01 mm per hour as per the industry standard (GB/T50123, 1999). e calcareous sand was subjected to compression tests at seven different levels of vertical stress (50, 100, 200, 400, 800, 1600, and 3200 kPa). e sample was sieved after the completion of compression that lasted for 36 h. e screening of calcareous sand particles below 0.25 mm in size was ignored.
Scanning electron microscope (SEM) tests were conducted to observe the microscopic images of the original calcareous sand with different particle sizes (4, 3, 2, 1, 0.5, and 0.25 mm). In this experiment, the sputtering speed is 10 mm/min and the sputtering time is 60 s. SEM diagrams of samples are shown in Figures 4 and 5.

Test Results
e particle size distribution of the calcareous sand samples before and after tests was obtained. Figure 6 represents the comparison between the particle size distribution under different stress levels.
It was found that the percentage of coarser particles decreased (while increasing the amount of finer fraction) as the stress increased. is indicated the increased amount of breakage in coarser particles as the stress level increased. Under each incremental stress level, the fraction of particles smaller than 1 mm increased mainly attributing to the breakage of bigger particles. For the stress below 400 kPa, the percentage mass of medium-sized particles (size 1-2 mm) appeared to vary, while as the stress exceeded 800 kPa, the amount of medium-size particles increased. e large particles were crushed after being compressed and split into several medium-to smaller-sized particles. e large-sized particles undergo more breakage compared with smaller particles. is is due to their primary role in stress transfer and development of large contact stresses resulting from the relatively smaller number of interparticulate contact areas available for the force-transfer. Under the low-stress levels below 400 kPa, the finer grains occupy the empty void spaces around coarser particles during particle reorientation and rearrangement thereby inducing grain breakage. Besides, the presence of a large number of asperities evident in coarse angular particles makes them prone to breakage due to intensive stress concentrations developed around the asperities. e dependence of particle shape on the particle size has been discussed earlier [33,34]. According to the data plotted in Figure 6, the coarser particles above 2 mm were crushed even at the low-stress level of 50 kPa. With the increase in stress, the larger amount of finer fraction implied the substantial particle crushing. is is consistent with the results of existing studies [35].
Figures 4 and 5 are the SEM diagrams of calcareous sand particles with single particle size. It was found that the surfaces of calcareous sands were irregular and the internal structure was porous. is was evident from the compression test data in which the increased stresses led to a larger finer fraction. e particle breakage at low-stress level was in the form of attrition or crushing caused by friction and slip between particles (refer to the location indicated in Figure 4), as well as the particle splitting as a result of specimen compression occurred at the high-stress level.

Particle Breakage Index
In order to evaluate the degree of particle breakage of calcareous sands after compression, two parameters, viz., particle fractal dimension and Einav's relative breakage rate, were selected. e particle fractal dimension increased with the increased fine fraction. For the calcareous sand subjected to elevated stress, the particles with large size reduced due to evolution of breakage. us, the change in the fractal dimension of a particle could indicate the measure of its breakage.
e calcareous sands used in this paper are granular material, and the relative breakage rate proposed by 2 Advances in Civil Engineering Einav [27] is suitable for describing the breaking degree of calcareous sands.

Fractal Dimension.
A large number of studies have shown that geotechnical media represents obvious fractal features [36][37][38][39][40][41][42][43]. e use of the weight distribution of the particle sizes is therefore appropriate. Based on the Sierpinski carpet (Sierpinski carpet) and the Menger sponge (Menger sponge) fractal model, Tyler and Wheatcraft [44] proposed the relationship between the soil smaller than certain particle size and the fractal dimension of particles: where R is the particle size, R max is the maximum particle size, M is the mass of the particle size range of 0∼R, and M T is the total mass of soil. By taking the logarithm of both sides of Equation (1), the fractal model could be rewritten as By plotting ln(M(r < R)/M T ) against ln(R/R max ), (3 − D) can be evaluated from the gradient k of the graph and then the fractal dimension can be determined as D � 3 − k.
e fitting results are shown in Table 2.
To obtain the relationship between the particle fractal dimension of the sample under different stresses, it is assumed that the particle fractal dimension before crushing is D b , the particle fractal dimension after crushing is D a , and the stress is P. e linear regression analysis is then performed on D a /D b and P/P a where P a is the atmospheric pressure, as shown in Figure 7. It is shown that a significant linear relationship exists between D a /D b and P/P a (R 2 � 0.98). e ratio D a /D b can be expressed as follows: Below 0.25mm 0.25~0.5mm 0.5~1 mm 1~2mm 2~3mm 3~4mm

Einav's Relative Breakage.
e definition of Einav's relative breakage [27] is shown in Figure 8. Based on the relative breakage proposed by Hardin, Einav defined the area of initial distribution F 0 (d) to ultimate distribution F u (d) as B p . e area of initial distribution F 0 (d) to current distribution F (d) is expressed as B t . Einav's relative breakage can be determined by where dM is the maximum particle size and dm is the minimum particle size.

