• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Dedication
 Table of Contents
 List of Tables
 List of Figures
 List of Symbols
 Introduction
 Theory for vibrations in granular...
 Materials, equipment, calibration...
 Previous work of other investi...
 Presentation of results
 Discussion of results
 Conclusions
 Bibliography
 Appendix
 Biographical sketch
 Copyright














Title: Effect of amplitude on damping and wave propagation in granular materials.
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Title: Effect of amplitude on damping and wave propagation in granular materials.
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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Dedication
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
    List of Symbols
        Page xiv
        Page xv
        Page xvi
        Page xvii
        Page xviii
    Introduction
        Page 1
        Page 2
    Theory for vibrations in granular materials
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
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        Page 40
    Materials, equipment, calibration and procedure for the present investigation
        Page 41
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        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
    Previous work of other investigations
        Page 75
        Page 76
    Presentation of results
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
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    Discussion of results
        Page 125
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    Conclusions
        Page 149
        Page 150
        Page 151
    Bibliography
        Page 152
        Page 153
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    Appendix
        Page 155
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    Biographical sketch
        Page 173
        Page 174
    Copyright
        Copyright
Full Text










EFFECT OF AMPLITUDE ON DAMPING

AND WAVE PROPAGATION IN

GRANULAR MATERIALS










By
JOHN RUSSELL HALL, JR.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA
August, 1962















ACKNOWLEDGMENTS


The author wishes to extend his appreciation to all those

who contributed help in any way towards the completion of this disser-

tation. He is deeply indebted to Professor F. E. Richart, Jr. for

his suggestion of the problem and guidance throughout the course of

this research. Appreciation is expressed to Professor J. H. Schmertmann,

Mr. J. Lysmer and Mr. G. Kao for their valuable assistance and sug-

gestions with the experimental work. He wishes to thank Mrs. D. Bruce

for typing the dissertation and its rough drafts.

He especially wishes to express his gratitude to his wife Enid

for her continued encouragement and understanding during the course

of this work.

The author also wishes to thank the National Science Foundation

and the Waterways Experiment Station, Vicksburg, Mississippi, for making

this investigation financially possible.
































To Enid
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . . . . . ii

LIST OF TABLES ..................... vii

LIST OF FIGURES .... . ................ viii

LIST OF SYMBOLS ... . . . . . . . . xiv

Chapter

I. INTRODUCTION . . . . . . . . 1

Purpose and Scope . . . . . . . 1

II. THEORY FOR VIBRATIONS IN GRANULAR MATERIALS .3

Duffy and Mindlin . . . . .... 4
Biot . . . . . . . . .. 5
Pisarenko . . . . . . .. 25
Theories for the Experimental Determination of
Velocity and Damping . . . . . .. 28
Velocity . . . . . . . . 28
Damping . . . . . . . . 30
Shape of Resonance Curve . . . 34
Decay of Vibrations. . . . .34
Exciting Force . . . . . 36

III. MATERIALS, EQUIPMENT, CALIBRATION AND PROCEDURE
FOR THE PRESENT INVESTIGATION . . .. 41

Materials . . . . . . . . . 41
Ottawa sand ........... .41
Glass beads No. 2847 . . . .. 41
Glass beads No. 0017 . . . .. 41
Novaculite No. 1250 . . . .. 43
Tests . . . . . . . . .... 43
Group I . . . . . . . 43
Group II . . . . . . . 43
Group III . . .. . . . 43









Page


Equipment of Previous Investigators .
Wilson and Dietrich (1960) . .
Hardin (1961) . . . . .
Equipment for the Present Investigation
Compression Apparatus . . .
Torsion Apparatus . . . .
Commercial Apparatus . . .
Calibration ... . . . ..
Compression Pickup . . . .
Torsion Pickup . . . . .
Camera . . . . . . .
Wide Range Oscillator . . .
Oscilloscope . . . . .
Electrical Measurements . . . .
Procedure . . . . . . . .
Preparation of Membranes . . .
Preparation of the Specimen . . .
Recording of Data . . . . . .

