# Effects of Strain Rate and Initial Density on the Dynamic Mechanical Behaviour of Dry Calcareous Sand

Authors: Zhangyong Zhao; Yanyu Qiu; Mingyang Wang
Journal: Shock and Vibration
 (2019)

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## Abstract

The dynamic compressive behaviour of dry calcareous sand under rigid confinement was characterised using a split-Hopkinson pressure bar (SHPB). Sand samples were confined inside a sleeve of hardened stainless steel and capped by a pair of aluminium cylindrical rods. This assembly was subjected to repeated dynamic compaction to attain precise bulk mass densities. It was then sandwiched between the incident and transmission bars of SHPB for dynamic compression testing. Sand specimens of three initial mass densities, namely, 1.26 g/cm3, 1.35 g/cm3, and 1.42 g/cm3, were loaded by incident pulses applying a stress of 35 MPa, 71 MPa, and 143 MPa, respectively. Experimental results show that in the strain rate range of 335 s−1 to 1253 s−1, the dynamic mechanical behaviours of dry calcareous sands exhibited no significant strain rate effect. The Lundborg model and the Murnaghan model could be used to describe the deviatoric and volumetric behaviours of calcareous sand with different initial densities, respectively.


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## Body

## 1. Introduction

Calcareous sand is generally regarded as an oceanic sediment which contains an appreciable amount (>30%) of CaCO3, leading to certain mechanical properties compared with continental sediments such as quartz sand [1, 2]. It usually originates from reworked shell fragments and skeletal debris of marine organisms and is widely distributed on coral reefs and seashores throughout the world such as in the South China Sea, Red Sea, west continental platform of Australia, and the Bass Strait [3, 4]. With the rapid development of the offshore construction, the mechanical response of calcareous sand has attracted the attention of engineers. Generally, it is found that calcareous sand includes large amounts of irregular crushable particles, leading to a higher compressibility and more significant grain crushing at elevated stresses than continental sediments [5–8]; however, compared with extensive studies under quasistatic condition, few studies consider the dynamic compressive behaviour of calcareous sand, which is important in some special engineering problems such as dynamic compaction and mine blast situations, and in air-raid shelters.Farr used a uniaxial strain device (WES 0.1 msec) to study the effect of loading rate on the one-dimensional compressibility of dense partially saturated Enewetak calcareous sand [9]. The test results show that when the strain rate of calcareous sand sample increases from 10−3 s−1 to 300 s−1, the constrained modulus can be increased by 120%. In recent years, the split-Hopkinson pressure bar (SHPB) has been widely used to study the dynamic mechanical behaviour of sand [10–12]. Bragov computed the lateral stress component of the specimen based on measurements of circumferential strain, which, together with obtaining the longitudinal stresses in the specimen using the SHPB method, make it possible to find the deviatoric stress and the hydrostatic pore water pressure in the specimen; however, in most studies, using this method, the soil under test is a quartz sand [13–18]. No experimental research has yet been aimed at the study of the deviatoric and volumetric behaviour of calcareous sand. Lv et al. studied the one-dimensional dynamic compressive behaviour of dry calcareous sand and dry silica sand with the same relative density and similar strain rate using the SHPB method [19], and the experimental results show that the apparent modulus of calcareous sand under dynamic loading is approximately 10% of that of silica sand; however, the effects of strain rate and initial density of the sample on the dynamic mechanical behaviour of dry calcareous sand under stiff confinement were not investigated in their studies. These effects will be studied by using the SHPB method in the present research, and the deviatoric and volumetric behaviours will be analysed based on the experimental results.


## 2. Experimental Work

### 2.1. Sand Samples

The white calcareous sand used in this research is from the South China Sea. The micrographs of calcareous sands obtained by using the Hitachi S-4800 scanning electron microscope are shown in Figure1, which shows that the sand particles are mostly angular with many interparticle voids in the calcareous sand particles, leading to its complex microstructure and the low strength of the particles.Figure 1
SEM micrographs of the calcareous sands. (a) 50x magnification. (b) 100x magnification.
(a)
(b)All the sand samples used in this research were first oven-dried. The grain size distribution curve of the sample is shown in Figure2, which was determined by using the laser-diffractometry instrument called Mastersizer 3000E. According to the result, the average particle size and the uniformity coefficient of the calcareous sand are 0.348 mm and 8.84, respectively. The particle density of sand particle was determined to be 2.81 g/cm3. The minimum and maximum dry density of the sand is 1.12 g/cm3 and 1.47 g/cm3; these properties were tested based on ASTM methods [19–22].Figure 2
Particle size distribution curve of both sands.To study the effect of the initial mass density of the sample on the dynamic mechanical behaviour of calcareous sand, three types of samples with initial dry density of 1.42 g/cm3, 1.35 g/cm3, and 1.26 g/cm3 were designed. The relative densities of these samples are 89%, 72%, and 47%, respectively.


