Study on creep mechanism of coral sand based on particle breakage evolution law

1. Introduction

Coral sand usually refers to special geotechnical media rich in calcium carbonate or other carbonate substances caused by marine organisms, and is commonly used as natural filling materials for island reef construction and foundation of marine structures [1]. Compared with common continental and marine sediments, coral sand has high porosity, irregular shape, low particle strength, easy breakage, easy cementation between particles, and very special engineering mechanical properties [2, 3]. At present, a great deal of research work has been carried out on its macro and micro structural characteristics [4-6], static properties [7, 8], cyclic load [9, 10] and mechanical behavior under explosion impact load [11, 12], and fruitful research results have been obtained. According to the relationship between the change rule of mechanical properties of coral sand and the degree of particle breakage found in shear or compression tests of coral sand by Sun Jizhu [13], Zhang Jiaming [14], Wang Yiqun [15], He Jianqiao and others [16], it can be seen that the change of particle-size distribution (PSD) caused by particle breakage under external force is the main reason why the mechanical properties of coral sand are different from those of continental sand [17, 18]. In view of this, many scholars [19-21] have also proposed different quantitative indicators of fragmentation to explain the change law of macro-mechanical properties of sand through PSD changes before and after the test. However, the current research on the correlation between PSD state change of coral sand particles and its creep deformation rule under long-term constant load and the mechanism of creep of coral sand is relatively weak. In the existing literature, Lade et al. [22, 23] put forward the concept of stress drop-creep effect for the time effect of deformation of crushed coral sand, and showed that creep can reshape the structure, and studied the creep characteristics under different confining pressures, initial deviatoric stresses and strain rates. Lv et al. [24] pointed out that quartz sand shows dilatancy during triaxial creep while calcareous sand always shows shear shrinkage. With the increasing number and scale of reef projects, the long-term stability of reef projects has become one of the focuses of people’s attention. The settlement of coral sand foundation with time is an important aspect that affects the long-term stability of artificial reefs and their affiliated buildings [25]. In view of this, one-dimensional compressive creep test was carried out on the main particle size groups of coral sand, and the creep deformation characteristics and particle breakage evolution law after creep were analyzed. The microscopic morphology changes of particles were observed by scanning electron microscopy on the samples after creep test, revealing the creep mechanism of coral sand, which can provide reference for foundation reinforcement in the later stage of existing projects and construction of island reef projects in the future.

2. Test introduction

2.1. Test materials

The sample is taken from an island reef in the South China Sea, which is unconsolidated loose body with white color mixed with red impurities, as shown in Fig. 1. The composition of the sand sample was analyzed by using Japan-made D/MAX-2500/PC X-ray polycrystalline powder diffractometer. The results show that the mineral composition is mainly aragonite and calcite, with CaCO₃ content of 90.63 % and MgCO₃ content of 8.16 %. Take a proper amount of sand samples, wash them with clear water to remove salt, bake them in an oven at 105℃ for 24 hours, and then screen them with the geotechnical standard sieve. The sample particle splitting curve is shown in Fig. 2, and the basic physical parameters are shown in Table 1. From Table 1 and Fig. 2, it can be seen that the curvature coefficient of the sample satisfies $C_c=$ 1-3, but don’t satisfy the condition of non-uniformity coefficient $C_u$ > 5, so it is medium sand with poor gradation.

Fig. 1Coral sand of sample

Fig. 2Particle size distribution of sample

Table 1Basic physical parameters of sample

$e_{min}$	$e_{max}$	$G_s$	$d_{10}$ / mm	$d_{50}$ / mm	$d_{60}$ / mm	$C_u$	$C_c$
0.78	1.14	2.77	0.13	0.32	0.37	2.85	1.20

3. Analysis of creep characteristics of coral sand

3.1. Comparative analysis of compression deformation laws

The strain-time curves of coral sand with particle sizes of 0.25-0.5 mm, 0.5-1 mm, 1-2 mm and 2-5 mm for 50 days are shown in Fig. 3. The strain-time curve of ISO standard sand for 50 days under the same particle size groups and test conditions are shown in Fig. 4.

