Particle breakage evolution of coral sand using triaxial compression tests

Gang Wang, Zhaonan Wang, Qinguo Ye, Jingjing Zha

a Key Laboratory of New Technology for Construction of Cities in Mountain Area, Chongqing University, Chongqing, 400045, China

b School of Civil Engineering, Chongqing University, Chongqing, 400045, China

c National Joint Engineering Research Center for Geohazards Prevention in the Reservoir Areas, Chongqing, 400045, China

Keywords:Particle breakage Coral sand Triaxial compression Relative breakage Breakage model

ABSTRACT Particle breakage continuously changes the grading of granular materials and has a significant effect on their mechanical behaviors.Revealing the evolution pattern of particle breakage is valuable for development and validation of constitutive models for crushable materials.A series of parallel triaxial compression tests along the same loading paths but stopped at different axial strains were conducted on two coral sands with different particle sizes under drained and undrained conditions.The tested specimens were carefully sieved to investigate the intermediate accumulation of particle breakage during the loading process.The test results showed that under both drained and undrained conditions, particle breakage increases continuously with increasing axial strain but exhibits different accumulating patterns, and higher confining pressures lead to greater particle breakage.Based on the test results, the correlations between particle breakage and the stress state as well as the input energy were examined.The results demonstrated that either the stress state or input energy alone is inadequate for describing the intermediate process of particle breakage evolution.Then, based on experimental observation, a path-dependent model was proposed for particle breakage evolution, which was formulated in an incremental form and reasonably considers the effects of the past breakage history and current stress state on the breakage rate.The path-dependent model successfully reproduced the development of particle breakage during undrained triaxial compression using the parameters calibrated from the drained tests,preliminarily demonstrating its effectiveness for different stress paths.

1.Introduction

Particle breakage (or grain crushing) continuously changes the grading during the loading process and has a significant effect on the hydraulic and mechanical behaviors of granular materials.Therefore, particle breakage is of great concern in situations in which notable breakage might occur,such as rockfills in a high dam(Marsal,1967;Lade et al.,1996;Xiao et al.,2014),rail ballasts under heavy traffic loading (Indraratna et al., 1998; Suiker et al., 2005;Anderson and Fair, 2008; Sun et al., 2016), sliding zones of landslides(Sassa et al.,2004),and carbonate sands in coastal and ocean engineering (Coop,1990; Al-Douri and Poulos,1992).

The most direct consequence of particle breakage is the increase in soil compressibility (Lee and Farhoomand,1967; Hagerty et al.,1993; Yamamuro et al., 1996; Nakata et al., 2001; Wang et al.,2011) and the decline for the locus of critical state line (CSL)(Coop, 1990; Al-Douri and Poulos, 1992; Brandes, 2011;Hassanlourad et al., 2014; Xiao et al., 2014; Ciantia et al., 2019),which makes the soil appear looser.More comprehensive investigations (e.g.Indraratna and Salim, 2002; Luzzani and Coop,2002; Muir Wood and Maeda, 2007; Bolton et al., 2008;Kikumoto et al., 2010; Bandini and Coop, 2011; Miao and Airey,2013; Xiao et al., 2016; Yu, 2017a; Ciantia et al., 2019) revealed the mechanism by which the soil exhibits a looser response once particle breakage occurs.Briefly, it is because particle breakage broadens the grading of the soil,which leads to a more efficient and compact arrangement of particles,and as a result,the critical state void ratio decreases.