### 2.2. Experimental Setup

Dynamic experiments are carried out with a standard split-Hopkinson pressure bar apparatus (Figure3), including a set of solid bars and a strain data acquisition system. The striker, the incident bar, and the transmission bar are made of aluminium alloy with a diameter of 37 mm and lengths of 600 mm, 2000 mm, and 2000 mm, respectively. The density, Young’s modulus, Poisson’s ratio, and bar wave speed of the aluminium material used are 2.8 g/cm3, 73 GPa, 0.324, and 5107 m/s, respectively. As its impedance is lower than that of steel, the aluminium material is suitable for the SHPB tests associated with low-impedance soil. The sand samples were loaded by the striker and the incident bar, and their dynamic strain and stress during loading were obtained by calculating the stress wave velocity in the incident bar and transmission bar based on elastic wave theory [23].Figure 3
Schematic of the SHPB apparatus used for dynamic compaction of sand.To study the strain rate effect on the dynamic mechanical behaviour of calcareous sand, the sand samples were loaded by the striker with three nominal velocities, namely, 5 m/s, 10 m/s, and 20 m/s. Based on the elastic wave theory [24], the corresponding nominal stresses applied via the incident impulse are 35 MPa, 71 MPa, and 143 MPa, respectively. The actual velocity of the striker was measured by using a laser velocimeter with 0.1 m/s resolution.The sample container consists of a sleeve, a pair of sealing rods, and a pair of locating rods. The sleeve was made of stainless steel and was designed to provide rigid confinement for the samples. The inner diameter, the outer diameter, and height of the sleeve are 37.05 mm, 47.05 mm, and 90 mm, respectively. The sealing rods together with the locating rods were used to locate and control the initial height and density of the sample. They were made of the same aluminium alloy as the pressure bar with a diameter of 37 mm and length of 20 mm. The sealing rod has a sealing groove with a width of 4.2 mm and a depth of 5 mm for mounting the Glyd Ring, which was used to fix the sealing rod in the sleeve during sample preparation and prevent the loss of sand particles during loading.Figure4 schematically shows the sample height and its position in the container. As is shown, the initial height of the sand sample is 10 mm, so the initial volume of the sample is 10.75 cm3. The mass of the sample that is needed to fill the container could be calculated through multiplying its initial density and volume. To prepare a sample with a given density, a locating rod and a sealing rod were firstly put into the vertically placed sleeve. Then, the sample with the corresponding mass was poured into the container, and another sealing rod and locating rod were placed in the container. A 1.25 kg hammer was continuously dropped from a height of 300 mm to strike the top surface of the locating rod until it was flushed with the top surface of the sleeve. Finally, two locating rods were taken out of the container, and a bolt was screwed into the hole in the top sealing rod to prevent the loss of sand particles under loading. The container, together with the sand sample, could then be sandwiched between the incident bar and the transmission bar (Figure 3).Figure 4
Schematic diagram of the sample in the container.A strain gauge was attached to the external surface of the sleeve to measure the circumferential strain of the sleeve, and the strain will be used to calculate the lateral stress on the specimen based on elastic mechanics [25].For the sands tested here, the sample diameter is the same as the bar diameter; when the sample is in a state of uniform stress, the strain and stress histories of the sample could be calculated as [14](1)ε˙t=C0Lsεit−εrt−εtt,εt=C0Ls∫0tεit−εrt−εttdt,σt=E02εit+εrt+εtt,where εi(t), εr(t), and εt(t) are incident, reflected, and transmitted strain histories measured by using the axial strain gauges on the solid bars, respectively; E0 and C0 are Young’s modulus and elastic wave speed of the bar material, respectively; Ls is the initial length of the specimen. Therefore, once the incident, reflected, and transmitted signals are measured, the axial strain-stress data for the sand under investigation can be obtained by eliminating the time variable of the strain and stress histories of the sample.