The comparison between Fig. 3 and Fig. 4 shows that under different constant stresses, the creep curves of coral sand are similar in shape and have obvious phases, all of which have experienced instantaneous deformation, accelerated deformation and slow deformation phase tending to stability. However, ISO standard sand only undergoes instantaneous deformation and accelerated deformation in a very short period of time, and the deformation of each sample is stable within 3 days, after which the deformation no longer increases with time.

Statistical analysis is carried out on the strain of coral sand and ISO standard sand in different deformation stages under the same test conditions and the same particle size group respectively, as shown in Table 3 and Table 4. Comparing Table 3 and Table 4, it can be seen that the final strain of coral sand is about 3-6 times that of ISO standard sand, and the deformation stability time is about 14-39 times that of ISO standard sand. Compared with ISO standard sand, the deformation of coral sand is still not finished after experiencing instantaneous deformation and accelerated deformation stages, and the deformation time in this stage is positively correlated with the test load, with the shortest time being 21 days and the longest 45 days, accounting for 41.83 % and 89.98 % of the total deformation time respectively. Obviously, under the action of constant load, the creep characteristics of coral sand are obviously different from ISO standard sand. Coral sand needs a longer time to stabilize, and its deformation is more time-effective, and it presents obvious nonlinear attenuation creep characteristics.

Fig. 3Axial strain-time relationship curves of coral sand with different particle sizes

a) 0.25-0.5 mm

b) 0.5-1 mm

c) 1-2 mm

d) 2-5 mm

Fig. 4Axial strain-time relationship curves of ISO standard sand with different particle sizes

a) 0.25-0.5 mm

b) 0.5-1 mm

c) 1-2 mm

Table 3Deformation amount of coral sand in different deformation stages

Stress / kPa	Particle size /mm	Instantaneous strain / %	Acceleration strain / %	Creep strain / %	Total strain / %	Creep strain / total strain / %
50	0.25-0.5	0.30	0.10	0.11	0.51	21.78
	0.5-1.0	0.41	0.05	0.23	0.68	33.82
	1.0-2.0	0.49	0.07	0.15	0.71	21.28
100	0.25-0.5	0.71	0.06	0.08	0.84	9.52
	0.5-1.0	0.89	0.13	0.27	1.28	20.78
	1.0-2.0	1.1	0.05	0.21	1.36	15.5
200	0.25-0.5	1.13	0.12	0.24	1.48	15.93
	0.5-1.0	1.55	0.17	0.27	1.98	13.38
	1.0-2.0	1.56	0.19	0.3	2.05	14.67
400	0.25-0.5	1.49	0.11	0.3	1.89	15.61
	0.5-1.0	2.5	0.22	0.24	2.96	8.12
	1.0-2.0	2.95	0.16	0.41	3.52	11.66
800	0.25-0.5	3.55	0.26	0.53	4.34	12.23
	0.5-1.0	3.87	0.19	0.43	4.48	9.49
	1.0-2.0	4.55	0.61	0.77	5.93	12.91

Table 4Deformation amount of ISO standard sand in different deformation stages

Stress / kPa	Particle size / mm	Instantaneous strain / %	Acceleration strain / %	Total strain / %
50	0.25-0.5	0.08	0.08	0.16
	0.5-1.0	0.14	0.1	0.24
	1.0-2.0	0.19	0.03	0.22
100	0.25-0.5	0.16	0.14	0.3
	0.5-1.0	0.24	0.08	0.31
	1.0-2.0	0.29	0.05	0.34
200	0.25-0.5	0.24	0.21	0.45
	0.5-1.0	0.3	0.13	0.43
	1.0-2.0	0.35	0.1	0.45
400	0.25-0.5	0.43	0.26	0.69
	0.5-1.0	0.49	0.15	0.64
	1.0-2.0	0.55	0.07	0.62
800	0.25-0.5	0.63	0.24	0.87
	0.5-1.0	1.29	0.4	1.69
	1.0-2.0	1.91	0.05	1.96

3.2. Relationship between creep deformation and time of coral sand

According to the research on the creep law of indoor soil, the empirical theory is based on the deformation reaction of soil under actual stress. The general form is simple and convenient for practical application. At the same time, its results also have some enlightening effects on the study of creep mechanism and structural model, and it is one of the main approaches for indoor research [28-32]. Therefore, based on the analysis of the deformation process of the three particle size groups of 0.25-0.5 mm, 0.5-1 mm and 1-2 mm, the function formula of coral sand strain-time is constructed by using the empirical theoretical method, which can be expressed as:

1

$\epsilon =\alpha t^{\beta },$

where $\alpha$ and $\beta$ are creep parameters and $t$ is creep time.