IV. PREVIOUS WORK OF OTHER INVESTIGATORS . .


V. PRESENTATION OF RESULTS . . . .


Stress Wave Velocities
Group I . . .
Group II . . .
Group III .. ...
Damping . . . .
Group I . . .
Group II ..
Group III .. ...
CFS-Test . . . .
Group III .. ...

VI. DISCUSSION OF RESULTS .
Results for Velocity .
Groups I and II .
Effect of Ampli


77
77
78
78
96
96
116
117
117
117


. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .

. . . .
. . . .
. . . .
tude . .


Effect of Confining Pressure
Effect of Mode of Vibration
Group III ... . . . . .
Effect of Amplitude . .
Damping . . . . . . .
Group I . . . . . .
Effect of Amplitude . .
Effect of Confining Pressure
Effect of Density . . .
Effect of Mode of Vibration
Effect of Pore Fluid . ..


. . .








Page


Group II . . . .. . 140
Effect of Amplitude . . .. 140
Effect of Confining Pressure ... 141
Effect of Mode of Vibration . . 141
Effect of Pore Fluid . . ... .141
Group III . . . . . . . .. 141
Effect of Amplitude . . . .. 141
Effect of Confining Pressure . .. 142
CFS-Tests . . . . . . . .. 142
Comparison of Results with Those of Previous
Investigators . . . . . .... .143
Effect of Confining Pressure . 144
Effect of Amplitude . . ... .144
Effect of Frequency ..... . 144
Comparison with Theoretical Results . . 147
Effect of Confining Pressure . . 147
Effect of Amplitude . . .. .148

VII. CONCLUSIONS . . . . . . .... .149

Ottawa Sand and Glass Beads . . . 149
Novaculite No. 1250 . . . .... .151

BIBLIOGRAPHY . . . . . . . . . . 152

APPENDIX . . . . ... . . . . . 155
















LIST OF TABLES



Table Page

1. Summary of Tests .................. 44

2. Typical Data Sheet ................. 68

3. Typical Data for Velocity .. .. . . . . 69

4. Typical Data for Damping .............. 70


vii














LIST OF FIGURES


Figure Page

1. Fluid Flow in a Two-Dimensional Duct . . . 14

2. Theoretical Variations of Amplitude with
Frequency for Torsional Vibrations of
a Shaft as Given by Theory of Pisarenko . . 27

3. Model Representing Theoretical Conditions
in Present Research . . . .... 30

4. Graphical Solution to Eq. (102) for the First
Mode of Vibration . . . . . .. 31

5. Graphical Solution to Eq. (102) for the Second
Mode of Vibration .... .. ..... . . 32

6. Graphical Solution to Eq. (102) for the Third
Mode of Vibration .............. 33

7. Various Relationships for Damping . . . . 35

8. Single Degree of Freedom System . . . . 37

9. Conversion to a Concentrated Mass System . . 38

10. Grain Size Curves for the Materials Used in
the Present Research . . . . . .. 42

11. Schematic Diagram of Apparatus Used by Wilson
and Dietrich . . . . . . . 46

12. Drawing of Vibration Mechanisms Used by Hardin
(a) Shear Wave Apparatus (b) Compression
Wave Apparatus . . . . . ...... 47

13. Model Representing Apparatus Used by Hardin . 48

14. Vibration Mechanisms Used in Present Research
(a) Shear Wave Apparatus (b) Compression
Wave Apparatus ............. . 51