## 2.1. Sand Samples

The white calcareous sand used in this research is from the South China Sea. The micrographs of calcareous sands obtained by using the Hitachi S-4800 scanning electron microscope are shown in Figure1, which shows that the sand particles are mostly angular with many interparticle voids in the calcareous sand particles, leading to its complex microstructure and the low strength of the particles.Figure 1
SEM micrographs of the calcareous sands. (a) 50x magnification. (b) 100x magnification.
(a)
(b)All the sand samples used in this research were first oven-dried. The grain size distribution curve of the sample is shown in Figure2, which was determined by using the laser-diffractometry instrument called Mastersizer 3000E. According to the result, the average particle size and the uniformity coefficient of the calcareous sand are 0.348 mm and 8.84, respectively. The particle density of sand particle was determined to be 2.81 g/cm3. The minimum and maximum dry density of the sand is 1.12 g/cm3 and 1.47 g/cm3; these properties were tested based on ASTM methods [19–22].Figure 2
Particle size distribution curve of both sands.To study the effect of the initial mass density of the sample on the dynamic mechanical behaviour of calcareous sand, three types of samples with initial dry density of 1.42 g/cm3, 1.35 g/cm3, and 1.26 g/cm3 were designed. The relative densities of these samples are 89%, 72%, and 47%, respectively.


## 2.2. Experimental Setup

Dynamic experiments are carried out with a standard split-Hopkinson pressure bar apparatus (Figure3), including a set of solid bars and a strain data acquisition system. The striker, the incident bar, and the transmission bar are made of aluminium alloy with a diameter of 37 mm and lengths of 600 mm, 2000 mm, and 2000 mm, respectively. The density, Young’s modulus, Poisson’s ratio, and bar wave speed of the aluminium material used are 2.8 g/cm3, 73 GPa, 0.324, and 5107 m/s, respectively. As its impedance is lower than that of steel, the aluminium material is suitable for the SHPB tests associated with low-impedance soil. The sand samples were loaded by the striker and the incident bar, and their dynamic strain and stress during loading were obtained by calculating the stress wave velocity in the incident bar and transmission bar based on elastic wave theory [23].Figure 3
Schematic of the SHPB apparatus used for dynamic compaction of sand.To study the strain rate effect on the dynamic mechanical behaviour of calcareous sand, the sand samples were loaded by the striker with three nominal velocities, namely, 5 m/s, 10 m/s, and 20 m/s. Based on the elastic wave theory [24], the corresponding nominal stresses applied via the incident impulse are 35 MPa, 71 MPa, and 143 MPa, respectively. The actual velocity of the striker was measured by using a laser velocimeter with 0.1 m/s resolution.The sample container consists of a sleeve, a pair of sealing rods, and a pair of locating rods. The sleeve was made of stainless steel and was designed to provide rigid confinement for the samples. The inner diameter, the outer diameter, and height of the sleeve are 37.05 mm, 47.05 mm, and 90 mm, respectively. The sealing rods together with the locating rods were used to locate and control the initial height and density of the sample. They were made of the same aluminium alloy as the pressure bar with a diameter of 37 mm and length of 20 mm. The sealing rod has a sealing groove with a width of 4.2 mm and a depth of 5 mm for mounting the Glyd Ring, which was used to fix the sealing rod in the sleeve during sample preparation and prevent the loss of sand particles during loading.Figure4 schematically shows the sample height and its position in the container. As is shown, the initial height of the sand sample is 10 mm, so the initial volume of the sample is 10.75 cm3. The mass of the sample that is needed to fill the container could be calculated through multiplying its initial density and volume. To prepare a sample with a given density, a locating rod and a sealing rod were firstly put into the vertically placed sleeve. Then, the sample with the corresponding mass was poured into the container, and another sealing rod and locating rod were placed in the container. A 1.25 kg hammer was continuously dropped from a height of 300 mm to strike the top surface of the locating rod until it was flushed with the top surface of the sleeve. Finally, two locating rods were taken out of the container, and a bolt was screwed into the hole in the top sealing rod to prevent the loss of sand particles under loading. The container, together with the sand sample, could then be sandwiched between the incident bar and the transmission bar (Figure 3).Figure 4
Schematic diagram of the sample in the container.A strain gauge was attached to the external surface of the sleeve to measure the circumferential strain of the sleeve, and the strain will be used to calculate the lateral stress on the specimen based on elastic mechanics [25].For the sands tested here, the sample diameter is the same as the bar diameter; when the sample is in a state of uniform stress, the strain and stress histories of the sample could be calculated as [14](1)ε˙t=C0Lsεit−εrt−εtt,εt=C0Ls∫0tεit−εrt−εttdt,σt=E02εit+εrt+εtt,where εi(t), εr(t), and εt(t) are incident, reflected, and transmitted strain histories measured by using the axial strain gauges on the solid bars, respectively; E0 and C0 are Young’s modulus and elastic wave speed of the bar material, respectively; Ls is the initial length of the specimen. Therefore, once the incident, reflected, and transmitted signals are measured, the axial strain-stress data for the sand under investigation can be obtained by eliminating the time variable of the strain and stress histories of the sample.