As shown in Table 5, the fitting correlation coefficient is between 0.9026 and 0.9826, so there is a power function relationship between them. Eq. (1) can well describe the relationship between creep deformation of coral sand and time. In addition, it can be seen from Table 5 that under the same particle size group, the creep parameter $\alpha$ is positively correlated with the test load, and the creep parameter $\beta$ is negatively correlated with the test load. Regression analysis is carried out on creep parameters $\alpha$ and $\beta$ and stress respectively, as shown in Fig. 5 and Table 6. As can be seen from Fig. 5(a), creep parameter $\alpha$ is linearly related to stress. As can be seen from Fig. 5(b), creep parameter $\beta$ is exponentially related to stress.

Table 5Fitting parameters of creep time of coral sand with different particle sizes

Particle size / mm	Fitting equation $\epsilon =\alpha t^{\beta }$	50 / kPa	100 / kPa	200 / kPa	400 / kPa	800 / kPa
1-2	$\alpha$	0.0049	0.0107	0.0167	0.0296	0.0490
	$\beta$	0.0329	0.0196	0.0165	0.0139	0.0164
	$R^2$	0.9629	0.9059	0.9348	0.9112	0.9285
0.5-1	$\alpha$	0.0038	0.0087	0.0160	0.0259	0.0386
	$\beta$	0.0530	0.0321	0.0173	0.0107	0.0125
	$R^2$	0.9518	0.9643	0.9026	0.9343	0.9114
0.25-0.5	$\alpha$	0.0032	0.072	0.0111	0.0151	0.0369
	$\beta$	0.0403	0.0324	0.0229	0.0192	0.0127
	$R^2$	0.9826	0.9214	0.9525	0.9634	0.9423

Fig. 5Relationship between creep parameters α, β and stress

a) Creep parameter $\alpha$

b) Creep parameter $\beta$

Table 6Fitting α, β with σ for different particle sizes

Creep parameter	Particle size / mm	Fitting equation	$R^2$
$\alpha$	0.25-0.5	$\alpha =$ 4.26E-5$\sigma$+0.0015	0.9666
	0.5-2	$\alpha =$ 4.48E-5$\sigma$+0.0047	0.9480
	1-2	$\alpha =$ 5.74E-5$\sigma$+0.0049	0.9855
$\beta$	0.25-0.5	$\beta =$ 0.0137+0.0354e-0.0062σ	0.9589
	0.5-2	$\beta =$ 0.0116+0.08254e-0.0138σ	0.9958
	1-2	$\beta =$ 0.0155+0.0722e-0.0284σ	0.9641

4. Analysis on evolution law of creep crushing of coral sand

4.2. Description of crushing state of coral sand-Weibull distribution function

The sample mass of the 2-5 mm particle size group is 85.62 g, particle screening data after creep tests under stress of 800 kPa, 1600 kPa, 2400 kPa, 3200 kPa and 4000 kPa are shown in Table 7.

Substituting the data in Table 7 into Eq. (1) to calculate particle breakage probability $p$, substituting into Eq. (5) to perform data fitting, and the results are shown in Table 8 and Fig. 6. As can be seen from Fig. 6 and Table 8, under the condition of one-dimensional compression, the particle size distribution state of smaller particles produced by crushing coral sand with single particle size conforms to Weibull function distribution.

Fig. 7 is a fitting curve of Weibull distribution parameters a, b, breakage probability p and stress $\sigma$, and the fitting parameters are shown in Table 9. As can be seen from Fig. 7 and Table 9, with the continuous increase of stress, the values of parameters $a$ and $b$ gradually decrease, and the breakage probability $p$ gradually increases. All of them have a good exponential relationship with stress and tend to evolve to a fixed value.