Experimental work is the basis for investigating the characteristics and influencing factors of particle breakage.Extensive laboratory tests have been conducted on various granular materials such as quartz sands (e.g.Yamamuro et al., 1996; Nakata et al.,2001; Brandes, 2011; Hassanlourad et al., 2014; Yu, 2017b),carbonate sands(e.g.Coop,1990;Qadimi and Coop,2007;Donohue et al., 2009; Miao and Airey, 2013; Shahnazari and Rezvani, 2013;Yu, 2018a), and coarse aggregates (e.g.Marsal, 1967; Indraratna et al., 1998; Suiker et al., 2005; Anderson and Fair, 2008; Xiao et al., 2014; Indraratna et al., 2016; Sun et al., 2016).These tests include one-dimensional compression tests (e.g.Hagerty et al.,1993; Yamamuro et al., 1996; Nakata et al., 2001; Altuhafi and Coop, 2011; Miao and Airey, 2013; Lv et al., 2019; Xiao et al.,2019), direct shear tests (e.g.Al-Douri and Poulos, 1992; Wang et al., 2017), monotonic and cyclic simple shear tests (e.g.Mao and Fahey, 2003; Porcino et al., 2008; Brandes, 2011; Shahnazari et al., 2016; Hubler et al., 2017), ring shear tests (e.g.Luzzani and Coop, 2002; Coop et al., 2004; Wei et al., 2018), triaxial compression tests (e.g.Lee and Farhoomand, 1967; Marsal, 1967; Coop,1990; Lade and Yamamuro, 1996; Indraratna et al., 2015; Ovalle et al., 2015; Yu, 2017c, 2018b), and cyclic triaxial compression tests(e.g.Hyodo et al.,2002;Donohue et al.,2009;Indraratna et al.,2016; Salem et al., 2013).Despite the differences in testing materials and methods, a consensus that particle breakage depends on particle characteristics (mineralogical composition, size and shape), packing properties (void ratio, grading, coordination), and loading conditions (stress magnitude, loading path) has been reached(Coop,1990;Hagerty et al.,1993;Lade et al.,1996;Nakata et al.,1999;Leleu and Valdes,2007;Shahnazari and Rezvani,2013).

It is worth noting that such a testing procedure was often adopted for simplicity to investigate the effect of particle breakage on mechanical behaviors of granular materials (e.g.Lade et al.,1996; Xiao et al., 2014; Wang et al., 2018).First, a specimen is loaded to observe the stress-strain behavior along the loading path, and then the specimen is taken out and sieved to examine particle breakage.This testing procedure can only obtain the final amount of particle breakage and ignores its intermediate accumulation.Consequently,only the final amount of particle breakage is available for interpreting the stress-strain behavior in the entire loading path.Although recent studies indicated the success of X-ray tomography (Cil and Alshibli, 2014; Alikarami et al., 2015; Karatza et al., 2018, 2019) in revealing the evolution of deformation and breakage for granular materials, it is still limited by testing site or expense,leading to this technique difficult to be a common test tool in laboratory.

For the conventional test method that lacks intermediate observations of particle breakage, some mechanical indices determined at an early phase in the stress-strain curve (e.g.initial modulus, dilatancy ratio, and peak strength) have to be related to the final amount of particle breakage.For example, for sand in triaxial compression tests,the peak strength was often mobilized at an axial strain less than 5%,but it was correlated to the final amount of particle breakage at an axial strain often greater than 20%(Xiao et al., 2014; Wang et al., 2018).This is not rational, and it explains why the correlation between peak strength parameters and final particle breakage was quite scattered if data from different loading paths were included (Wang et al., 2018).Therefore, more efforts should be put on the evolution of particle breakage along specific loading paths.Regretfully, only a few experimental works(Indraratna and Salim 2002;Coop et al.,2004;Wei et al.,2018;Yu,2017c,2018a)involved the intermediate particle breakage during a specific loading process.For the ring shear or triaxial compression test, the amount of particle breakage increases with the shear strain, but at a decreasing rate until approaching a constant value(Indraratna and Salim,2002;Coop et al.,2004;Wei et al.,2018),and the particle breakage continuously develops even after the deviatoric stress reaches its peak value(Yu,2017c,2018a).For the impact load test of carbonate sand, the volumetric strain and particle breakage progressively increase with the increase in blow number,and the void ratio converges to be a constant value as the particle grading tends to the fractal state(Xiao et al.,2017).However,there are also some deficiencies need to be improved, such as without giving a specific mathematical description for evolution, and the testing pressure being too high to suitable for the practical engineering project.

This study aims at providing a valuable database to improve the understanding of the intermediate accumulation of particle breakage during triaxial compression loading paths.Two types of coral sands of different particle sizes were tested under drained and undrained triaxial compressions.To observe the intermediate particle breakage for each loading path and initial state,a group of specimens were prepared.The specimens in one group were loaded to different axial strains and then sieved to obtain the grading corresponding to their final axial strains.Based on the experimental results,the characteristics of particle breakage evolution,as well as its descriptions, were discussed.