## 3. Experimental Results

### 3.1. Axial Strain-Stress Relationship of Dry Calcareous Sand

In this research, three independent replicates were performed for each test group with different sample densities and striker velocities, and the corresponding basic results are summarised in Table1.Table 1
Experimental results of calcareous sand.
Test group number
Initial sample density (g/cm3)
Average striker velocity (m/s)
Average sample strain rate (s−1)
Average sample peak strain
Average sample peak stress (MPa)
CS1
1.26
5.3
385
0.104
6.4
CS2
10.5
748
0.192
18.5
CS3
19.2
1253
0.314
61.5
CS4
1.35
5.1
412
0.097
10.2
CS5
9.6
693
0.177
24.7
CS6
19.9
1189
0.293
77.1
CS7
1.42
4.0
335
0.080
12.5
CS8
9.8
487
0.118
17.8
CS9
21.4
1128
0.273
90.8Figure5 shows a typical set of strain records of calcareous sand in test group CS9. It could be found that the rise time of the incident pulse is very short, so the dispersive effects of the pulses need to be corrected. In this research, Bussac’s and Tyas’s correction methods were used to correct for dispersive effects [26–28]. After correction, the strain records could be used to calculate the strain and stress histories of the sample based on equation (1).Figure 5
Typical set of strain records of calcareous sand (test group number: CS9).Dynamic stress equilibrium was checked by the “2-wave,” “1-wave” method as used by Song [14] (Figure 6). The stress history of sample front end was calculated by adding the incident stress wave to the reflected stress wave, while the stress history of sample back end equals the transmitted stress wave. It can be seen from the figure that the stress histories at both ends of the sample almost overlapped except during the initial stage of loading, indicating that the stress in the specimen was uniform under most load conditions.Figure 6
Dynamic stress equilibrium in the sand sample (test group number: CS9).After the dynamic stress equilibrium is satisfied in the sample, the strain-stress curve of the experiment calculated by equation (1) is valid. Following the similar procedure, the mean curves with error bars for samples of the same initial density are plotted (Figures 7(a)–7(c)). There are oscillations in the stress-strain curves at small strains. By comparing the stress from the stress-strain curves and the corresponding stress histories at both ends of the sample (Figure 6), it was concluded that the oscillations were caused by out-of-equilibrium stresses in the sand sample. Despite the stress nonequilibrium at the initial deformation stage, the results showed no significant differences under overlapping stress ranges with different strain rates, indicating that, in the strain rate range of 335 s−1 to 1253 s−1, the dynamic compression characteristics of dry calcareous sand exhibited no significant rate effect.Figure 7
Effect of strain rate on the dynamic axial response of calcareous sand. The average axial strain-stress curves of calcareous sand with initial densities of (a) 1.26 g/cm3, (b) 1.35 g/cm3, and (c) 1.42 g/cm3.
(a)
(b)
(c)After discarding the nonequilibrium stage of the experimental result, the axial strain-stress curves of the sample with different initial densities could be obtained (Figure8(a)). The stress-strain curves shift upwards with the increase of initial mass density. The unloading curve shows an initial steep reduction in stress without recovery of deformation, followed by a further reduction in stress with a slight recovery of deformation.Figure 8
Effect of initial density on the dynamic axial response of calcareous sand. (a) The axial strain-stress curves of calcareous sand. (b) Thee-log σ curves of calcareous sand.
(a)
(b)The rigid confined compression test is oedometric, and thee-lop σ curve is usually used to describe the deformation of the sample, where e is the void ratio of the sample and σ is the axial stress on the sample. In a one-dimensional compression test, the lateral strain is approximately zero when compared with the axial strain, so the void ratio is given by(2)e=GSρ−1,ρ=ρ01−εz,where ρ and ρ0 are the current and initial densities of the sample, respectively; Gs is the density of the soil particle; and εz is the axial strain.According to equation (2), the axial strain-strain experimental results shown in Figure 8(a) could be transferred into e-log σ curves (Figure 8(b)). As the sand particles are crushable, the initial skeleton of calcareous sand is unstable under high stresses. With increasing stress, the initial skeleton structure of the calcareous sand was gradually destroyed and a stable structure formed. So at a certain stress, the e-log σ curve is independent of the initial sample skeleton properties and the initial density of the sample. So, all the loading curves of calcareous sands with different initial densities in Figure 8(b) merge into a same straight line at an axial stress of about 40 MPa. The slope of the line is the compression index and is 0.239 for this calcareous sand under dynamic compression. Luo et al. studied the dynamic compression behaviour of quartz sand, and the compression index is 0.281 to 0.312 [15], indicating the calcareous sand is “softer” than quartz sand.