15. Calibration Curve for the Compression Pickup . 55


viii









Figure Page

16. Typical Calibration Curve for the Torsion
Pickup . .... . . . . . 56

17. Schematic Diagram of Electrical Equipment . .... 59

18. Detail of High Pass Filter . . . ... 60

19. Detail of Attenuater and Phase Shifter . . . 60

20. Detail of Triggering and Time Delay Switch . .. 61

21. End Pieces Used with Specimens (a) Bottom Cap
(b) Top Cap . . . . . . . . .. 63

22. Mold Used for Preparation of the Specimen . . 64

23. Resonance Curves Obtained with Torsion Apparatus. 71

24. Typical Decay Curves .............. 73

25. Variation of Velocity with Amplitude for Ottawa
Sand Dry and Saturated with Water . . . 79

26. Variation of Velocity with Amplitude for Ottawa
Sand Dry and Saturated with Water . . . 80

27. Variation of Velocity with Amplitude for Ottawa
Sand Dry and Saturated with Water . . .. 81

28. Variation of Velocity with Amplitude for Ottawa
Sand Dry and Saturated with Water . . . 82

29. Variation of Velocity with Amplitude for Ottawa
Sand in the Dry Condition . . . . . 83

30. Variation of Velocity with Amplitude for Ottawa
Sand in the Dry Condition . . . . .. 84

31. Variation of Velocity with Amplitude for Ottawa
Sand in the Dry Condition . . . ... 85

32. Variation of Velocity with Amplitude for Ottawa
Sand Saturated with Water . . . . . 86

33. Variation of Velocity with Amplitude for Ottawa
Sand Saturated with Dilute Glycerin . . . 87

34. Variation of Velocity with Amplitude for Ottawa
Sand Saturated with Dilute Glycerin . . . 88

35. Variation of Velocity with Amplitude for Glass
Beads No. 2847 in the Dry and Water Saturated
Condition . . . . . . . . 89

ix









Figure


36. Variation of Velocity with Amplitude for Glass
Beads No. 0017 in the Dry and Water Saturated
Condition ....... . ... . .

37. Variation of Velocity with Amplitude for Glass
Beads No. 2847 in the Dry and Water Saturated
Condition . . . . . . . . .

38. Variation of Velocity with Amplitude for Glass
Beads No. 0017 in the Dry Condition . . ...

39. Variation of Velocity with Amplitude for
Novaculite No. 1250 Consolidated to
2030 lb./ft. and 4100 lb./ft.2 . .

40. Variation of Velocity with Amplitude for
Novaculite No. 1250 Consolidated to
7270 lb./ft. and Rebounded to 4130 lb./ft.2
and 2050 lb./ft.2 . . .

41. Comparison of the Variation of Logarithmic
Decrement with Amplitude for Dry and Saturated
Ottawa Sand ...................

42. Comparison of the Variation of Logarithmic
Decrement with Amplitude for Dry and Saturated
Ottawa Sand . . . . . . . .

43. Comparison of the Variation of Logarithmic
Decrement with Amplitude for Dry and Saturated
Ottawa Sand . . . . . . . ....


44. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand in the Dry
Condition . . . . . ..

45. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Water . . . . ....

46. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Water . . . . . . .

47. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand in the Dry
Condition . . . .. ....

48. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand in the Dry
Condition . . . . . . .


* . .



* . .



* . .



. . .



. . .


100


101


Page









Figure


49. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Water . . . . . . ...

50. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Water. . . . . . . .

51. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Dilute Glycerin . . .

52. Variation of Logarithmic Decrement with
Amplitude for Ottawa Sand Saturated
with Dilute Glycerin . . . ..

53. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 2847 in
the Dry Condition . . . . . .

54. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 2847
Saturated with Water .. .....

55. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 0017 in
the Dry Condition . . . ....

56. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 0017
Saturated with Water . . . .

57. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 2847 in
the Dry Condition . . .....

58. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 2847
Saturated with Water . . . . .

59. Variation of Logarithmic Decrement with
Amplitude for Glass Beads No. 0017 in
the Dry Condition . . . . .

60. Variation of Logarithmic Decrement with
Amplitude for Novaculite No. 1250
Consolidated to 2010 lb./ft.2 . . .

61. Variation of Logarithmic Decrement with
Amplitude for Novaculite No. 1250
Consolidated to 4100 lb./ft.2 ....


. .



. .



. .



. S *



. .



* .



* .



. S


108


109


110


114


115


* . S .

* S e


118


Page









Figure


62. Variation of Logarithmic Decrement with
Amplitude for Novaculite No. 1250
Consolidated to 7250 lb./ft.2 . . . .

63. Variation of Logarithmic Decrement with
Amplitude for Novaculite No. 1250 after
Rebounding from 7270 lb./ft.2 to 4130 lb./ft.2. .


