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Final Report

Vibration Induced Liquefaction of a Granular Material

Group #7 2 May 2007

ME 241

Gerald AbtGlenn DillerHeather Howard

Abstract

We investigated properties of a liquefied granular material subjected to horizontal

shaking. The density of the glass beads we used was = 2523 kg/m3; upon liquefaction, we

found an average density of = 2021 kg/m3. We found that viscosity of the liquefied beads

varies linearly with angular frequency, with the relation (f) = -349.7f + 6722.76 where

frequency f is in rad/s and viscosity is in kg/ms


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Introduction

Liquefaction is the process by which a granular material transforms from a solid state to a

liquid state. Because of its connection with seismic activity, the liquefaction of soils has long

been an area of interest. Many of the catastrophic effects of earthquakes are related to the

behavior of liquefied sand. Several reports have been published describing the onset of

liquefaction; the causes of the phenomenon are generally understood. However, to our

knowledge very little research has been done on the properties of fluidized soil. Our goal was to

determine the effective density of liquefied sand when exposed to various frequencies of

vibration, using a system of small glass beads subjected to horizontal shaking as a model for

earthquake-induced vibrations.

Last year, a group of students in ME 241 investigated the conditions required for the

onset of liquefaction [1]. They used a dimensionless parameter, , to measure this critical point.

is the dimensionless acceleration and is defined by

g

2= (1)

where A is the amplitude, is the angular frequency, and g is the gravitational acceleration.

Their determination of the critical value for liquefaction corroborated the work done by

Metcalfe et al [2]. We built on the work of these groups to test the properties of a liquefied

medium.

Formulation & Procedure

We had originally planned to use a peanut can filled partially with lead shot to determine

the density of the liquefied beads. However, initial experimentation showed that when the

shaker table was turned on the can would sink into the beads at an angle. This would have made

calculations of the submerged volume extremely difficult so we decided to change the object

used to a ping-pong ball (Figure 1). This allowed us to determine the volume of the submerged

portion of the ball, no matter how it sank into the beads, using the equation for the volume of a

spherical cap

=

3

2 hrhVsubmerged (2)


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where Vsubmergedis the submerged volume, h is the height of the submerged portion of the ball,

and ris the balls radius. We filled the ping-pong ball with lead shot to vary the effective density

of the ball.

After we found the volume of the ball that was submerged under the surface, we

determined the density of the liquefied beads using Archimedes Principle (3),

SubmergedB gVF = (3)

whereFb is the buoyant force, is the effective density of the fluid,gis gravitational

acceleration and VSubmergedis the volume submerged.

Our shaker table was initially set up to simply run from an input voltage. To allow us to

directly control frequency from LabVIEW, we determined the voltage-frequency relationship

using a stopwatch to time ten rotations at each voltage (Figure 2). This allowed us to modify

our LabView virtual instrument for frequency input, rather than voltage input (Figures 3, 4). We

then calibrated a Linear Variable Displacement Transducer (LVDT) to determine the relationship

between LVDT voltage and the displacement of the shaker tables moving platform (Figure 5).

The LVDT output a plot of displacement versus time to the computer, and enabled us to

determine the amplitude of shaking. Using frequency and the amplitude calculated with the

LVDT, we were able to quantitatively determine when we reached cr.

To measure viscosity, we attached the LVDT vertically to the fish tank using C Clamps

(Figure 6). The LVDT was then screwed into a nut that had been glued to a ping-pong ball. We

modified our LabView setup to output displacement and time as the ball sank, allowing us to

obtain data for the rate at which the ball sank into the beads. Initially we performed the test by

setting the ball on top of the sand and using the flattest portion of the displacement curve to

calculate velocity. We then used Stokes Law (4) for a sphere sinking in an infinite fluid [3]

gagVVa d )3

4(6 3 =+ (4)

where is the viscosity, is the density of the material in which the object is sinking, Vis the

terminal velocity of the sphere, a is the radius of the sphere, andgis the acceleration of gravity.

However, the viscosities we obtained from this method were extremely high, causing us to

question the accuracy of our results.