### 3.2. Deviatoric and Volumetric Behaviours of the Sand

For the sand sample confined by the sleeve within the elastic range, the lateral stressσr is given by(3)σr=12κrori2−1Esεθ,where ro and ri represent the outer and inner radii of the sleeve; Es is Young’s modulus of the sleeve; εθ is the circumferential strain on the sleeve; and κ is a correction coefficient for the effect of the smaller sample length compared with that of the sleeve, which could be derived from the numerical method [29, 30].Here, the correction coefficient is given by equation (4), which is obtained by numerical analysis [30], where l is the current thickness of the sample:(4)κ=0.0598l+0.0162.Figure9 shows the axial stress and lateral stress histories of calcareous sand in test group CS9. The lateral stress coefficient k0 is defined as(5)k0=σrσz,where σr is the lateral stress and σz is maximum the axial stress. To improve the precision of k0, it could be calculated using equation (5) by using the maximum values of the stresses. The mean experimental results of k0 in different test groups and corresponding error bars are summarised in Figure 10. The results show that the lateral stress coefficient increases as the axial stress increases.Figure 9
Typical axial stress and lateral stress histories of calcareous sand (test group number: CS9).Figure 10
Relationship between lateral stress coefficient and axial stress.In a confined SHPB experiment, the hydrostatic pressure (isotropic stress) componentp and equivalent von Mises stress σY are given by [15](6)p=13σz+2σr,σY=σz−σr.Based on equation (5), the aforementioned equations could be transformed to give(7)p=1+2k03σz,σY=1−k0σz.The Drucker–Prager yield function is usually used to describe the deviatoric behaviour of soils under low pressure, in which the von Mises stress is linearly dependent on hydrostatic pressure. Based on this yield function, the lateral stress coefficient is shown to be a constant according to equation (7), which is inconsistent with the experimental results from tests on calcareous sand. To describe why k0 increases as the axial stress increases, the Lundborg model is used to describe the deviatoric behaviour of calcareous sand, whose yield function is expressed as [31](8)σY=μp1+μp/σYmax,where μ and σYmax are the material constants. Substituting equation (7) into the above equation, the relationship between lateral stress coefficient and axial stress is given by(9)σz=2μ+3k0+μ−3μ1−k01+2k0σYmax.The material constantsμ and σYmax could be obtained by fitting the experimental results from equation (9), and the corresponding results are μ = 1.21 and σYmax = 363 MPa. The fitting curve is shown in Figure 10, and the yield function is shown in Figure 11.Figure 11
Lundborg yield surface of the calcareous sand.Based on the axial strain-stress experimental results (Figure8(a)), the volumetric behaviour of dry calcareous sand in the form of density-pressure relationship could be obtained by using equations (2), (7), and (9), and the results are shown in Figure 12. To describe how the slope of the curves increases as the pressure increases, the Murnaghan equation [32, 33] was used to model the volumetric behaviour of dry calcareous sand:(10)p=knkρρ0nk−1,where k and nk are the material constants, which could be obtained by fitting the equation with experimental data.Figure 12
Effect of initial density on the density-pressure relationship of dry calcareous sand.The experimental results shown in Figure12 show that the density-pressure curves of samples with different initial densities could be separated into two segments at the point where the density is 1.75 g/cm3. In the first segment, the curves are dependent on the initial density of the sample, whereas in the second segment, the curves are independent of the initial density of the sample; therefore, the curves should be fitted separately.For the first segment, the relationships between the material constants using data fitted to equation (10) with the initial sample density are summarised in Figures 13(a) and 13(b). The results show that the material constants are related to the initial density of the sample in a quasilinear manner as follows:(11)k=203.2ρ0−239.1,nk=14.4−6.4ρ0.Figure 13
Relationship between material constants and initial density of the sample. (a) Material constantk. (b) Material constant nk.
(a)
(b)The second segment of the density-pressure curve is independent of the sample initial density, soρ0 in equation (10) could be chosen as 1.42 g/cm3. The corresponding material parameter fitting results are nk = 5.46 and k = 50.1.In summary, the density-pressure relationship of dry calcareous sand with any initial densityρ0 could be summarised as(12)p=knkρρ0nk−1,ρ≤1.75g/cm3,50.15.46ρ1.425.46−1,ρ>1.75g/cm3,where the material constants k and nk could be calculated by using equation (11) and the comparison between experimental results and those arising from the use of equation (12) is shown in Figure 14. By comparing the calculation results and the test results shown in the figure, it can be seen that equation (12) could well describe the dynamic volumetric behaviour of dry calcareous sand.Figure 14
Comparison between experimental and calculated results. Comparison of (a) the first segment and (b) the second segment.
(a)
(b)


