Investigation of particle size effect on particle breakage of carbonate sand under one-dimensional compression

Previous studies on the particle-size effect have indicated that larger particles tend to exhibit lower strength under equivalent loading conditions; however, the underlying mechanism remains unclear. This study investigated the influence of particle size on the macrocompression behavior of carbonate sand by performing one-dimensional compression tests on various uniformly graded samples. The variations in particle size and shape resulting from particle breakage during compression were quantitatively analyzed. In addition, X-ray micro-tomography (μCT) is employed to obtain microstructural insights into the intra-particle porosity of carbonate sand particles, and its role in the particle size effect was further discussed.


Introduction
In the numerical modeling of granular materials, idealized particles of different sizes are typically assumed to have the same single-particle strength.However, particles are heterogeneous in nature with physical differences in their shapes and internal structures.The tensile strength of particles decreases with increasing particle size, as observed in many studies on granular materials [1,2].The tensile strength is known to follow a Weibull distribution, and the characteristic tensile stress corresponding to the survival of 37% of the particles is a function of the particle size [3].The size dependency also holds for the macromechanical behavior.This is because the grain tensile strength is related to the yield stress and breakage index [1,4].The effect of particle size on the breakage behavior of granular assemblies involves a complex process that depends on the interparticle coordination number and contact conditions [5][6][7][8] as well as material flaws [2].Internal flaws are an inherent characteristic of granular materials, leading to stress concentration in the solid particle skeleton and a decrease in the particle failure strength [9].These structural weaknesses may trigger various forms of tensile, shear, and bending cracks that determine fracture patterns [10].Although the particle-size effect is commonly attributed to a higher probability of flaws existing within larger particles [11], there remains a lack of quantitative analysis regarding the microstructural properties of the particles.
This study presents a detailed investigation of the macroscale compression behavior with respect to the particle size effect.The particle breakage behavior associated with changes in particle size and shape throughout the compression process was tracked.Quantitative characterization of the internal structure, specifically the intraparticle pores, was achieved using image processing technology, which provided valuable insights into the intricate micromechanical behavior of particle crushing.

Test material and procedure
A steady state of particle breakage is typically characterized by a stable distribution of particle size, which requires a very high pressure to be applied to the sample.In this study, a novel high-stress onedimensional compression apparatus was developed and used to conduct particle crushing tests.It comprised three primary components: a loading system, piston, and sample container.The loading system was equipped with an MTS E45.305 device, which, through the piston, could apply a pressure up to 150 MPa on a sample of 50 mm in diameter, and this apparatus satisfies the stress requirements for breakage of particles.Both the piston and sample container were constructed from high-strength steel to minimize the deformation of the components under high-pressure conditions.The test material was carbonate sand retrieved from the South China Sea.The physical properties of the tested carbonate sands are summarized in table 1.Three uniformly graded samples with particle sizes of 0.5, 2.5, and 5 mm were tested at a stress level of 64 MPa (figure 1), and the loading rate of all samples was set to 1 mm/min.
To obtain microstructural information on particle breakage, post-test samples were subjected to sieve analysis to identify changes in particle sizes.In addition, a QICPIC instrument was employed to track and analyze the evolution of the particle shape during compression.This instrument can help capture and store morphological data from vertically falling particle flows and provide precise shape measurements through the associated software.

Particle shape and measurement
Performing objective geometric measurements of particle shapes can provide useful insights.Three particle-shape parameters were introduced for each particle (AR), convexity (C), and sphericity (S).
The aspect ratio is defined as the ratio of the maximum and minimum ferret diameters.The convexity was calculated as the particle area divided by the area filled by any concave surface of the particle.The sphericity is defined as the ratio of the circumference of a circle with the same area as the projected area of the particle to its actual circumference.All these shape measures exhibited a trend wherein particles were rotated to a greater extent with increasing shape value.Yang and Luo [12] proposed a new shape descriptor, the overall regularity (OR), to characterize the regularity of the particle shape.It is expressed as

Compression behavior at the macro scale
The compression behavior of the carbonate sand is shown in figure 2(a).A clear particle-size effect was observed, with a significant difference in the compression curves.Samples with larger particle sizes present a higher volume deformation during compression, as evidenced by their final porosity ratios at 64 MPa for 5 mm samples, which were 0.14 and 0.26 greater than those of the 2.5 and 0.5 mm samples, respectively.The yield stress (σy), referred to as the onset of marked particle crushing, is plotted in figure 2(b).A good correlation was observed between the mean particle size (d50) and yield stress.The yield stress decreased with increasing d50, in accordance with the phenomenon observed in the compression curves, where larger samples experienced a faster and more pronounced descent.