64. Variation of Cohesion and Friction with
Axial Compressive Strain for Novaculite
No. 1250 Consolidated to 4100 lb./ft.

65. Variation of Cohesion and Friction with
Axial Compressive Strain for Novaculite
No. 1250 Consolidated to 4100 lb./ft.2.


* S



* .


122


66. Variation of Cohesion and Friction with
Axial Compressive Strain for Novaculite
No. 1250 Consolidated to 7180 lb./ft.2
for 2 Hr. and Rebounded to 4100 lb./ft.2.


67. Variation of Cohesion and Friction with
Axial Compressive Strain for Novaculite
No. 1250 Consolidated to 7180 lb./ft.'.

68. Variation of Amplitude with Frequency for
Longitudinal Vibrations of Ottawa Sand
in the Dry Condition Under a Confining
Pressure of 619 lb./ft.2 . .


. . 123


. .




. S 0


69. Variation of Shear Wave Velocity with
Confining Pressure for Ottawa Sand . .. .


70. Variation of Compressive Wave Velocity
with Confining Pressure for Ottawa Sand


* . &


71. Variation of Compressive and Shear Wave
Velocity with Confining Pressure for
Glass Beads No. 2847. . . .. . .. .

72. Variation of Compressive and Shear Wave
Velocity with Confining Pressure for
Glass Beads No. 0017. . . . . .. .

73. Variation of Shear Wave Velocity with
Confining Pressure and Void Ratio
for Dry and Saturated Ottawa Sand . . ....

74. Variation of Compressive Wave Velocity
with Confining Pressure and Void Ratio
for Dry and Saturated Ottawa Sand . . . .


146


xii


Page









Figure


75. Properties of the Shear Wave in Water-Saturated
Ottawa Sand as Given by the Biot Theory . .

76. Properties of the Shear Wave in Glycerin-
Saturated Ottawa Sand as Given by the
Biot Theory . . . . . . . ..

77. Properties of the Fluid Wave in Water-Saturated
Ottawa Sand as Given by the Biot Theory . .

78. Properties of the Fluid Wave in a Glycerin-
Saturated Ottawa Sand as Given by the
Biot Theory . . . . . . . . .

79. Properties of the Frame Wave in Water-Saturated
Ottawa Sand as Given by the Biot Theory . .

80. Properties of the Frame Wave in Glycerin-
Saturated Ottawa Sand as Given by the
Biot Theory . . . . . . ..


81. Variation of Damping and Velotity with
Pressure for Water-Saturated Ottawa
Given by the Blot Theory . .

82. Variation of Damping and Velocity with
Pressure for Water-Saturated Ottawa
Given by the Biot Theory ..


* .



* .


. . 168



. . 168


. . 169



. . 169


Confining
Sand as


Confining
Sand as
. 0 . . .


83. Variation of Damping and Velocity with Confining
Pressure for Glycerin-Saturated Ottawa Sand as
Given by the Biot Theory . .. . . . .


xiii


Page














LIST OF SYMBOLS


CL = characteristic pore dimension

A = amplitude of vibration

b = coefficient of dissipation in the Biot theory

B 9 + L B9 parameters in the Biot theory related
to shear wave velocity

S-= bulk modulus of the fluid

B bulk modulus of the solid

Cv viscous damping coefficient

C = fluid compressibility

CL arbitrary constants

D = elastic coefficient in the Biot theory

e = void ratio

E o Young's modulus of elasticity

7 frequency of vibration

o = resonant frequency

n = undamped natural frequency

c characteristic frequency

F = exciting force for vibrations

F, ) F real and complex parts of the function F(K)
Gr shear modulus

G = shear modulus of the solids

C1, shear modulus of the membrane


XLv








Gs = shear modulus of the specimen
H T+ R +2Q


L unit vector in the X -direction

I mass polar moment of inertia of the specimen
Io = mass polar moment of inertia of the driver and the pickup

j unit vector in the y -direction

J- polar moment of inertia
To Bessel's function of zero order

J *- polar moment of inertia of the membrane

s polar moment of inertia of the specimen

coefficient of permeability
unit vector in the Z -direction
K factor for the determination of apparent mass,
1 length of the specimen
L wave length

mass
M a damping parameter for dilitational waves
l porosity
-0 fluid pressure

= exciting force
- force component acting on the solid per unit volume of
material in the L -direction
Q elastic coefficient in the Biot theory