2.Testing material and methods

Coral sand is often employed to study particle breakage,as it can produce notable particle breakage under the normal stress levels of common laboratory apparatuses.Past studies (e.g.Nakata et al.,2001; Coop et al., 2004; Altuhafi and Coop, 2011) have shown that particle breakage for uniformly graded sand is greater than that for well-graded sand.To enlarge the amount of particle breakage to facilitate quantity comparisons,two types of uniformly graded coral sands of different particle sizes are adopted in this study.As shown in Table 1,SAND-I has a particle size of 0.6-0.8 mm and SAND-II has a particle size of 0.5-0.6 mm.Both sands are sieved from the deposits on a coral reef in Xisha Island,South China Sea.The grain specific gravity Gs, maximum and minimum void ratios emaxand emin, and the coefficients of uniformity and curvature Cuand Ccof the two coral sands are listed in Table 1.The testing of the soil properties follows the Chinese Specification of Soil Tests SL237-1999 (1999).

The triaxial compression tests were performed using a fully automated triaxial testing system (GDS triaxial testing system),which has resolutions of pressure measurement of 0.1 kPa and volume measurement of 0.1 mm3.The triaxial specimens have a height of 80 mm and a diameter of 39.1 mm.The specimens were prepared by the air pluvial deposition of dry sand,and saturated by circulating CO2gas and applying a back pressure of 100 kPa.Before the consolidation phase,the Skempton’s B value of the sample was measured, and the saturation of sample is completed until B is higher than 0.98.Then,the saturated specimens were loaded under consolidation drained (CD) and undrained (CU) triaxial compression.

As no cheap and reliable technique is available to monitor the grading change continuously during a triaxial loading process, the sieving method is still employed herein to obtain the intermediate particle size distributions (PSDs) at different axial strains.Because the specimen has to be taken out for sieving and the loading process has to be terminated,a group of specimens at the same initial states has to be prepared and loaded along the same path but stopped at different axial strains.As the specimens in the same group are treated as identical specimens that are terminated at different stages of a loading process, the specimens in the samegroup are called parallel specimens herein,and the tests conducted on the parallel specimens are called a group of parallel tests.A group of parallel tests can obtain the PSD curves at different axial strains of a loading path, and the resolution of the intermediate grading changing process depends on the number of the parallel specimens.Fig.1 shows the grading and morphology of the coral sands.It can be seen that the particles are in angular and irregular shapes and have surface cavities and intra-particle pores.

Table 1Basic physical parameters of the two coral sands.

3.Breakage characteristics

In this study,it is necessary to examine the susceptibility of the coral particles to breakage and the corresponding breakage characteristics.Isotropic and triaxial compression tests were conducted on the specimens of SAND-II with a relative density of Dr＝65%to examine the changes in the final PSD.

Fig.2 shows thePSD curves after isotropic compression to the mean pressures of p′＝ 400 kPa,800 kPa,1600 kPa and 3200 kPa.It can be seen that very few fine particles were generated even under a relatively high pressure of 3200 kPa, indicating that the coral particles have a high resistance to breakage under isotropic compression.In addition, the PSD curves of 400 kPa, 800 kPa and 1600 kPa almost coincide,demonstrating that this change in PSD is mainly induced by the fragmentation and uncertainties in the sieving process.Based on these two experimental facts, we can confidently assume that isotropic compression-induced breakage with a pressure less than 1600 kPa is negligible for the coral sand employed.

Fig.3 shows the PSD curves after a drained triaxial compression test(CD test)and an undrained triaxial compression test(CU test)with a total confining pressure of σ3＝ 400 kPa compared with the PSD curve of the isotropic compression test under a mean pressure of 3200 kPa.Despite the fact that the pressure level(400 kPa)in the triaxial compression is far less than that(3200 kPa)in the isotropic compression, the content of fine particles after the triaxial compression is much larger than that after the isotropic compression, demonstrating that shear is more efficient than compression in producing particle breakage.Therefore, even in triaxial compression tests with moderate confining pressure levels,particle breakage should also be carefully considered.

Fig.3 shows that particle size and PSD do not change significantly except for the increase in the content of fine particles.The changing characteristics of the PSD curves are related to the breakage patterns of particles.Fig.4 illustrates three typical particle breakage patterns: splitting, corner breakoff and surface abrasion(Daouadji and Hicher,2009).For splitting,the whole particle breaks into smaller ones of similar sizes.For corner breakoff,the rupture of sharp corners breaks the particle into a slightly smaller particle and several much smaller ones.For surface abrasion, the abrasion of surface asperities produces very fine particles, while the original particle shows almost no change in size.For a specific situation,thebreakage pattern is related to the stress level in addition to the mineral composition and particle size(Daouadji and Hicher,2009;Xiao et al., 2019).Fig.5 displays the images of the coral particles before and after the CD test under confining pressure of 400 kPa.The images clearly show that a considerable number of fine particles were produced after the test,and almost no mid-sized particles were produced, indicating that the primary breakage patterns of the coral particles were corner breakoff and surface asperity abrasion during triaxial compression at moderate confining pressure levels.