## 3.1. Axial Strain-Stress Relationship of Dry Calcareous Sand

In this research, three independent replicates were performed for each test group with different sample densities and striker velocities, and the corresponding basic results are summarised in Table1.Table 1
Experimental results of calcareous sand.
Test group number
Initial sample density (g/cm3)
Average striker velocity (m/s)
Average sample strain rate (s−1)
Average sample peak strain
Average sample peak stress (MPa)
CS1
1.26
5.3
385
0.104
6.4
CS2
10.5
748
0.192
18.5
CS3
19.2
1253
0.314
61.5
CS4
1.35
5.1
412
0.097
10.2
CS5
9.6
693
0.177
24.7
CS6
19.9
1189
0.293
77.1
CS7
1.42
4.0
335
0.080
12.5
CS8
9.8
487
0.118
17.8
CS9
21.4
1128
0.273
90.8Figure5 shows a typical set of strain records of calcareous sand in test group CS9. It could be found that the rise time of the incident pulse is very short, so the dispersive effects of the pulses need to be corrected. In this research, Bussac’s and Tyas’s correction methods were used to correct for dispersive effects [26–28]. After correction, the strain records could be used to calculate the strain and stress histories of the sample based on equation (1).Figure 5
Typical set of strain records of calcareous sand (test group number: CS9).Dynamic stress equilibrium was checked by the “2-wave,” “1-wave” method as used by Song [14] (Figure 6). The stress history of sample front end was calculated by adding the incident stress wave to the reflected stress wave, while the stress history of sample back end equals the transmitted stress wave. It can be seen from the figure that the stress histories at both ends of the sample almost overlapped except during the initial stage of loading, indicating that the stress in the specimen was uniform under most load conditions.Figure 6
Dynamic stress equilibrium in the sand sample (test group number: CS9).After the dynamic stress equilibrium is satisfied in the sample, the strain-stress curve of the experiment calculated by equation (1) is valid. Following the similar procedure, the mean curves with error bars for samples of the same initial density are plotted (Figures 7(a)–7(c)). There are oscillations in the stress-strain curves at small strains. By comparing the stress from the stress-strain curves and the corresponding stress histories at both ends of the sample (Figure 6), it was concluded that the oscillations were caused by out-of-equilibrium stresses in the sand sample. Despite the stress nonequilibrium at the initial deformation stage, the results showed no significant differences under overlapping stress ranges with different strain rates, indicating that, in the strain rate range of 335 s−1 to 1253 s−1, the dynamic compression characteristics of dry calcareous sand exhibited no significant rate effect.Figure 7
Effect of strain rate on the dynamic axial response of calcareous sand. The average axial strain-stress curves of calcareous sand with initial densities of (a) 1.26 g/cm3, (b) 1.35 g/cm3, and (c) 1.42 g/cm3.
(a)
(b)
(c)After discarding the nonequilibrium stage of the experimental result, the axial strain-stress curves of the sample with different initial densities could be obtained (Figure8(a)). The stress-strain curves shift upwards with the increase of initial mass density. The unloading curve shows an initial steep reduction in stress without recovery of deformation, followed by a further reduction in stress with a slight recovery of deformation.Figure 8
Effect of initial density on the dynamic axial response of calcareous sand. (a) The axial strain-stress curves of calcareous sand. (b) Thee-log σ curves of calcareous sand.
(a)
(b)The rigid confined compression test is oedometric, and thee-lop σ curve is usually used to describe the deformation of the sample, where e is the void ratio of the sample and σ is the axial stress on the sample. In a one-dimensional compression test, the lateral strain is approximately zero when compared with the axial strain, so the void ratio is given by(2)e=GSρ−1,ρ=ρ01−εz,where ρ and ρ0 are the current and initial densities of the sample, respectively; Gs is the density of the soil particle; and εz is the axial strain.According to equation (2), the axial strain-strain experimental results shown in Figure 8(a) could be transferred into e-log σ curves (Figure 8(b)). As the sand particles are crushable, the initial skeleton of calcareous sand is unstable under high stresses. With increasing stress, the initial skeleton structure of the calcareous sand was gradually destroyed and a stable structure formed. So at a certain stress, the e-log σ curve is independent of the initial sample skeleton properties and the initial density of the sample. So, all the loading curves of calcareous sands with different initial densities in Figure 8(b) merge into a same straight line at an axial stress of about 40 MPa. The slope of the line is the compression index and is 0.239 for this calcareous sand under dynamic compression. Luo et al. studied the dynamic compression behaviour of quartz sand, and the compression index is 0.281 to 0.312 [15], indicating the calcareous sand is “softer” than quartz sand.


