Journal of Rock Mechanics and Geotechnical Engineering. 2011, 3 (Supp.): 415–420

Investigation on relations between grain crushing amount and void ratio change of granular materials in one-dimensional compression and creep tests Zhechao Wang1, Ron. C. K. Wong2, Liping Qiao3* 1 Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, 250061, China 2 Department of Civil Engineering, University of Calgary, Calgary, T2N1N4, Canada 3 Department of Engineering Mechanics, Shandong University, Jinan, 250061, China

Received 16 August 2010; received in revised form 3 May 2011; accepted 8 October 2011

Abstract: Grain crushing plays an important role in one-dimensional (1D) compression and creep behaviors of granular materials under high stress. It is clear that the macro-properties of granular materials are closely related to the micro-fracture properties of grains in 1D compression and creep tests. In this paper, a series of 1D compression and creep tests were performed on Ottawa sand to investigate the deformation and grain crushing properties of granular materials, and it shows that the void ratio is correlated to the grain crushing amount (the quantity of crushed grains) for granular materials subjected to grain crushing. The test results, combining with the existing test data related to grain crushing of granular materials, were used to verify the relation. Moreover, the implications of these relations on the yield of granular material, and the equivalent effect of stress and time in changing soil fabric are presented. Key words: granular material; grain crushing; geometrical aspect; one-dimensional (1D) compression and creep; yield; soil fabric

1 Introduction

In geotechnical engineering, it is observed that deep

shafts may experience earth pressures up to 70 MPa, and soils under the tips of deep driven-pile foundations may experience stress up to 350 MPa [1]. Meanwhile, in well drilling and petroleum extraction, soils experience extremely high pressures [2]. For these problems, it is important to understand the instantaneous and long-term behaviors of soils under high stresses.

When granular materials are subjected to high stresses, the deformation is usually accompanied with grain crushing. It has been found that the mechanical properties of granular materials depend on the integrity of grains or grain crushing amount in the materials [3–8]. For instance, the yield stress of

Doi: 10.3724/SP.J.1235.2011.00415 *Corresponding author. Tel: +86-531-88392448; E-mail: qiaolp@sdu.edu.cn Supported by Natural Sciences and Engineering Research Council of Canada, Alberta Energy Research Institute and the Department of Civil Engineering at University of Calgary

granular soil in 1D compression corresponds to the stress, at which grain crushing occurs in the soil [7]. The dilatational component of friction strength of dense sand decreases with the mean effective stress due to the fact that the grain crushing amount increases with the applied stresses at high stress levels [4]. The change in permeability of sands is related to the grain crushing amount in sands [9, 10]. In addition, Leung et al. [9] investigated the role of grain crushing in pile creep in sand using physical model tests. They conducted 1D creep tests on the remodeled sand, and found that the grain crushing amount increases with the duration of creep test.

Energy method has been used to theoretically investigate the effect of grain crushing on strength and deformation of granular materials [11–13]. In the pioneer work by McDowell and Bolton [14], the energy release in grain crushing was quantified as the surface energy stored in the newly created surfaces or fracture surfaces of grains. In this manner, a micromechanical 1D compression model was proposed, in which the compression index is expressed in terms of micro-fracture parameters of grains. On the other hand, numerical techniques, e.g. discrete element

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method, are widely used to numerically investigate the influence of grain crushing on the macro-properties of granular materials [15, 16]. It has been found that the yield of granular material coincides with the onset of grain crushing, and the yield stress is proportional to the tensile strength of all grains statistically [15]. By varying the average grain size of a material in simulations, Shi et al. [13] found that the higher the average grain size is, the lower the yield stress of the material is.

However, there are few studies on investigation of geometrical quantities of granular material subjected to grain crushing. In this paper, a series of 1D compression and creep tests are performed on Ottawa sand. The grain size distribution curves of the samples after 1D compression and creep tests are obtained. Hereby, the void ratio change is correlated to the grain crushing amount for granular materials. The existing 1D compression test data related to grain crushing are collected to verify the relation. The implications of the relations on the yield of granular materials, equivalent effect of stress and time in changing soil fabric are discussed as well.

2 1D compression and creep tests on Ottawa sand 2.1 Test material, setup and program

The material used in the tests is unground ASTM 20/30 silica sand, which was mined at Ottawa, Illinois, USA. In the grain solid, the content of silicon dioxide is around 99.8%. The sand has a specific gravity of 2.65. A batch of grains with the size in the range of 0.595 to 0.841 mm was used to prepare samples.