Q =- force component acting on the fluid per unit volume of
material in the L -direction
r = radius

R = elastic coefficient in the Biot theory








4 = hysteresis parameter

SG = specific gravity of the soil solids

t = time

tm = thickness of the membrane

T D + aG

Tvn 3 torque resisted by the membrane

Ts m torque resisted by the specimen
U pore pressure

Ui = displacement components of the solid material

UL displacement components of the fluid material

U- displacement vector of the solid material

L4 displacement vector of the fluid material

/b' = velocity

f = characteristic velocity

S =- velocity of the rotational wave

t a velocity of the disturbed fluid wave

/3n velocity of the disturbed frame wave
W a total energy available in one stress cycle

S= coordinate in the X -direction

X = acceleration caused by an external volume force

= dimensionless velocity ratio

HI = complex root of the equation for dilatation waves

Z=n complex root of the equation for dilatation waves

OC = attenuation constant

OCj = non-dimensional elastic properties

OC phase angle between force and displacement


xvi








jc= non-dimensional dynamic properties

a hysteresis parameter

w coefficient of fluid content

P unit weight of the fluid

= logarithmic decrement

0 = unjacketed compressibility coefficient

small parameter

ij- = strain components

6 = dilatation of the solid material

6 dilatation of the fluid material

- structural factor

-= fluid viscosity

K non-dimensional frequency parameter

K coeffient of unjacketed compressibility

A Lame coefficient


S Poisson's ratio

/ =- Poisson's ratio of the solid material

=5 sinuosity correction factor for damping

TT 3.14159...

P -mass density

Pq apparent mass density

pff mass density of the fluid

Sa mass density of the solids per unit volume of material
P mass density of the fluid per unit volume of material

ij =- mass density coefficients in the Biot theory

CT = confining pressure
xvii








CMj = stress components

T shear stress due to viscosity

= friction angle measured by the CFS-test

twist of a circular shaft
CU circular frequency

C = rotational vector of the solids

-0 = rotational vector of the fluids

K.E. = kinetic energy
= m operator defined by t + 2 +z z

eXP OC a operation defined as c where e is the base of
the natural logarithms

r real part of

= Z real part of 2
i = complex part of \Z7

o = complex part of \Z-r


xviii















CHAPTER I


INTRODUCTION


In recent years it has become more and more important to be able

to determine the behavior of soils under dynamic loads. The design of

foundations and underground structures for transient and dynamic loads

has become very important from the standpoint of protection against

nuclear blasts. Underground missile bases and tracking stations must

be constructed to critical specifications and tolerances of motion;

consequently, an understanding of the characteristics of soil under

dynamic loads is necessary for the design of such structures.

Another significant field requiring knowledge of the behavior

of soil under dynamic conditions is that of steady state vibrations

of machine foundations. Richart (1960) has shown that analytic so-

lutions may be used to determine the dynamic characteristics of this

type foundation if the elastic parameters of shear modulus and Poisson's

ratio of the soil are known. A knowledge of the damping character-

istics of the soil is needed in order to determine the transmission of

the effects of a vibrating foundation to nearby structures.



Purpose and Scope


The purpose of the present research was to study the velocity

of shear and longitudinal waves and their damping in a column of granular


-1-





-2-


material. The effects of confining pressure, density, pore fluid (air,

water and dilute glycerin) amplitude of vibration and frequency were to

be considered. This is an extension of the work by Hardin (1961) to

include the effects of amplitude and the measurement of damping by use

of decay curves.