Fig.2.PSD curves after isotropic compression for SAND-I.

Fig.3.PSD curves of coral sand under CD and CU conditions.

Fig.1.Images of the coral particles: (a) The grading and particle image of the coral sand; and (b) Scanning electron microscope (SEM) image to show surface cavities and intraparticle pores.

Fig.4.Illustration of typical breakage types of particles.

4.Test results

Considering the amount of experimental work involved, only one relative density is tested for each sand: Dr＝75% for SAND-I,and Dr＝65% for SAND-II.Four levels of total confining pressure σ3including 50 kPa,100 kPa, 200 kPa and 400 kPa were selected.For each confining pressure, both drained and undrained conditions are considered.Then, there are eight groups of parallel tests for each sand as shown in Tables 2 and 3.Each group corresponds to one type of drainage condition,one level of total confining pressure σ3and several levels of final axial strain ε1.As it is impossible to make the parallel specimens identical in practical experimental operations,the void ratios of all the specimens are listed in Tables 2 and 3 for error estimation.

4.1.Examining similarity of parallel specimens

Since the parallel specimens in a group are assumed to have the same behavior,it is important to check their similarities in stress-strain behavior.Figs.6 and 7 compare the measured stress-strain curves of some parallel specimens for SAND-I and SAND-II,respectively.In these figures, the mean pressure p′and deviatoric stress q are defined as follows:

Table 2Summary of the parallel tests conducted on SAND-I.

Fig.5.SEM images of the sand particles before and after the CD test under σ3 ＝ 400 kPa.

Table 3Summary of the parallel tests conducted on SAND-II.

As can be seen from Figs.6a,b and 7a,b for the CD tests, the stress-strain curves and volumetric strain curves in each group almost coincide, demonstrating the good repeatability of the parallel specimens.Figs.6c,d and 7c,d show that the stress-strain curves in the CU tests agree with each other, and apart from the 10% axial strain of I-CU200, other curves have the better repeatability.The reason for this discrepancy is that in CU tests, a small difference in volumetric strain εvwould cause a relatively large difference in the effective confining stress, which then results in a relatively large deviation in the axial stress-strain curves.Considering this reason, the similarities in the stress-strain response of the parallel specimens in the CU tests are generally acceptable.

4.2.Evolution of PSD curves with axial strain

Figs.8 and 9 show the PSD curves of all testing groups for SANDI and SAND-II, respectively, and each plot in these figures corresponds to a testing group in Tables 2 and 3.To make the differences in the PSD curves at different axial strains clearer, the doublelogarithmic coordinate was used.Moreover, the doublelogarithmic coordinate plot has another advantage: if the grading after excessive breakage tends to be fractal(McDowell et al.,1996;Einav, 2007), the PSD curves would rotate upward around the top point and become a straight line (McDowell and Bolton, 1998;Konrad and Salami, 2018).

It can be seen from Figs.8 and 9 that as the axial strain increases,the content of fine particles obviously increases while the grading curve only rotates upward slightly.The PSD curves can then be divided into two segments: one for newly generated fine particles and the other for abraded original particles,which agrees with the main breakage patterns of the particles.

It is worth nothing that the PSDs in Fig.9 have a gap-graded distribution, but it is not the case in Fig.8.This is possibly due to the fact that SAND-I has a larger particle size (mean diameter d50＝ 0.7 mm)than that of SAND-II(mean diameter d50＝ 0.55 mm),resulting in the lower coordination number within SAND-I.Once with the low contact number, the whole particle will easily tend to produce corner breakoff even splitting,as a result of the compressive forces imposed at their contacts with their neighbors(Karatza et al.,2019).

