Static fatigue of sand particle - University of Toronto T ... · Journal: Canadian Geotechnical Journal Manuscript ID cgj-2017-0543.R2 Manuscript Type: Note Date Submitted by the

Draft

Static fatigue of sand particle

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2017-0543.R2

Manuscript Type: Note

Date Submitted by the Author: 27-Feb-2018

Complete List of Authors: Liu, Su; City University of Hong Kong, Department of Architecture and Civil Engineering Wang, Jianfeng; City University of Hong Kong,

Is the invited manuscript for consideration in a Special

Issue? : N/A

Keyword: Static fatigue, sand particles, strength degradation, uniaxial compressive load

https://mc06.manuscriptcentral.com/cgj-pubs

Canadian Geotechnical Journal

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1

Static fatigue of sand particle

Su Liu1 and Jianfeng Wang

2

1 Research Associate

Department of Architecture and Civil Engineering,

City University of Hong Kong, Hong Kong

2 Associate Professor

Department of Architecture and Civil Engineering,


Corresponding Author

Dr. Jianfeng Wang

Department of Architecture and Civil Engineering


Tel: (852) 34426787, Fax (852) 27887612

E-mail: [email protected]

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Abstract

Static fatigue is of particular concern in studying the time-dependent mechanical

behavior of brittle geo-materials. In this regard, the time-dependent strength behavior

of individual particles is essential for understanding the creep behavior of sand. In this

study, short-term strength test and static fatigue test of individual sand particles

subjected to uniaxial compressive load were carried out using a mini-loading

apparatus and modified oedometer frames, respectively. The sand particles in the

static fatigue test were loaded in an incremental manner, and the load at each stress

level was maintained for a fixed period of time. Scatter of the strength of sand

particles was described using the Weibull distribution. Long-term strength from the

static fatigue test of individual sand particles is found to be less than the short-term

strength.

Keywords: Static fatigue; sand particles; uniaxial compressive load; strength

degradation

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Introduction

Static fatigue (or delayed fracture) is the phenomenon of a material failing at

stresses smaller than the short-term strength after a period of constant loading

(Callister 2005). The short-term strength is measured from tests under an

instantaneous loading, whereas the long-term strength is determined from a failure

test of a specimen after a period of sustained loading. For granular materials, the

macro-scale long-term strength of an assembly of particles depends on many factors,

including particle rearrangement, interparticle friction, and the time-dependent

strength of individual particles at the micro-scale (Lade and Karimpour 2010). When

the loading stress is greater than the long-term strength and less than the short-term

strength of an individual particle, cracks become unstable and start propagating due to

the gas and moisture adsorption. A sudden fracture occurs when the crack has grown

to such a size that the applied stress can propagate a new surface (Wiederhorn et al.

2002). The time to delayed failure decreases as the loading stress increases.

The phenomenon of static fatigue has been observed in a few types of

homogenous and continuous materials (e.g., glasses and ceramics). About one third of

the short-term fracture stress is enough to produce delayed fracture in glass if the load

is maintained for a number of weeks (Orowan 1944). Water vapor corrosion is the

most important cause of the delayed fracture in glass (Wiederhorn et al. 2011). Based

on the Griffith's criterion, the time-dependent strength can be related to the

rate-dependent strength for ceramics. Therefore, a strength-probability-time (SPT)

diagram can be generated by simple measurements of the strain rate dependence of

fracture strength (Davidge et al. 1973). This theory has been adopted by Kwok and

Bolton (2013) in DEM simulations of the creep behavior of sand due to progressive

crush. The deformation during the time to delayed failure is negligible in glasses and

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ceramics, both of which are brittle materials.

Static fatigue is also found in natural earth materials (e.g., rocks and sands),

which usually contain multiple types of minerals and discontinuities. These properties

result in a higher variability in strength. For rocks, static fatigue is mainly caused by

the growth of pre-existing internal microcracks to a critical length (Erarslan and

Williams 2012). An exponential function was adopted by Schmidtke and Lajtai (1985)

to fit the time-stress data of Lac du Bonnet Granite in a log-log space. The existence

of a long-term strength given by the exponential function was further confirmed by

Damjanac and Fairhurst (2010) through in situ, laboratory and numerical analyses.