Table 7The mass of each particle size of coral sand (2-5 mm) after the test

Stress / KPa	Quality of each particle size group after test / g
	5-2	2-1	1-0.5	0.5-0.25	0.25-0.075	< 0.075
800	78.45	5.14	1.60	0.32	0.09	0.02
1600	71.89	9.12	3.47	0.71	0.29	0.06
2400	66.09	12.11	5.49	1.33	0.49	0.11
3200	60.78	14.42	7.39	2.01	0.84	0.18
4000	55.97	15.68	9.59	2.73	1.53	0.3

Table 8Values of Weibull distribution parameters a, b and breakage probability p under different stresses

Stress / kPa	800	1600	2400	3200	4000
$p$	0.0837	0.1596	0.2281	0.2901	0.3463
$a$	2.1802	1.9281	1.7370	1.5628	1.4173
$b$	1.4811	1.4136	1.3702	1.3304	1.2717
$R^2$	0.9967	0.9972	0.9957	0.9989	0.9994

Fig. 6Fitting relationships X and Y

Fig. 7Evolution of parameters a, b and p

Table 9Fitting parameters of Weibull distribution parameters a, b, breakage probability p and stress σ

Parameters	Fitting equation	Correlation coefficient / $R^2$
$a$	$a=0.6831+1.7844e^{-2.21\times {10}^{-4}\sigma }$	0.9857
$b$	$b=0.6213+0.9151e^{-8.36\times {10}^{-5}\sigma }$	0.9952
$p$	$p=0.8825\left(1-e^{-2.21\times {10}^{-4}\sigma }\right)$	0.9986

In addition, theoretical values of particle breakage probability p of coral sand at 50 kPa, 100 kPa, 200 kPa, 400 kPa, 800 kPa can be calculated from Table 9. Compared with the actual value of particle breakage probability p of each particle size group under the action of each stress, as shown in Table 10. It can be seen from Table 10 that the particle breakage probability p of coral sand is related to the stress level, and has nothing to do with the particle size. Therefore, for coral sand of any single particle size group, the particle breakage state conforms to the above Weibull distribution evolution rule.

Table 10Comparison of theoretical value and experimental value of p of coral sand for different particle sizes

Stress / kPa		50	100	200	400	800
$P_{theoretical}$		0.0055	0.0109	0.0217	0.0429	0.0837
0.25-0.5 mm	$P_{experimental}$	0.0054	0.0107	0.0216	0.0428	0.0836
	Relative error	2.35 %	1.42 %	0.43 %	0.31 %	0.08 %
0.5-1 mm	$P_{experimental}$	0.0057	0.0110	0.0219	0.0432	0.0839
	Relative error	3.39 %	1.30 %	0.73 %	0.58 %	0.25 %
1-2 mm	$P_{experimental}$	0.0058	0.0112	0.0218	0.0430	0.0838
	Relative error	4.65 %	2.60 %	0.48 %	0.31 %	0.07 %
Note: $\text{Relative error}=\left|P_{experimental}-P_{theoretical}\right|/P_{experimental}\times 100\%$

5. Analysis of creep mechanism of coral sand

There are two main creep factors for geotechnical granular materials: one is the continuous rotation and movement of soil particles with time under the action of load, that is to say, the adjustment of particle position; the other is the time-related fatigue fracture of geotechnical particles under the continuous action of load, that is to say, particle breakage [35, 36]. For ISO standard sand, the particle breaking stress is 12.8 MPa [37]. Within the stress range of this test, the particle will not be broken, so its deformation is mainly caused by the adjustment of particle position. However, coral sand particles have low strength and are easy to be broken. In the following, taking coral sand with a particle size of 1-2 mm as an example, the creep mechanism of coral sand is discussed from the point of view of particle breakage and combination with the change of particle morphology after creep.

In order to quantify the degree of particle breakage, the relative breakage rate $B_r$ proposed by Hardin [19] is used as the quantitative index, which is currently the most widely used, and its calculation formula is as follows:

where $B_t$ is the total amount of crushing, that is to say, the area enclosed by the particle gradation curve before and after crushing and the vertical line segment with particle size of 0.074 mm; $B_p$ is the crushing potential, that is to say, the area enclosed by the initial grading curve and the vertical line segment with particle size of 0.074 mm.