New apparatus was constructed for the measurement of the velocity

of shear and longitudinal stress waves at amplitudes of vibrations corre-

sponding to those of machine vibrations. The largest amplitudes for a

twelve-inch specimen fixed at one end were 5 x 10"4 in. longitudinally

and 1.5 x 10-3 rad. torsionally. This is on the order to ten times that

attainable with the previous apparatus. The new equipment was also de-

signed to permit evaluation of damping by measuring the vibration decay

from the steady state condition after the driving mechanism was turned

off. Hardin measured damping mainly by the shape of the amplitude vs.

frequency curve. The pickups used on the new apparatus were calibrated

so that the amplitude of vibrations could be measured at any time during

a test.

Theoretical solutions of several persons have been presented for

the behavior of stress waves and damping in porous materials. The theory

by Biot (1956) may be used for theoretical evaluation of the stress wave

properties of velocity as well as the viscous damping associated with

saturated materials. The theory given by Duffy and Mindlin (1957) may

be used to calculate theoretical stress-strain relationships for granular

materials as a function of confining pressure and also the friction damp-

ing associated with contact stresses. The theory of Pisarenko predicts

the behavior of a material with non-linear stress-strain relationships in

torsion. The above theoretical solutions are given since they should

provide a comparison with experimental results.















CHAPTER II


THEORY FOR VIBRATIONS IN GRANULAR MATERIALS



Theories dealing with vibrations of granular materials are varied

and different approaches have been made by different investigators. Some

theories are based upon a specific configuration of the granular particles

while others consider a porous frame. When a fluid-filled porous ma-

terial is considered, three types of waves are found to exist. One of

these waves is a shear wave and the other two are dilatational waves.

In one of the dilatational waves the solid and fluid particles move in

phase while in the other the two particle movements are out of phase.

When the system is considered as an evacuated porous material then only

one dilatational wave will exist.

Biot (1956a) has solved the problem of a fluid-saturated porous

material and found expressions for rotational and dilatational wave

velocities as well as the viscous damping associated with each. Duffy

and Mindlin (1957) have derived equations for the velocity of longi-

tudinal waves in a face-centered cubic array of perfect spheres of equal

size. They have considered both normal and tangential contact forces

which give rise to a friction damping. Pisarenko (1962) has derived

equations governing vibrations of a material with a non-linear stress-

strain curve. His theory is derived for elastic materials but should

be applicable to dry granular materials.


-3-








Duffy and Mindlin

Duffy and Mindlin (1957) derived a differential stress-strain

relation for a medium composed of a face-centered cubic array of elastic

spheres in contact. They retained the classical Hertz theory for normal

forces and used the theories of Cattaneo (1938), Mindlin (1949) and

Mindlin and Deresiewicz (1953) to include the tangential components of

the forces at the contacts between the spheres. The theory predicts

that when a tangential force is applied to two spheres in contact, slip

occurs at the circumference of the contact surface. Because of this

fact the stress-strain relation depends upon the entire stress history

of the material. This phenomenon gives rise to a frictional dissipation

of energy which does not occur when only normal forces are considered.

Solutions are found for the case of a small increment of stress

applied to a medium under an initial isotropic plus uniaxial stress. The

resulting medium is anisotropic and the stress-strain relation depends

on the direction of orientation. Orientation which avoids coupling with

flexural waves gives a solution in the X or (1, 0, 0) direction and

the (1, 1, 0) direction. The solutions are


2 2 3


E
,oo, ( ,- s-) )


2 (4-3/)(8-77)_______ 3 &2 3 ]
(' o) ((4-3) (8-7, )(-) 2(l-)








where / and G are Poisson's ratio and shear modulus of the

individual spheres. The two values of E differ depending upon /u

At / = 0.25 the difference is about 2 per cent.

The theoretical frictional energy loss per cycle was found by a

summation of the energy dissipated at the individual contacts and by

taking a sinusoidal displacement distribution along the length of the

bar. The theory predicts an energy loss proportional to the cube of

the amplitude. For small amplitudes the total energy stored is pro-

portional to the square of the amplitude. This would indicate a

variation of logarithmic decrement which is proportional to amplitude.

The theory also predicts a minus one-third power variation of loga-

rithmic decrement with pressure.