4.3.Development of relative breakage with axial strain

The relative breakage(Hardin,1985)was employed to define the amount of particle breakage.Relative breakage, denoted as Br, is defined as the quotient of total breakage Btdivided by breakage potential Bp, i.e.

where Btis the area between the initial and current PSD curves,and Bpis the area between the initial PSD curve and the vertical line at the particle size 0.075 mm,as illustrated in Fig.10.According to this definition,the values of relative breakage for the PSD curves shown in Figs.8 and 9 were calculated and are listed in Tables 2 and 3.

Figs.11a and 12a show the development of the relative breakage with axial strain during drained triaxial compression processes.Note that the effective confining pressureremains constant during the compression process.It can be seen that the relative breakage increases with increasing axial strain at a gradually decreasing rate, and even after the peak deviatoric stress was reached,the relative breakage increases continuously.A comparison of the data points at different confining pressures shows that higher confining pressures lead to greater breakage.A similar phenomenon was also observed in the large-scale CD tests on rail ballasts(Indraratna and Salim, 2002) and the CD tests on quartz sand (Yu,2017c)and coral sand(Yu,2018a).

Figs.11b and 12b illustrate the development of the relative breakage with axial strain during undrained triaxial compression processes.These figures show that the relative breakage increases with increasing axial strain but not at a monotonically decreasing rate as that observed in the drained conditions.Figs.11b and 12b also show that a higher total confining pressure leads to greater breakage; however, the difference among the breakage amounts under different total confining pressures is much less than that observed in the drained conditions.This difference can be attributed to the different evolutions of effective stress paths in drained and undrained conditions.Note that although the total confining pressure σ3remains unchanged in the CU tests, the effective confining pressurevaries due to the generation of excess pore water pressure, which makes the breakage evolution pattern in undrained conditions much more complicated than that in drained conditions.

Fig.6.Comparison of the measured behaviors of the parallel specimens in the groups for SAND-I:(a)Deviatoric stress curves and(b)Volumetric strain curves in the CD tests;and(c) Deviatoric stress curves and (d) Stress-strain path curves in the CU tests.

Fig.7.Comparison of the measured behaviors of the parallel specimens in the groups for SAND-II:(a)Deviatoric stress curves and(b)Volumetric strain curves in the CD tests;and(c) Deviatoric stress curves and (d) Stress-strain path curves in the CU tests.

Fig.8.PSD curves at different axial strains for the testing groups of SAND-I: (a-c) Drained compression under different confining pressures; and (d-g) Undrained compression under different confining pressures.

Fig.9.PSD curves at different axial strains for the testing groups of SAND-II: (a-d) Drained compression under different confining pressures; and (e-h) Undrained compression under different confining pressures.

Fig.10.Definition of relative breakage and its value at fractal grading.d is the particle size, dM is the maximum particle size, α is the fractal dimension, F（d） represents the mass content finer than d,Br,f is the value of relative breakage at fractal grading,and S is the area of the polygon.

5.Mathematical descriptions for particle breakage evolution

5.1.Stress state correlation

Particle breakage has a strong correlation with the stress state.Hardin (1985) presented a stress correlation for particle breakage.He related the relative breakage Brto an effective breakage stressby a hyperbolic equation:

where nbandare the parameters that reflect the characteristics of soil particles; andis the breakage effective stress introduced to combine the effects of the mean stress p′and deviatoric stress ratio q／p′on the extent of particle breakage,which can be defined as

To examine the applicability of Hardin’s correlation, the test data shown in Figs.11 and 12 were simulated.First, the development of the effective breakage stresswas calculated by Eq.(5)according to the recorded stress histories (shown in Figs.6 and 7).Then, the developing curves of the relative breakage were obtained by Eq.(4)once the values of the parametersand nbwere given.Figs.13 and 14 show the development of the effective breakage stress and the comparisons between simulation and test data for SAND-I and for SAND-II,respectively.The values forand nbwere selected by a trial-and-error method for better fitting.As can be seen,the effective breakage stress decreases after its peak is reached, and as a result, the calculated relative breakage ceases to increase.Hardin’s breakage model failed in simulating the continuous development of particle breakage after the stress peak.

The formulation of the effective breakage stress, i.e.Eq.(5),builds up a convex crushing surface in the stress space (Kikumoto et al., 2010), and the particle breakage occurs only when the stress state pushes the crushing surface outside.Therefore,Hardin’s correlation is suitable only for compression-induced breakage (i.e.mean effective stress increases when shear stress ratio remains unchanged), for which if stress ceases to increase, strain and particle breakage ceases to develop as well.For shear-induced particle breakage such as that in triaxial compression, shear strain could develop continuously and induce further breakage with no increase in stress in the situations of strain softening or perfect plasticity(Indraratna and Salim, 2002; Coop et al., 2004; Yu, 2017c, 2018a;Wei et al.,2018).Therefore,stress alone is inadequate for describing the evolution of particle breakage induced by shear.