Compression behavior at the micro scale
To present a more distinct illustration of the evolution of both particle shape and size, the median particle size (d50) and normalized median overall regularity (OR50,n = OR50/OR50,initial, where OR50,ini represents the initial median OR value) are plotted in figure 3.As expected, d50 exhibited a rapid decrease, followed by a slower decline during continuous particle crushing.This phenomenon was more pronounced in samples with larger particle sizes, whereas the 0.5 mm sample exhibited minimal changes in the median size.The median values of ORn exhibited a monotonous decrease with increasing vertical stress, indicating that the geometry of the particles tended to be more irregular during particle breakage.The particle size effect was also observed in the evolution of particle shape, with the 5 mm sample experiencing the most significant reduction compared to the smaller-sized samples.Furthermore, the 0.5 and 2.5 mm samples reached a steady state at relatively low stress levels, whereas the 5 mm sample required higher stress levels to achieve a steady state.

Porosity of individual particle
To obtain a micro view of particle structure, randomly selected particles of carbonate sand were tested using the X-ray micro-tomography (μCT) technology.Visual data on the intraparticle pores were obtained through a series of image processing operations, including median filtering, threshold segmentation, rendering, pore filling, and pore extraction, as shown in figure 4. Porosity (φ) is an important parameter for characterizing pore volumes within particles.It is defined as the ratio of the pore volume to the sum of the solid and pore volumes.Pore network Modeling (PNM) analysis was conducted on particles with two different porosities, as depicted in figure 5. Figure 5(a) and 5 (b) show particles with porosities of 0.220 and 0.335, respectively.Particles with higher porosity exhibited a more complex PNM and a larger overall equivalent sphere volume.The maximum porosity of the tested carbonate sand particles was approximately 0.5.The relationship between porosity and particle size is shown in figure 6.Although the data points showed certain degree of variability, the intraparticle porosity demonstrated a good correlation with particle size, with a Pearson correlation coefficient of 0.575.This suggests that the porosity increases with increasing particle size, which decreases the strength of larger particles under compression.

Discussion
Figure 7(a) shows six idealized distribution patterns for three internal pores in a single particle under compression, and figure 7(b) shows three special cases of the uniform distribution of two internal pores in a particle.As evident, the main force propagation path, which was oriented in the vertical direction and located in the highly stressed area schematically described in figures 7(a) and 7(b), was more likely to completely pass through the pore entity in the case of a uniform distribution of three pores.However, it may not be able to pass through or may partially pass through the pore entity in the case of a uniform distribution of two internal pores.In other words, the probability of the main force propagation path passing through the pores in the case of a uniform distribution of the three pores is relatively high.Thus, the stress distribution and particle breakage behavior are more influenced by the internal pore structures in a particle with more internal pores.As revealed in the previous section, relatively large carbonate sand particles tended to have more internal pores.As illustrated in figure 8, the two idealized particle assemblies share the same packing pattern but are composed of particles of different sizes.It is evident that the force propagation pathways, which are denoted by the dotted lines, were more prone to penetrate through the pore entities for the assembly composed of large particles, as they contained more internal pores.Thus, particle breakage in the assembly of large particles is more affected by internal pores and is more likely to occur.

Conclusion
This study investigated the particle size effect on the compression behavior of carbonate sand using μCT technology to unveil its underlying mechanisms.The following key conclusions are drawn based on the test results and discussion: (1) The internal porosity of the particles was related to the size-dependent particle breakage behavior during compression.Larger particles exhibited greater susceptibility to breakage, leading to a reduction in particle size and an increase in compressibility.The maximum porosity of the carbonate sand particles was approximately 0.5.
(2) The particle sizes correlated well with the internal porosities, and the Pearson correlation coefficient for this correlation was determined to be 0.575.The pore network model offered an effective three-dimensional representation of intraparticle porosity.Models with lower porosities tended to be simpler, whereas those with higher porosities were generally more complex.The probability of the force propagation paths passing through the pore entities was higher for a particle with more internal pores, and the intraparticle porosity was positively correlated with the particle size.This accounted for the fundamental cause of the size dependency of the particle breakage behavior under compression.

Figure 2 .
Particle size effect on (a) compression curves and (b) yield stresses

Figure 4 .
Figure 4. Image processing procedure of three-dimensional images

Figure 6 .
Figure 6.Relationship between intra-particle porosity and particle size

Figure 7 .
Conceptual diagrams of single-particle compression under different settings of uniformly distributed pores: (a) three pores in a particle, (b) two pores in a particle.

Figure 8 .
Figure 8. Conceptual diagram of force transmission pathways in a particle assembly composed of particles with different sizes.

Table 1 .
Physical properties of tested sands Figure 1.Photo of tested carbonate sand