PSD test data of each sample after creep test of coral sand with particle size group of 1-2 mm are shown in Table 11. Under the same test conditions, the consolidation test of coral sand was completed within 2 days, PSD test data of each sample after the test are shown in Table 12. The data in Table 11 and Table 12 are substituted into Eq. (6) to calculate the relative breakage rate $B_r$ of the samples after each test, as shown in Table 13. As can be seen from Table 13, after the consolidation of coral sand is completed, its particle breakage is not completed. With the increase of stress acting time, coral sand continues to break, indicating that particle breakage has certain timeliness. However, ISO standard sand did not break within the stress range of this test, thus making the creep deformation of coral sand have stronger deformation timeliness than ISO standard sand.

Fig. 8SEM image of coral sand after creep test under different Stresses

a) 50 kPa stress

b) 200 kPa stress

c) 800 kPa stress-fracture

d) 800 kPa stress-filling

The microstructure of coral sand after creep under different stresses is shown in Fig. 8. As can be seen from Fig. 8(a), under the stress of 50 kPa, there are many scratches on the surface of particle due to the relative movement of the particles, and the particle breakage performance is extremely insignificant under this stress level, so the adjustment and rearrangement of the positions between the particles are the main reason of creep deformation of coral sand. As can be seen from Fig. 8(b), under the stress of 200 kPa, during the process of particle movement and recombination, the rough particle surface is worn and broken due to friction, resulting in the increase of fine particles. When the stress increases to 800 kPa, it can be seen from Fig. 8(c) and Fig. 8(d) that the particles are cracked and the internal pores are clearly visible, the crushed and refined fine particles slip to fill the pores of the particles themselves in the process of moving and recombining. Compared with 50 kPa stress, the particle breakage is more severe and the overall breakage degree of the sample increases, and the relative breakage rate B_r under the stress is 3.74 times that under 50 kPa, and the resulting final strain is 8.41 times that under 50 kPa. Obviously, under the higher stress conditions, the main reason of creep deformation is filling pores with crushed and refined coral sand particles.

In addition, Weibull distribution parameters a and b after creep crushing of coral sand under this stress range are calculated from Table 9, as shown in Table 14. According to the analysis of Fig. 8, Table 11 and Table 14, the smaller the parameter a, the more severe the particle breakage, the more fine particles produced, and the smaller the parameter $b$, the more uniform the particle composition distribution of the crushed sample. Further regression analysis is carried out on parameters a and b, the final total strain and the relative breakage rate $B_r$ respectively, as shown in Fig. 9 and Fig. 10. As can be seen from Fig. 9, Weibull distribution parameters a and b have negative linear correlation with the relative breakage rate $B_r$. As can be seen from Fig. 10, the final total strain has a positive correlation with the relative breakage rate $B_r$, and has a good logarithmic relationship. Obviously, Weibull distribution parameters a and b describe the overall broken degree of the sample after creep. The smaller the parameters a and b, the greater the overall broken degree of the sample, and the greater the creep deformation, which has a negative correlation with the final deformation amount.

Fig. 9Relationship between a, b and Br

Fig. 10Relationship between Br and total strain

6. Conclusions

In this paper, taking coral sand from an island reef in the South China Sea as a sample, one-dimensional compressive creep tests are carried out on its main particle size groups, which are compared with ISO standard sand. Through analysis of test data, the following conclusions are obtained:

1) Coral sand has stronger deformation timeliness than ISO standard sand, showing significant nonlinear attenuation creep characteristics. The creep of coral sand has obvious stages, which has experienced instantaneous deformation, accelerated deformation and slow deformation phase tending to stability. The strain-time relationship can be well described mathematically by power function. The creep parameter α has a linear relationship with stress, and the creep parameter $\beta$ has a exponential function relationship with stress.

2) The two-parameter Weibull distribution function can well describe the particle crushing state of any single particle size group of coral sand after creep. Weibull distribution parameters a and b have good exponential relationship with the stress, and have a negative linear correlation with particle breakage quantitative index B_r, and have a negative correlation with the final creep deformation.

3) Under the action of constant stress, coral sand particle breakage has a certain timeliness, and creep occurs under the combined action of particle movement recombination effect and particle breakage effect. Under the low stress condition, the amount of particle breakage is not obvious, and creep deformation is mainly caused by movement recombination and arrangement optimization among coral sand particles. Under the higher stress conditions, in the process of relative sliding of particles, the surface of rough particles is worn and broken, and some particles are broken, the main reason of creep deformation is that crushed and refined coral sand particles fill the pores.