5.2.Energy-based correlation

As particle breakage consumes extra energy in addition to internal friction, the energy input seems to be an appropriate parameter to correlate with particle breakage.Moreover, energy computation is independent of how the current strain and stress state was achieved, which means that the correlation between energy and particle breakage would be path-independent.This advantage is very attractive for constitutive modeling.Therefore,the correlation between the input energy and particle breakage has been discussed frequently in experimental investigations (Lade et al.,1996; Shahnazari and Rezvani, 2013; Yu, 2017c; Konrad and Salami, 2018; Xiao et al., 2019).

Theoretically, plastic work seems more relevant to particle breakage.Actually,there is little difference between total work and plastic work,because the elastic deformation range for sand is verysmall.As computing the total input energy is much easier,the total input energy was employed in this study.The total amount of energy per unit volume of soils during an increment of straining(δε1,δεv) in triaxial condition is given by Lade et al.(1996):

Fig.11.Development of the relative breakage against axial strain for SAND-I: (a) Drained triaxial compression; and (b) Undrained triaxial compression.

Fig.12.Development of the relative breakage against axial strain for SAND-II: (a) Drained triaxial compression; and (b) Undrained triaxial compression.

The amounts of the total input energy per unit volume of all specimens were calculated and are also given in Tables 2 and 3.Fig.15 shows the plots of the relative breakage against the total input energy.It can be seen that a larger input energy corresponds to a larger relative breakage,but the data points are quite scattered,especially for the data points at small axial strains.This scatter might be attributed to the difference in the energy efficiency in producing particle breakage under different loading paths and at different loading stages.From the stress-strain curves shown in Figs.6 and 7 and the particle breakage development shown in Figs.11 and 12,it can be seen that an equal strain increment at the early stage consumes a smaller input energy but produces a larger increase in particle breakage than those during the late stage,demonstrating different energy efficiencies of breakage at different loading stages.Therefore, it seems controversial to establish aunique correlation between particle breakage and the input energy regardless of the intermediate stress-strain evolution.Lade et al.(1996) showed that the correlation between particle breakage parameters and the total input energy was quite satisfactory for triaxial compression and extension tests on three different densities of Cambria sand under drained and undrained conditions.It is worth noting that in their studies,all the specimens achieved their failure states, and only the final amounts of particle breakage and input energy were involved in their plots.

Fig.13.Comparison of the simulation of Hardin’s breakage model with the test results for SAND-I.

Fig.14.Comparison of the simulation of Hardin’s breakage model with the test results for SAND-II.

Fig.15.Relative breakage versus total input energy during the process of triaxial compression: (a) SAND-I and (b) SAND-II.

Fig.16.Performance of the proposed hyperbolic equation for fitting the change of the relative breakage with axial strain during the drained triaxial compression with constant effective confining pressure: (a) SAND-I and (b) SAND-II.

5.3.Stress-strain path-dependent correlation

Fig.16 shows that a good relationship exists between the relative breakage and axial strain under a constant effective confining pressure.The accumulation rate of particle breakage (?Br／ ?ε1),which is called the breakage rate herein, decreases with the increasing axial strain, implying that there is an upper limit for particle breakage under monotonic shear.Coop et al.(2004) also demonstrated through ring shear tests that carbonate sand will reach a stable grading at very large shear displacements.Therefore,a hyperbolic curve, as shown in Fig.16, is employed to fit the data points for each confining pressure,in the form of

Fig.17.Illustration of the variation in Br with effective confining pressure

where RB0equals the initial slope of the hyperbolic curve, named the initial breakage rate; andrepresents the upper limit of the hyperbolic curve, named the breakage limit.

As shown in Fig.16,the initial breakage rate RB0increases with increasing effective confining pressure.The following equation is employed to determine the change in RB0with:

where c1and c2are the fitting parameters; and pais the atmospheric pressure employed for normalization, and pa＝101 kPa.

where A is a fitting parameter; andis the value of relative breakage at the ultimate PSD,which represents the ultimate extent that particle breakage could be generated under extremely high pressure and successive shearing.