To confine samples, a special apparatus was fabricated. As shown in Fig.1, the apparatus consists of a confining cell, a pedestal, two compression plates, a loading guider and a piston. The confining cell has a diameter of 6.35 cm and a height of 2.54 cm. The vertical force is applied by using material test system (MTS), with a capacity of 100 kN. An external laser displacement transducer is used to measure the displacement of samples. The accuracy of the transducer is 0.001 mm.

The sand samples were prepared by using the air pluvial method. The initial void ratio of the samples is about 0.543. A loading rate of 10 kN/min was used in the tests. After each 1D compression and/or creep test, the sample was collected for grain size analysis. The

Fig.1 Compression apparatus for 1D compression tests.

grain size analysis was conducted by using a stack of sieves consisting of opening size No.30, 35, 45 and 60 (0.595, 0.500, 0.354 and 0.250 mm), respectively. A scale of high accuracy (±0.001 g) was used to measure the mass of grains from each sieve. The 1D creep tests on Ottawa sand were conducted at three stress levels, i.e. 10, 18 and 28 MPa. The durations of tests at each stress level were 15, 30, 60 and 120 minutes. 2.2 1D compression test results

The bilinear relation between grain crushing amount and vertical stress of Ottawa sand in 1D compression tests is shown in Fig.2. The grain crushing amount is the percentage of the mass of grains passing through the sieve No.30 (0.595 mm) out of the mass of the whole sample. In low stress, the grain crushing amount is quite low or undetectable, while in high stress, the grain crushing amount is high and increases with increasing vertical stress. The transition of the two conditions occurs at the vertical stress of about 10 MPa.

Fig.2 Bilinear relation between grain crushing amount and

vertical stress.

Figure 3 shows the grain size distribution curves of samples before and after tests at the vertical stresses of 10, 18 and 28 MPa. It can be seen that the grain

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Fig.3 Grain size distribution curves of Ottawa sand after 1D

compression tests.

crushing amount increases with increasing vertical stress. Figure 4 shows the relation between void ratio and grain crushing amount. It is shown that, in 1D compression, the higher the grain crushing amount is, the lower the void ratio of the sand is. The test results can be fitted into a linear equation:

Fig.4 Relation between void ratio and grain crushing amount of Ottawa sand.

c0.529 0.5e (1)

where e is the void ratio and c is the grain crushing amount. Equation (1) indicates that the void ratio at yield of Ottawa sand is 0.529, at which the grain crushing amount is quite low or undetectable. It should be noted that a detailed derivation of Eq.(1) from fracture mechanics and fractal geometry can be found in Refs.[17, 18].

Figure 5 shows the compression curves of Ottawa sand. Based on the predicted void ratio at yield by Eq.(1), the yield stress is estimated to be about 10 MPa, at which the compression index of the sand increases slightly.

(a)

(b)

Fig.5 Compression curves of Ottawa sand in 1D compression.

2.3 1D creep test results

Figure 6 shows the creep curves of Ottawa sand subjected to different vertical stresses. The legends in Fig.6 denote the applied vertical stress and the elapsed time for creep stage. The creep rate of Ottawa sand decreases with time, but increases with vertical stress. In terms of axial strain, the average creep rates in the first 30 minutes are about 6.3×107, 9.3×107 and 9.3×106 per minute in the tests at 10, 18 and 28 MPa, respectively. These creep rates are much higher than those observed in silica sandstone samples by McDowell and Khan [19]. In their uniaxial tests, it was found that the average creep rate of silica sandstones is from 5.0×1011 to 2.5× 1010 per minute at the compressive stress of 45 MPa. This comparison indicates that the creep deformation observed in this study is mainly due to the grain crushing, rather than the creep in solid grains themselves.

Figure 7 shows the grain size distribution curves for samples at different vertical stresses. It is found that the grain size distribution curves after 1D creep tests are similar to those after 1D compression tests.

(a)

0

1

2

3

4

5

7

0 30 60 90 120

Time (min)

Dec

reas

e in

voi

d ra

tio

(10

5 )

10 MPa, 15 min 10 MPa, 30 min 10 MPa, 60 min 10 MPa, 120 min 18 MPa, 15 min 18 MPa, 30 min 18 MPa, 60 min 18 MPa, 120 min

6

Sieve opening (mm)

c0.529 0.5e

Voi

d ra

tio,

e

Grain crushing amount, c (%)

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(b)

Fig.6 1D creep curves of Ottawa sand at different vertical

stresses.