We then tried the same test on a substance for which we knew the viscosity, 10,000

centistokes silicone oil, to estimate the error involved in our procedure. Unfortunately, the tests


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with the oil failed to give us the expected viscosity. To improve our method, we modified our

starting position so that the entire ping-pong ball began fully submerged in the liquid. This was

done to better simulate an infinite fluid, bringing the conditions of the test closer to those that can

be modeled by the Stokes flow equation. This test produced a result of 4227.31 centistokes for

the oil. After comparing the 4227.31 cS found to the correct 10,000 cS, our tests appeared to be

off by a factor of 2.37. We then tested the glass beads with an initially submerged ball and found

the viscosities of the beads at different frequencies. Then we adjusted these values with the error

factor of 2.37 to predict the actual viscosity of the beads.

Results

Using the angular frequency of the table and the amplitude of shaking, we were able to

calculate the critical value of our system. Based on a qualitative observation of when the top

layer of beads appeared to liquefy, we found crto be 0.57. This compares favorably with the

value of cr= 0.52 found by Metcalfe et al [2], as liquefaction could possibly have occurred at a

lower frequency then we observed, which would have brought our value closer to 0.52.

To evaluate whether our values for density of the liquefied beads made sense, we first

found the density of the stationary beads. We used an optical microscope to measure the

diameters of ten individual beads and found an average diameter of 1.133 mm. We weighed

beads individually and used our data to calculate a density of = 2523 kg/m3

for the beads

themselves.

With the density of the stationary beads determined, we began to measure the density of

the liquefied beads. Using balls of mass 43.43 g (Ball G) and 19.36 g (Ball B), we found the

densities to be = 2103 kg/m3

and = 1950 kg/m3, respectively (Figures 7, 8). This resulted in

an average density of = 2021 kg/m3. The standard deviation was 165 kg/m

3, which is small in

comparison to the average value (about 8%).

We also showed that a low-density object experienced upward motion when placed

beneath the surface of the beads. To demonstrate this, we placed an empty ping-pong ball on the

bottom of the tank, underneath the beads. When the beads were fully liquefied, the ball rose

quickly to the top and remained on the surface.

Our final viscosities ranged from 465.95 to 1517.40, varying linearly with frequencies

ranging from 15 Hz to 18 Hz. We used Microsoft Excels trendline option to find a relationship


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(f) = -349.7f + 6722.76 (Figure 9) between the angular frequency f and the viscosity . The

RMS error value was approximately 56.96.

Conclusions/Discussion

The average density of the liquefied beads we measured is significantly less than that of

the stationary beads. This makes sense as, with the notable exception of water, the density of

materials in their liquid state is less than the density of the solid state. As shown in Figures 7

and 8, there was a significant variation in the measured density of the liquefied beads across the

range of frequencies tested. This was mostly due to the human error in our measurement method

of using a ruler to determine how much of the ball was initially and finally submerged in the

beads. The smallest scale on the ruler was in millimeters, and even a small error in the

measurement would have had a significant effect on our calculations when the ball itself was

only 38 mm in diameter. However, we discovered that when we input a frequency into the

control and turned the table on, the initial sudden movement caused the ball to sink in further

than if a gradual frequency increase was used. To avoid this error, we modified our procedure

and used a slow increase in the frequency to obtain our data.

At high frequencies the rapid movement of the tank caused waves to form in the beads.

This appeared to be amplified by the small size (two gallons) of our tank. The high vibration

frequencies also caused the ball to travel towards the ends of the tanks where it encounters the

most severe wave action. As a result, the ball may have sunk deeper into the beads than if it had

not been affected by waves. To test whether the waves had a significant impact on the ball we

submitted a request for more beads to Flex-o-Lite, the manufacturer. This would have allowed

us to utilize a larger tank where the ball could have been placed further from the tank edges and

presumably be less affected by the wave action. Unfortunately, despite the company

representatives assurance that the new beads were on their way, the package did not arrive in

time for us to perform this additional testing.

We encountered some confusion while evaluating the density of our liquefied material.