The value of the ultimate breakagedepends on the hypothesis of what the ultimate PSD would be.Hardin (1985) postulated that under extremely high stresses, all particles would be crushed into fine particles, and then＝1.However, another viewpoint is becoming more widely accepted, i.e.the PSD tends to be fractal during excessive loading (McDowell et al., 1996; McDowell and Bolton 1998; Coop et al., 2004).The fractal distribution illustrated in Fig.10 is expressed as (Einav, 2007):

Referring to the ring shear test results by Coop et al.(2004),α in Eq.(10) is assumed to be 2.57, and then it can be easily calculated that

Table 4Parameters of the path-dependent breakage model for the coral sands.

In the process of drained axial straining,the effective confining pressure is constant.However, for more common situations, the effective confining pressure may not be constant because of the generation of excess pore water pressure or the change in the total confining pressure.To predict the evolution of particle breakage in situations for a variable effective confining pressure, the breakage rate RBat the current material and stress states is employed herein.Combination of Eqs.(7)-(9) and then differentiation with respect to ε1gives

This equation shows that the breakage rate depends on the current effective confining pressure and accumulated particle breakage during the past loading history.

Although Eq.(11) is deduced from the experimental results of the CD tests, we postulate that it would work for other situations and use the experimental results of the CU tests to validate the postulation preliminarily.Ignoring the particle breakage induced by compression, the shear-induced particle breakage in situations of variable effective confining pressure can be calculated step by step by the following simple explicit integration:

Fig.18 also shows the particle breakage evolution during undrained triaxial compression.The breakage rate RBdecreases with the accumulated amount of Brand increases with the current effective confining pressure.At the initial phase,since Bris small,the effect ofon the accumulation of RBprevails.The increase or decrease incauses RBto increase or decrease correspondingly,leading to a gentle or sharp change in the accumulating curves of relative breakage.In the latter phase, as the soil approaches the critical state,andhas little change,thus,the effect of Brprevails.Then, Brgradually approachesat a decreasing rate.

6.Conclusions

A comprehensive database for investigating the intermediate particle breakage evolution was presented through a series of parallel CD and CU tests on two coral sands along the same loading paths but stopped at different axial strains.Based on the database,the characteristics of particle breakage evolution and the corresponding descriptions were discussed.The main findings are summarized as follows:

(1) Under isotropic compression with a high pressure of up to 3200 kPa, the coral sands produced negligible particle breakage; in contrast, under triaxial compression with a moderate confining pressure of 400 kPa, considerable breakage was observed, indicating that shear is much more efficient than compression for producing particle breakage.Under triaxial compression with moderate confining pressures, the main particle breakage patterns are sharp corner break-off and surface abrasion.Accordingly, the change in the PSD is manifested as an increase in fine particle content and no considerable change in the sizes of original particles.

(2) During drained triaxial compression with a constant effective confining pressure,particle breakage increases with the axial strain at a gradually decreasing rate and continues to increase even after the peak deviatoric stress is reached.Higher confining pressures lead to greater breakage.During undrained triaxial compression with a constant total confining pressure, particle breakage increases continuously with the increasing axial strain but not at a continuously decreasing rate.Higher total confining pressures also lead to greater particle breakage, but the gaps among different total confining pressures under undrained compression are much smaller than those under drained compression.

Fig.18.Comparison of the prediction of the path-dependent breakage model with the test data under undrained triaxial compression.

(3) During triaxial compression, shear is the main mechanism inducing breakage.For the situations of strain softening or perfect plasticity,shear strain can develop continuously and induce further breakage without an increase in stress.Therefore,stress alone is inadequate for describing the evolution of particle breakage induced by shear.Although the energy input combines the stress state and strain development together, the plots of the input energy versus the relative breakage are quite scattered when involving intermediate particle breakage data due to the difference in the energy efficiency of breakage under different loading paths and at different loading stages.

(4) The development of the relative breakage with the axial strain under constant effective confining pressures can be well fitted by the hyperbolic equation.Based on this experimental observation,a path-dependent model was proposed for particle breakage evolution that reasonably considers the effect of the past breakage history and current stress state on the breakage rate.The path-dependent model successfully reproduced the development of particle breakage during undrained triaxial compression using the parameters calibrated from the CD tests, preliminarily demonstrating its effectiveness for different stress paths.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant Nos.51679016 and 52079012).