(a)

(b) Fig.7 Grain size distribution curves of Ottawa sand in 1D creep tests for different periods of elapsed time at vertical stresses of 10, 18 and 28 MPa.

The relation between the creep rate in terms of void ratio and grain crushing rate of Ottawa sand in 1D creep tests is fitted by using a linear function:

v ce C (2)

where e is the creep rate in terms of void ratio; c is the grain crushing rate, which is defined as the grain

crushing amount divided by creep duration; and vC is the coefficient between the creep rate and grain crushing rate and termed as viscous crushability index.

Figure 8 shows the relation between creep rate and grain crushing rate of Ottawa sand in 1D creep tests. It is concluded that the creep rate of Ottawa sand is linearly proportional to the grain crushing rate in 1D creep tests. However, it is noted that the conclusion is drawn for the tests performed at the stress levels beyond the yield stress of Ottawa sand, i.e. 10 MPa.

Fig.8 Relation between creep rate and grain crushing rate of

Ottawa sand in 1D creep tests.

Figure 9 shows the relations between grain crushing amount and vertical stress in 1D compression and creep tests. The bilinear relation between the grain crushing amount and vertical stress in the compression behavior was also found in the creep behavior of Ottawa sand. In the condition of low stresses (<10 MPa), the grain crushing rate is small or undetectable; while in the condition of high stresses (10 MPa), the grain crushing rate increases with both the vertical stress and time. In the creep tests, it is observed that the creep rate and grain crushing rate change significantly beyond the yield stress. Therefore, the yield stress of granular soil marks the abrupt onset of increase in the grain crushing amount on level III, as suggested by Mesri and Vardhanabhuti [6].

Fig.9 Relations between grain crushing amount and vertical

stress in 1D compression and creep tests on Ottawa sand.

0

1

2

3

4

5

6

7

0 30 60 90 120 Time (min)

Dec

reas

e in

voi

d ra

tio

(10

5 )

28 MPa, 15 min

28 MPa, 30 min

28 MPa, 60 min

28 MPa, 120 min

0

1

2

3

4

5

0 5 10 15 20 25 30

Vertical stress (MPa)

Gra

in c

rush

ing

amou

nt (

%) Compression

Creep, 15 min Creep, 30 min

Creep, 60 min Creep, 120 min

0 1 2 3 4

6.0

4.5

3.0

1.5

0.0

Grain crushing rate, c (106 min1)

c

2

0.123 8

( 0.908 4)

e

R

Cre

ep r

ate,

e (

105

min

1)

.

0.0

0.5

1.0

1.5

2.0

0.2 0.3 0.4 0.5 0.6

Sieve opening (mm)

Per

cent

fin

er b

y w

eigh

t (%

)

Initial 10 MPa, 0 min

10 MPa, 15 min

10 MPa, 30 min

10 MPa, 60 min

10 MPa, 120 min

18 MPa, 0 min

18 MPa, 15 min

18 MPa, 30 min

18 MPa, 60 min

18 MPa, 120 min

0

1

2

3

4

5

0.2 0.3 0.4 0.5 0.6

Sieve opening (mm)

Per

cent

fin

er b

y w

eigh

t (%

) Initial 28 MPa, 0 min

28 MPa, 15 min

28 MPa, 30 min

28 MPa, 60 min

28 MPa, 120 min

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3 Verification of fitting equation The 1D compression test data of granular materials

in literatures were collected to verify Eq.(1), as listed in Table 1. The materials in Table 1 cover from highly crushable petroleum coke [19] and pasta [17] to quartz sands [1, 3]. Figure 10 compares the predictions from Eq.(1) with the test data in literatures. The predictions are consistent with those test data.

Table 1 1D Compression test data of granular materials related to grain crushing.