Biot

Biot (1956a) presented a theory dealing with the propagation of

elastic waves in a fluid-saturated porous solid assuming that the solid

was elastic and that the fluid was compressible and viscous. He also

assumed that the walls of the main pores (interconnected pores) are

impervious and that the pore size is concentrated around some average

value.

Equations are first derived assuming that the viscosity of the

fluid is zero and thus obtaining expressions for rotational and dila-

tational waves in the undamped case. This represents the condition of

wave propagation for very high frequency as will be shown later.

The total force acting on the fluid on one face of the cube di-

vided by the area of that face is defined as the fluid stress. The








total force acting on the solid on one face of the cube divided by the
area of that face is defined as the stress acting on the solid.
By assuming a statistically isotropic, elastic material he
obtains the stress strain relationships

TK = 2 G Ex + DE + Q

Oy, = 2 G E,3 + DE Q

Tz = 2 G Ez + D +6 + Q

(2)


^Z = G cxt
oGz= GoCe

-U = Q + RC


where (Cj represents the stress components, G6j represents the
strain components, C is the dilatation of the solid, & is the
dilatation of the fluid, LA is the fluid stress or pore pressure,
G is the shear modulus of the porous material and D ,
and R are constants relating stresses to volume changes of the
solid-fluid system.
In the absence of viscosity the kinetic energy of the system
can be expressed by assuming the X Y and Z directions
equivalent and uncoupled dynamically as


[K E]- [.( (t t)2 atl)












J k-/ jl-- r t/- -,



where K. E. is the kinetic energy, aU represents the displacement
component of the solid, Lc represents the displacement component of
the fluid, and t represents time. The mass density coefficients

PA,, p and pz are defined by

,, = P + = +P = (14)

where f is the mass of the solid per unit volume of material, j
is the mass of the fluid per unit volume of material and pa is an
additional or apparent mass. Thus

p,, + 2 f + )p = f (5)

which is the mass density of the fluid-saturated material.
If we let Lx equal the net force acting on the solid per unit
volume in the X -direction and Qx the net force acting on the
fluid per unit volume in the X -direction then

!2ao0' + aCzx + a0tx
ax az ay
(6)
Qx =

From Eq. (3) we get
- a- 7- K. uE. = t- J 'P" u,,
(7)

K-a f E K'









where Lx denotes D-/x /t
By combining Eqs. (6) and (7) and substituting Eqs. (2) we get the
equilibrium equations for the X -direction.

QV2 Ux +(D- )- + C-- Q@ p +6 (P, ,

S+ Rae = ( u- + p,


where V9 == C/X / 2 + z2/ .
Two similar equations exist for the y and Z directions. By
changing to vector notation the equation can then be written as

^6 nd[2 Id-I ) of QF 2 (P f +)
VC)


a t LP + f12 2

where grad a /59X + J /t9! +A4- /' 2 J and A
being unit vectors in the X y and Z directions. The vectors
( and U are the displacement vectors of the solid and fluid
respectively. Equations (9) are the dynamic equilibrium equations for
a fluid-saturated porous material written in terms of displacements.
The equations governing the rotation of the material can be obtained by
applying the curl operator and the equations governing the dilatation
can be obtained by applying the divergence operator. Thus, for the
rotational waves




O = +rd + F2 )
+IO








where W is the rotation vector of the solid and CO is the rotation
vector of the fluid. By combining Eqs. (10) we get

Gr V 0 1)



This has a solution for the velocity of rotational waves given by

2


or
G- I



We also find that

S= (13)



which means that the rotation of the fluid and solid are proportional
and in the same direction. It is interesting to note that the greatest
contribution of the pore fluid to the mass density when Pa. = P
gives an equivalent material mass of p + which is an addition
of only half of the mass of the pore fluid. However, this is assuming
that the fluid has no viscosity.
The equations for dilatational waves are found by applying the
divergence operator to Eqs. (9) and are

v (2Te+ ) = (p ,- + )
04)

v2 (Q + RT) = j^ (^ r + f^ e





-10-


where T= D 2G-.
Let H = T+R + ZQ.

Also, let


T

OCII= H


Q
C 2 = -,
H(i5)


P P
Equations (15) define non-dimensional quantities of the elastic, a'ij

and dynamic, f,- properties of the material.