The macro-scale time-dependent behavior of sand samples, which stems mainly from

a repetitive cycle of particle crushing, interparticle contact fatigue, rearrangement of

particles and redistribution of contact stresses (Takei et al. 2001), has been reported

by a number of authors (Kuwano and Jardine 2002; Baxter and Mitchell 2004; Kiyota

and Tatsuoka 2006; Lade et al. 2009; Gao et al. 2012). The time-dependent particle

crushing was identified by comparing the grain size distributions obtained before and

after creep or stress relaxation tests (McDowell and Khan 2003; Lade and Karimpour

2010, 2016; Chen and Zhang 2016). Not until recently, the micro-scale process of

static fatigue at the silica sand contacts was revealed by scanning electron microscope

(SEM) observations (Michalowski and Nadukuru 2012; Wang and Michalowski 2015;

Michalowski et al. 2017). This contact fatigue results in the fracture of asperities and

a time-dependent increase of contact stiffness. However, in their studies, no effort was

made to investigate the fracture and the time-dependent strength behavior of

individual sand particles. In this regard, the objective of this work is to explore the

static fatigue phenomenon of individual sand particles subjected to uniaxial

compressive load. Single-particle crushing and static fatigue tests were carried out on

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Leighton Buzzard sand (LBS), a typical quartzitic sand, to yield understanding of the

time-dependent strength behavior of sand particles. The static fatigue tests on

individual grains were performed in a stable humidity environment, to minimize the

influence of water vapor. The original contribution of this study stems from an

experimental testing program on static fatigue behavior of a quartz sand leading to an

improved understanding of the time effects, especially from a statistical point of view,

of static fatigue on the strength of single sand particles.

Material and Method

Material Tested

The LBS particles used in this study are mainly composed of quartz. The grain

size ranges from 0.9 mm to 1.1 mm (Fig. 1). To minimize any possible shape effects

on the static fatigue behavior, particles with similar shapes were selected by

hand-picking. Nevertheless, a high variability in particle strength may still exist due to

the irregular particle shape and inhomogeneous mass distribution. This high

variability makes the behavior of individual particles very complicated, and results in

difficulties in static fatigue test due to the uncertainty of individual particle strength.

Short-term Strength Tests

A mini-loading apparatus developed by Zhao et al. (2015) was used for single

particle short-term strength test. The sand particle was loaded between two copper

platens under a quasi-static loading condition by moving the lower platen at a

displacement rate of 0.1 mm/min. In order to avoid unstable particle rotation during

loading, sand particles in both short-term strength and static fatigue tests were placed

and measured in the steadiest possible condition, under which the particle is in static

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equilibrium with the minimum gravitational potential. Force and displacement were

measured by a load cell and a linear variable differential transformer (LVDT),

respectively. Loading continued until the particle was fractured entirely.

Fig. 2 shows a typical force-displacement relationship obtained from one of 37

short-term strength tests. It can be seen that the force is roughly proportional to the

displacement with a slowly increasing slope of the curve, till a displacement of 0.043

mm, where a sudden drop of the force occurs as the particle is entirely fractured.

Jaeger (1967) defined the single particle strength as:

ff 2

F

dσ = , (1)

where Ff is the peak force causing the particle fracture (Nakata et al. 1999), d is the

particle diameter, defined as the initial distance between the platens before the start of

the particle crushing test. Fig. 3 shows the curves for the survival probability of a

particle under a given stress σf (and the corresponding force Ff) sufficient to cause

failure. Because the diameter of every testing particle is approximately 1 mm, the

curves almost overlap each other. About 56% of the data points fall within the range

of 30 N to 90 N.

Static Fatigue Tests

Static fatigue tests were carried out using the front-loading oedometer frame.

Particles were loaded between two copper cylinders (Fig. 4). LVDTs were read every

10 seconds to record the displacement. Therefore, the time to failure at the ultimate

failure stress could be determined with a precision of 10 s. A flow chart showing the

procedure of a static fatigue test is shown in Fig. 5.

As shown in Fig. 3, the highest particle strength (i.e., 208.6 MPa) is about ten

times greater than the lowest (i.e., 20.6 MPa) from a typical short-term strength test.