Reference Material Grain shape

Grain size (mm)

Initial void ratio

Void ratio at yield

Eq.(1) Empirical

Roberts and de Souza

[1]

Ottawa sand (OS1)

Round 0.425–0.85 0.6 0.6 0.55

Hagerty et al. [3]

Ottawa sand (OS2)

Round 0.425–0.85 0.7 0.63 0.61

Hagerty et al. [3]

Slag (BBS) Angular 0.425–0.85 0.8 0.79 0.69

Bard [17] Petroleum coke (PC)

Angular 5–10 2.3 2.02 2.1

Nakata et al. [5]

Silica sand (SS1)

Angular 1.4–1.7 0.63 0.62 0.54

Nakata et al. [5]

Silica sand (SS2)

Angular 0.25–2 0.6 0.61 0.48

McDowell and Khan

[19] Pasta (PS) Flaky 6.25–15 2.40 1.63 2.2

This study Ottawa

sand (OS3) Round 0.591–0.85 0.543 0.53 0.53

Fig.10 Relations between void ratio and grain crushing amount

of different granular materials in 1D compression tests.

Table 1 also presents the void ratios at yield estimated by using the Casagrande empirical method, in which the yield point corresponds to the maximum curvature point of the relation curve of void ratio versus logarithmic vertical stress. The estimated void ratios at yield of OS2 and PC are close to those predicted by Eq.(1). However, the estimated void ratios at yield of other materials by using the empirical method are lower than those predicted by Eq.(1).

There is few data collected in literatures on the deformation and grain crushing properties of granular materials.

4 Implications 4.1 Yield of granular materials

The damage of grains could be classified into three levels: (1) level I: abrasion or grinding of grain surface, (2) level II: breakage of grain corners and edges, and (3) level III: fracturing, splitting or shattering of grains at different stress levels, according to Mesri and Vardhanabhuti [6]. For granular materials, the yield stress is commonly defined at the maximum curvature point of relation curve of void ratio versus logarithmic vertical stress, which marks the abrupt onset of level III damage. However, for some granular soils, it is observed that there is no significant change in compressibility, although level III damage of grains occurs in the soils. In the latter situation, it was suggested that the yield of soil should be defined according to the onset of level III damage [6].

In this study, it was observed that there is no significant change in compressibility before and after grain crushing occurs. However, based on the bilinear relation between grain crushing amount and vertical stress, the yield stress could be determined to be about 10 MPa. Beyond this yield stress, the damage of grains is of level III and the grain crushing amount increases significantly. For other materials in Table 1, it is also shown that the onset of level III damage could provide a more general definition for the yield of granular material than the point at the maximum curvature of relation curve of void ratio versus logarithmic vertical stress [15, 18, 20]. 4.2 Equivalent effect of stress and time

As shown in Fig.9, under higher stresses (10 MPa), the grain crushing amount increases with both the vertical stress and time. Thus, it is expected that, for any time in a creep test, there is a corresponding vertical stress, at which the grain crushing amount is the same as that in the creep test. That is, the effects of time and vertical stress are equivalent in changing the soil fabric from one state to the other state, characterized by the same grain size distribution. Therefore, the equivalent effect substantiates the same potential of plastic and creep flow rules proposed by Lade and Liu [21]. It is important to note that the equivalent effect is only valid for the test materials and testing conditions used in this study.

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5 Conclusions

In this study, a series of 1D compression and creep tests have been performed to investigate the deformation and grain crushing properties of granular materials. The void ratio is correlated to the grain crushing amount for granular materials in 1D compression and creep tests. From this study, the following conclusions could be drawn:

(1) A bilinear relation is found between the grain crushing amount and vertical stress in 1D compression and creep tests. In the condition of low stress, the grain crushing amount is quite low or undetectable, while in the condition of high stress, the grain crushing amount is high and increases with increasing vertical stress. The transition of the two conditions occurs at about 10 MPa. At low stress levels (<10 MPa), most damage of grains in the sand samples is of levels I (abrasion or grinding of grain surface) and II (breakage of grain corners and edges), while at high stresses (10 MPa), damage of grains is of level III (fracturing, splitting or shattering of grains).

(2) In 1D compression, the higher the grain crushing amount is, the lower the void ratio of the sand is. For the tested Ottawa sand, the change in void ratio is linearly proportional to the amount of damaged grains in 1D compression tests.

(3) The grain size distribution curves after 1D creep tests are similar to those after 1D compression tests.

(4) The yield stress is estimated to be about 10 MPa, which marks the abrupt onset of the increase in the amount of damaged grains on level III.

(5) In 1D compression and creep tests, the effects of time and vertical stress are equivalent in changing granular soil fabric from one state to the other state, characterized by the same grain size distribution.

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