## 3.2. Deviatoric and Volumetric Behaviours of the Sand

For the sand sample confined by the sleeve within the elastic range, the lateral stressσr is given by(3)σr=12κrori2−1Esεθ,where ro and ri represent the outer and inner radii of the sleeve; Es is Young’s modulus of the sleeve; εθ is the circumferential strain on the sleeve; and κ is a correction coefficient for the effect of the smaller sample length compared with that of the sleeve, which could be derived from the numerical method [29, 30].Here, the correction coefficient is given by equation (4), which is obtained by numerical analysis [30], where l is the current thickness of the sample:(4)κ=0.0598l+0.0162.Figure9 shows the axial stress and lateral stress histories of calcareous sand in test group CS9. The lateral stress coefficient k0 is defined as(5)k0=σrσz,where σr is the lateral stress and σz is maximum the axial stress. To improve the precision of k0, it could be calculated using equation (5) by using the maximum values of the stresses. The mean experimental results of k0 in different test groups and corresponding error bars are summarised in Figure 10. The results show that the lateral stress coefficient increases as the axial stress increases.Figure 9
Typical axial stress and lateral stress histories of calcareous sand (test group number: CS9).Figure 10
Relationship between lateral stress coefficient and axial stress.In a confined SHPB experiment, the hydrostatic pressure (isotropic stress) componentp and equivalent von Mises stress σY are given by [15](6)p=13σz+2σr,σY=σz−σr.Based on equation (5), the aforementioned equations could be transformed to give(7)p=1+2k03σz,σY=1−k0σz.The Drucker–Prager yield function is usually used to describe the deviatoric behaviour of soils under low pressure, in which the von Mises stress is linearly dependent on hydrostatic pressure. Based on this yield function, the lateral stress coefficient is shown to be a constant according to equation (7), which is inconsistent with the experimental results from tests on calcareous sand. To describe why k0 increases as the axial stress increases, the Lundborg model is used to describe the deviatoric behaviour of calcareous sand, whose yield function is expressed as [31](8)σY=μp1+μp/σYmax,where μ and σYmax are the material constants. Substituting equation (7) into the above equation, the relationship between lateral stress coefficient and axial stress is given by(9)σz=2μ+3k0+μ−3μ1−k01+2k0σYmax.The material constantsμ and σYmax could be obtained by fitting the experimental results from equation (9), and the corresponding results are μ = 1.21 and σYmax = 363 MPa. The fitting curve is shown in Figure 10, and the yield function is shown in Figure 11.Figure 11
Lundborg yield surface of the calcareous sand.Based on the axial strain-stress experimental results (Figure8(a)), the volumetric behaviour of dry calcareous sand in the form of density-pressure relationship could be obtained by using equations (2), (7), and (9), and the results are shown in Figure 12. To describe how the slope of the curves increases as the pressure increases, the Murnaghan equation [32, 33] was used to model the volumetric behaviour of dry calcareous sand:(10)p=knkρρ0nk−1,where k and nk are the material constants, which could be obtained by fitting the equation with experimental data.Figure 12