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Therefore, it is impossible to predict the strength of an untested LBS particle, and then

load it at a specific stress level proportional to it. A convenient way to obtain the

demanded data of delayed fracture is to load particles at a low stress level, which is

maintained for a fixed period of time. Those particles surviving are then loaded in an

incremented manner until all have failed. To avoid large times to delayed failure, the

step-loading approach was commonly used in performing static fatigue tests for

ceramics and glass materials with high variability in strength (e.g., Davidge et al.

1973; Davis and Mould 1984; Bermejo et al. 2008). It should be noted that by using

this step-loading approach, the load is not steady throughout the loading process.

In the 1st stage of the test, the load on the particle is continuously increased from

0 N to 29.43 N (i.e., corresponding to 300 g of weight on the hanger of oedometer

frame) at an interval of 4.905 N. The duration between load increments within this

stage is negligible. According to Fig. 3, about 86.1% of particles would survive at a

load of 29.43 N. Then, the survived particles are stressed at 29.43 N for 12 hours

overnight in the 2nd stage. In the 3

rd stage, the load placed on the survived particles is

increased at an interval of 2.4525 N or 4.905 N for up to 24 times. The load is

sustained for half an hour at each stress level, leading to a total duration of 12 hours in

this stage. In the last stage, in order to save testing time, the load on the particle is

continuously increased at an interval of 0.981 N until the particle is entirely fractured.

This rapid loading would cause an instantaneous failure of the particle.

Results and Discussion

A typical load-time and displacement-time relationship from a static fatigue test

are shown in Fig. 6. For this test, the load was increased with an interval of 4.905 N in

the 3rd stage. The particle was fractured at an ultimate load of 68.67 N after sustaining

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the load for about 370 s.

Two groups of static fatigue tests were conducted simultaneously using two

modified oedometer frames at a temperature of 22±1 oC and a relative humidity of

70±2 %. In the 3rd stage, the load interval used for one group was 2.4525 N (SF-25),

and 4.905 N for the other group (SF-50). Assuming two identical sand particles are

loaded simultaneously, the different interval loads would lead to different ultimate

failure loads and different amount of time to the delayed failure in the 3rd stage.

Detailed test results from both groups are listed in Table 1. A failure time of zero

means the sample undergoes an instantaneous failure at the ultimate failure stress. The

probability of survival under a given stress σf for particles in both groups are then

compared with that from short-term strength tests in Fig. 7. Note, that the data from

tests with instantaneous failure mode are also included, because excluding these data

from the 4th stage (i.e., data from particles with higher strength), would result in an

unrealistic distribution of survival probability. The data for the short-term strength

tests in Fig. 7 is the same as that in Fig. 3. The obvious reduced strength for the two

groups of static fatigue test clearly suggests the dependence of the particle strength on

the loading time. The loading rate of SF-25 can be regarded as lower than that of

SF-50. This lower rate of loading leads to an overall lower distribution of the static

fatigue strength (σf) for SF-25.

The Weibull distribution, which is widely used to describe the scatter of the

strength in brittle materials, can be written as:

fs

f 0

exp

m

Pσσ

= −

, (2)

where Ps is the survival probability of a particle under a stress σf, σf0 is the

characteristic stress where 37% of the particles survive, and m is the Weibull modulus,

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which is used to describe the scatter of the strength in brittle material and increases

with the decreasing variability in particle strength. σf0 is 77.4 MPa, 74.7 MPa and 63.5

MPa for short-term strength, SF-50 and SF-25 tests, respectively. Equation (2) can be

rewritten as

f

s f 0

1ln ln lnmP

σσ

=

. (3)

The data from both short-term strength and static fatigue tests are plotted using

equation (3) in Fig. 8. The Weibull modulus m for the short-term strength test is 2.16,

which is lower than the value of about 3 reported by Wang and Coop (2016) for the

same type of LBS but with a larger particle size. The higher value of m for the static

fatigue test (i.e., m=2.34 for SF-50 and m=2.87 for SF-25) indicates a decreasing

variability in strength due to the time effects.