Advances in Civil Engineering
Combined with Equation (4), Einav's relative breakages under different stress were calculated. Table 3 shows the values of Einav's relative breakage corresponding to each stress level. Figure 9 shows the relationships between the fractal dimension, Einav's relative breakage, and the vertical stress, respectively. It was seen that the breakage indices, fractal dimension, and Einav's relative breakage showed a rapid increase at an initial stage of compression (the slope of this section is k 1 ). As the stress continued to increase to 400 kPa, the rate of increase in breakage reduced (the slope of this section is k 2 ). When the stress exceeded 800 kPa, the increasing rate of breakage declined gradually (the slope of this     Advances in Civil Engineering 5 section is k 3 ). It was evident that the slope of each following stage became steeper (i.e., k 1 >k 2 >k 3 ), which indicated that the increase in breakage was the most obvious during the initial stage of loading. However, the change in breakage index under subsequent incremental stress level was small. Under the small incremental stress, only a small portion of coarser particles underwent crushing (caused by friction and slip between particles), while the amount of finer particles remained relatively unchanged. It also indicated that within a certain stress range, the well-graded calcareous sand could resist crushing. But if the stress continued to increase, a large number of particles underwent breakage. However, as evident from Figure 9, the stress had a great influence on the initial state of samples (sudden change in value of breakage index), which also showed that calcareous sand at each stress level is accompanied by particle breakage, even at the lowstress level of 50 kPa. is could be verified by the variation in amount of large particles before and after compression testing of calcareous sand as evident from Figure 6. According to the breakage data listed in Tables 2 and 3, the relationship between Einav's relative breakage and fractal dimension is plotted in Figure 10. It could be seen that all test points fell in a much narrower band, indicating a unique relationship between Einav's relative breakage and the fractal dimension exist although the stress states of different test points are different. is relationship could be expressed using the following power function: where a and b are constant. For the calcareous sand, a is 2.119 and b is 7.901 × 10 −4 . It is evident that D and Br have some intrinsic relation; therefore, it is reasonable to take fractal dimension D as a quantitative index of particle breakage [45].
e relation between fractal dimension before crushing and Einav's relative breakage can be captured by substituting Equation (3) into (5): Einav's relative breakage of calcareous sand under different stresses can be estimated by Equation (6) when the fractal dimension before crushing and fitting parameters a and b is known. Because the fragility of calcareous sand particles posed challenges in the construction of airport runway and other projects on Southern China Sea islands and reefs, the present study is beneficial to the construction of islands and reefs in the Southern China Sea and the value of local materials.

Conclusions
Particle breakage of calcareous sands occurs under small stresses commonly encountered in the field. A series of laboratory tests were performed to determine the breakage law of calcareous sands under such small stresses. e fractal fitting of the laboratory test data and the determination of Einav's relative breakage were carried out. e main findings of the study are summarized as follows: (1) At the low-stress level, two modes of particle breakage, viz, attrition and crushing (caused by the friction and slippage between the particles) of particles occurred. In contrast, splitting in larger particles occurred at a relatively higher stress level. (2) e stress had a great influence on the initial state of calcareous sand (increased breakage at initial loading stage), which showed that calcareous sand at each stress level is accompanied by particle breakage, even at the low-stress level of 50 kPa. (3) e variation in the breakage indices, Einav's relative breakage, and fractal dimension both implied the increase in finer fraction. At the outset of testing, substantial particle breakage was evident which then continued to occur as the stress increased to 400 kPa. e amount of particle breakage was then gradually reduced as the stress increased beyond 800 kPa. e rate of increase became smaller, which showed that the well-graded calcareous sand could resist breakage.

Data Availability
e data used to support the findings of this study are included within the supplementary information files.

Conflicts of Interest
e authors declare that they have no conflicts of interest.