The density that we found for the liquefied beads was 2092 kg/m3, compared to 2523 kg/m

3for

the material of just the beads themselves. Initially this seemed impossible because the liquefied

density of 2092 kg/m3

is greater than the 1867 kg/m3

obtained from Gauss 74% maximum close

packing factor for spheres. When we examined the beads under an optical microscope, we found


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that the diameter of the beads was not uniform at all. We took measurements of 10 individual

beads and found an average diameter of 1.133 mm (Figure 10). This is only a rough average,

however, as there were all sorts of imperfections in our beads. Diameters in our sample of beads

ranged from 1.05 mm to 1.26 mm, and in addition there were some beads attached to other

beads. Since the beads were non-uniform, we hypothesized that the 74% maximum close

packing factor does not apply. We also discussed the matter with Professor Lambropoulos, and

he agreed with our assessment that with the uneven bead size, the liquefied density could be

greater than 74% of the density of the glass material [4]. The Gauss close-packing model only

applies to uniform beads, and in the non-uniform case the smaller beads can fit interstitially

between larger ones, packing closer and increasing the density.

Our viscosity calculation was based on a scaling factor that we assumed to be constant.

However, we did not prove that this scaling factor is accurate. The most appropriate way to

confirm our error factor of 2.37 would be to test another control fluid with a high, known

viscosity. If the viscosity calculated for this fluid could be scaled to the correct value using our

error factor, our result would be much more convincing. A fluid that is more viscous than

10,000 cS would be useful, as a slower sinking rate could allow us to take more data and confirm

the validity of our experimentation. The error factor may be a result from a problem with our

Stokes flow assumption. The Stokes model assumes an infinite fluid, but we had only a limited

supply of beads and control fluid. More beads would have made Stokes flow a better

approximation. Also, as mentioned before, a wave effect began at high frequencies, disrupting

the falling motion and giving inaccurate velocity values. Even with these inaccuracies, the RMS

error value of 56.96 is small compared to our viscosity values (at most 7%).

Also, Stokes flow only applies for low Reynolds numbers. We initially assumed low

Reynolds number and after calculating the viscosity with Stokes law, we solved for the Re for

the system. We found Re to be less than 0.001 using our experimentally determined viscosity

and density, confirming that we satisfied that condition for Stokes law.

References

[1] Dhillon, Tejpal. Greenwald, Jesse. Shay, Tom. Granular Flow as a Model for Earthquake

Induced Liquefaction of Soils, May 2006.


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[2] G. Metcalfe, S.G.K. Tennakoon, L. Kondic, D.G. Schaeffer, and R.P. Behringer,

Granular friction, coulomb failure, and the fluid-solid transition for horizontally shaken

granular materials, inPhysical Review E, vol. 65, pp 1-15, February 2002.

[3]Batchelor, G. K. An Introduction to Fluid Dynamics. Chapter 6. Cambridge University

Press (1967).

[4] Lambropoulos, J. C. Personal Communication.

Figures

Figure 1. Shaker Table.


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Voltage vs. Frequency

y = 53.872x + 0.1162

R2 = 0.9999

0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1 0.12

Shaker Table Voltage (V)

Frequency(Hz)

Figure 2. Frequency Calibration.

Figure 3. Block Diagram.


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Displacement vs. LVDT Voltage

y = 2.5732x - 7.9116

R2 = 1

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8

LVDT voltage (V)

Displacement(in)

Figure 4. Front Panel.

Figure 5. LVDT Calibration.


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Ball G

1500

1700

1900

2100

2300

2500

11 12 13 14 15 16 17 18

Angular Frequency (rad/s)

Density(kg/m3)

Figure 6. Viscosity Measuring Configuration.

Figure 7. Liquefied bead density measured with Ball G of mass 43.43 g.


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Ball B

1500

1700

1900

2100

2300

2500

10 11 12 13 14 15 16 17 18

Angular Frequency (rad/s)

Density(kg/m

3)

Viscosity vs. Frequencyy = -3.49699E+02x + 6.72276E+03

R2 = 9.74152E-01

400

600

800

1000

1200

1400

1600

14.5 15 15.5 16 16.5 17 17.5 18 18.5

Frequency

Viscosity(k

g/m*s)

Figure 8. Liquefied bead density measured with Ball B of mass 19.364 g.

Figure 9. Predicted Viscosity of Liquefied Beads vs. Frequency.


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Figure 10. Beads under an optical microscope.

7 liquefaction

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