The current study does not allow us to explore the micro-scale mechanisms of

static fatigue of sand particles. From the observations of X-ray CT scanning (Zhao et

al. 2015), there is no obvious initial internal microcrack in LBS particles. However, a

rich surface texture, which indicates a distribution of initial surface cracks, is

observed in the scanning election microscope (SEM) images (Wang and Michalowski

2015) of quartz sand. The time-dependent micro-fracturing of asperities has been

observed under constant oblique force, leading to a decrease in roughness and an

increase in stiffness (Michalowski et al. 2017). The phenomenon of delayed fracture,

therefore, could be the result of combined surface and internal crack propagation, at

different load levels.

Conclusions

This study endeavors to explore the static fatigue behavior of sand particles. A

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mini-loading apparatus and two modified oedometer frames were used to conduct the

single particle short-term strength and static fatigue tests, respectively. The

phenomenon of static fatigue does exist in the constant loaded individual LBS particle.

A clear tendency for strength degradation is found by comparing the data from static

fatigue tests with those from short-term strength tests. Moreover, static fatigue tends

to have a larger influence on lowering the strength when the particle has a higher

short-term strength. This results in a decreasing variability in strength due to the time

effects.

Acknowledgements

The study presented in this article was supported by the General Research Fund

CityU122813 from the Research Grant Council of the Hong Kong SAR, National

Science Foundation of China (NSFC) grant No. 51779213 and Shenzhen Basic

Research Grant No. JCYJ20150601102053063.

References

Baxter, C. D. P., and Mitchell, J. K. 2004. Experimental study on the aging of sands.

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Bermejo, R., Torres, Y., Anglada, M., and Llanes, L. 2008. Fatigue behavior of

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Callister, W. D. 2005. Fundamentals of materials science and engineering: an

integrated approach, 2nd edn. Hoboken: Wiley.

Chen, X., and Zhang, J. 2016. Effect of load duration on particle breakage and dilative

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behavior of residual soil. Journal of Geotechnical and Geoenvironmental

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crystalline rock. Rock Mechanics and Rock Engineering, 43(5): 513-531.

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relationships in ceramics. Journal of Materials Science, 8(12): 1699-1705.

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relationship to the fracture toughness of rocks. International Journal of Rock

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219-227.

Kiyota, T., and Tatsuoka, F. 2006. Viscous property of loose sand in triaxial

compression, extension and cyclic loading. Soils and Foundations, 46(5):

665-684.

Kuwano, R., and Jardine, R. 2002. On measuring creep behaviour in granular

materials through triaxial testing. Canadian Geotechnical Journal, 39(5):

1061-1074.

Kwok, C. Y., and Bolton, M. D. 2013. DEM simulations of soil creep due to particle

crushing. Géotechnique, 63(16): 1365-1376.

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Lade, P. V., and Karimpour, H. 2010. Static fatigue controls particle crushing and

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ageing of silica sand. Géotechnique, https://doi.org/10.1680/jgeot.16.P.321.

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List of Tables

1 Static fatigue data for LBS.

List of Figures

1 Photograph of LBS particles.

2 A typical load-displacement relationship from a short-term strength test.

3 Probability of survival considering peak stress and peak force.

4 A modified oedometer frame for static fatigue test.

5 Flow chart showing the procedure of a static fatigue test.

6 (a) Load-time and (b) displacement-time relationship of a static fatigue test.

7 Probability of survival for static fatigue and short-term strength tests.

8 Comparison of the data with Weibull function.

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Fig. 1 Photograph of LBS particles

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Fig. 2 A typical load-displacement relationship from a short-term strength test

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Force: N

Displacement: mm

Ff

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Fig. 3 Probability of survival considering peak stress and peak force

0 50 100 150 200 250

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250

Ff: N

Survival probability

σf: MPa

Stress

Force

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Fig. 4 A modified oedometer frame for static fatigue test

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Fig. 5 Flow chart showing the procedure of a static fatigue test

Yes No

4th stage,

quickly increase

the load until the

particle is

fractured.

Increase the

load by 0.981N

Fractured? Instantaneous

failure

2nd stage,

load=29.43N

3rd stage,

29.43N<load≤88.29N for SF-25

29.43N<load≤147.15N for SF-50

Yes

No

No

Yes

No

Yes

Sustain the

load for 0.5 h

Static

fatigue

Fractured?