Effect of initial density on the density-pressure relationship of dry calcareous sand.The experimental results shown in Figure12 show that the density-pressure curves of samples with different initial densities could be separated into two segments at the point where the density is 1.75 g/cm3. In the first segment, the curves are dependent on the initial density of the sample, whereas in the second segment, the curves are independent of the initial density of the sample; therefore, the curves should be fitted separately.For the first segment, the relationships between the material constants using data fitted to equation (10) with the initial sample density are summarised in Figures 13(a) and 13(b). The results show that the material constants are related to the initial density of the sample in a quasilinear manner as follows:(11)k=203.2ρ0−239.1,nk=14.4−6.4ρ0.Figure 13
Relationship between material constants and initial density of the sample. (a) Material constantk. (b) Material constant nk.
(a)
(b)The second segment of the density-pressure curve is independent of the sample initial density, soρ0 in equation (10) could be chosen as 1.42 g/cm3. The corresponding material parameter fitting results are nk = 5.46 and k = 50.1.In summary, the density-pressure relationship of dry calcareous sand with any initial densityρ0 could be summarised as(12)p=knkρρ0nk−1,ρ≤1.75g/cm3,50.15.46ρ1.425.46−1,ρ>1.75g/cm3,where the material constants k and nk could be calculated by using equation (11) and the comparison between experimental results and those arising from the use of equation (12) is shown in Figure 14. By comparing the calculation results and the test results shown in the figure, it can be seen that equation (12) could well describe the dynamic volumetric behaviour of dry calcareous sand.Figure 14
Comparison between experimental and calculated results. Comparison of (a) the first segment and (b) the second segment.
(a)
(b)


## 4. Conclusions

The dynamic mechanical behaviour of dry calcareous sand under rigid confinement provided by a thick-wall stainless steel sleeve was investigated using a split Hopkinson pressure bar (SHPB) at high strain rates (335 s−1 to 1253 s−1). A new sand specimen assembly by using Glyd Ring was developed, and the assembly relies on repeated dynamic compaction to prepare sand samples with precise initial mass densities. Sand samples with three initial mass densities (1.26 g/cm3, 1.35 g/cm3, and 1.42 g/cm3) were compressed at three nominal striker velocities (5 m/s, 10 m/s, and 20 m/s). A strain gauge was attached to the external surface of the sleeve to measure the circumferential strain therein, and hence, the deviatoric and volumetric behaviours of calcareous sand could be determined.Experimental results show that, in the strain rate range of 335 s−1 to 1253 s−1, the dynamic compression characteristics of dry calcareous sand exhibited no significant rate effect. The e-log σ curves of samples with different initial densities merge into one straight line whose slope is 0.239 at an axial stress of 40 MPa. The dynamic compression index of dry calcareous sand is much smaller than that of quartz sand, indicating that the dry calcareous sand is much “softer.” The lateral stress coefficient of calcareous sand increases with axial stress, so the Lundborg model could be used to describe its deviatoric behaviour. The density-pressure curves of dry calcareous sand could be separated into two segments when the density is 1.75 g/cm3; in the first segment, the curve is dependent on the initial density, whereas in the second segment, it is independent thereof. In both segments, the curve could be described by the Murnaghan equation.



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*Source: 3526727-2019-07-22.xml*