Fractured during 0.5 h?

Increase the load by

2.4525N or 4.905N

Instantaneous

failure

Sustain the load for 12 h, fractured

during 12 h?

Static

fatigue

Yes

No

1st stage

load<29.43N

Increase the load by

4.905N till load=29.43N

Start

Load the sand particle

with 4.905N of load

FracturedInstantaneous

failure

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(a)

(b)

Fig. 6 (a) Load-time and (b) displacement-time relationship of a static fatigue test

0

20

40

60

80

0 4 8 12 16

Load: N

Time: hour1st stage

2nd stage3rd stage

0

0.01

0.02

0.03

0.04

0 4 8 12 16

Displacement: mm

Time: hour1st stage

2nd stage

3rd stage

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Fig. 7 Probability of survival for static fatigue and short-term strength tests

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300

Survival probability

σf: MPa

Short-term

SF-25

SF-50

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Fig. 8 Comparison of the data with Weibull function

-5

-4

-3

-2

-1

0

1

2

3

-2 -1 0 1

ln[ln(1/P

s)]

ln(σf/σf0)

Short-term

SF-25

SF-50

Linear fit of short-term, m=2.16

Linear fit of SF-25, m=2.87

Linear fit of SF-50, m=2.34

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Table 1 Static fatigue data for LBS

No.

Load increased by 2.4525 N Load increased by 4.905 N

d (mm) Load (N) Failure

time (s) d (mm) Load (N)

Failure

time (s)

1 0.94 107.9100 20 1.10 98.1000 180

2 1.02 36.7875 590 0.92 29.4300 0

3 0.96 134.8875 0 0.95 68.6700 370

4 1.06 41.6925 300 1.00 68.6700 610

5 0.93 31.8825 880 1.09 103.0050 690

6 1.05 101.0430 0 1.03 122.6250 10

7 1.01 66.2175 700 0.97 39.2400 180

8 0.96 29.4300 0 1.00 68.6700 620

9 1.00 107.9100 0 0.98 34.3350 80

10 0.98 29.4300 21560 0.97 53.9550 160

11 1.10 88.2900 130 1.06 117.7200 400

12 1.09 73.5750 1060 1.08 29.4300 21620

13 0.96 41.6925 10 1.08 53.9550 1780

14 1.09 56.4075 460 0.96 83.3850 840

15 1.03 44.1450 640 1.04 73.5750 910

16 1.03 56.4075 550 1.02 44.1450 1720

17 1.00 29.4300 40 0.95 157.9410 0

18 1.06 66.2175 1150 0.96 53.9550 10

19 1.00 63.7650 820 0.96 98.1000 300

20 0.97 46.5975 760 0.95 78.4800 1660

21 1.01 112.8150 0 0.93 34.3350 170

22 1.05 85.8375 970 1.02 39.2400 30

23 0.93 61.3125 860 1.02 49.0500 860

24 1.01 39.2400 40 1.00 73.5750 1340

25 0.99 46.5975 220 0.9 53.9550 10

26 0.95 73.5750 1000 1.02 44.1450 10

27 0.96 19.6200 0 0.90 882.9000 30

28 0.93 36.7875 30 0.98 93.1950 540

29 1.04 83.3850 500 1.02 29.4300 10

30 1.02 63.7650 60 0.99 68.6700 10

31 0.99 29.4300 530 0.90 49.0500 130

32 0.92 58.8600 220 1.05 122.6250 110

33 0.99 76.0275 1090 1.00 98.1000 230

34 1.01 46.5975 1070 1.04 29.4300 0

35 1.06 80.9325 10 0.97 88.2900 80

36 0.97 46.5975 1780 1.00 137.3400 30

37 1.00 58.8600 20 1.01 63.7650 1450

38 1.05 34.3350 110 0.94 63.7650 10

39 1.01 41.6925 960 0.99 14.7150 0

40 1.06 51.5025 370 0.99 29.4300 940

41 0.94 76.0275 280 1.06 49.0500 390

42 0.97 73.5750 30

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Static fatigue of sand particle - University of Toronto T ... · Journal: Canadian Geotechnical Journal Manuscript ID cgj-2017-0543.R2 Manuscript Type: Note Date Submitted by the