A characteristic velocity is defined by


2 H
-= -F"


(I6)


Solutions to Eqs. (14) are taken as

= C, exp L (x

6= C p[ex P [( X


+wt )

4+ co t


where W is the circular frequency, C, and Ce are constants and
the wave velocities are given by /-

Equations (17) are substituted into Eqs. (14) and the determinant

of the coefficients of C, and Ca is set equal to zero. By letting
Z- = /v~/ the above gives

(c,,, ,, -c;)z2' (i,,,/ +2, ,, -2.A, )P
(1 8)
+ -4,;) = o.


The roots of this equation correspond to two velocities.

velocity is designated as the disturbed fluid wave, /vi


The higher

, and the


R





-11-


lower velocity is designated as the disturbed frame wave, nt Biot
shows that the solid and fluid are in phase for L-T and are in opposite
phase for /r .
The effect of dissipation due to the viscosity of the fluid is
introduced by adding the term b /at(& a U) to the first of Eqs. (9)
and subtracting from the second of Eqs. (9). This introduces a force on
the solid by the fluid which is proportional to the relative velocity
between the solid and fluid. The new equations are

G VE + +^[(D + G) +Q]

dt" 2p, I +pC 4- b ) a u } (1).
+'P,2 U 4- ( LI)




The equation defining rotational waves is found by applying the
curl operator to the modified equilibrium equations and substituting
solutions of the form

C I e x P + W t 0 x] (20)
J = cexP ex[ X x + ct)-cx]


where OC is the attenuation coefficient.
Logarithmic decrement is given by

o/-I, 2rr (21)
CU





-12-


A characteristic frequency is defined by

b (z2)
= = b ___(-(


After substituting Eqs. (20) into the equilibrium equations and
setting the determinant of the coefficients of C, and C2 equal
to zero the solutions are

(23)
2


6(= 27^-. 7 P (24)

The quantities BE- and BL are given by




S+ 2 (()
-2 + J 22.

__ ^ *^ ____ (26)
B fj


The equations for dilatational waves are found by applying the
divergence operator to the modified equilibrium equations and substituting
solutions of the form

E C, ex(P X +i )-o Xj
(27)

T = C exp[ X + C -)-o(




-13-


Setting the determinant of the coefficients of C, and C2
equal to zero gives a complex quadratic equation which is

(oc,,o o 2, ) Z (g,, i~. +-3 .- 2 ,L ) z
(28)
4+(e-)2) + b (- = (28



Two complex roots Zr and Z~ are obtained which correspond to
the disturbed fluid wave and the disturbed frame wave respectively.

Let

2 = + (2 q)


Then
^K-
= __ (30)


oC C (31)

21rr, GOr (32)


The above theory is based on the assumption of Poiseuille flow
and is valid for low frequency vibrations. Biot presents a second paper
(1956b) which deals with the investigation of higher frequencies. The
flow of a viscous fluid between parallel plates and in a circular tube
under an oscillatory pressure gradient is considered. These two cases
are studied in order to obtain an indication of the effect of cross-

sectional pore shape. A complex viscosity correction factor is derived
which is a function of f/ f where f, is the characteristic
frequency of the material as defined by Eq. (28).






-14-


In a two-dimensional duct as shown in Fig. 1 the equation of

motion is given by


,pi LA= -5x


(33)


The relative velocity between the boundary and the fluid is

defined as

U, = U u,. (34)


Substituting into Eq. (33) gives


2
+ a V a


y= a,


(35)


-/~~/~ / II /I/III/IIIii/i/II


LAi,


/I // f //// ////I/////I///I//'I711 1l 77ill////// Iillll//


Fig. 1. Fluid flow in a two-dimensional duct.



An equivalent to an external volume force is considered as defined


3 L
3 X


(36)


Equation (35) then becomes


y2
a U,


=


a2 L4
L 2'7
4- Y7 2


pf I, a


- P U"


h,


U-


(37)





-15-


where


By assuming sinusoidal functions of time we have

2 Y2


The solution to Eq. (38) with the boundary conditions that the
function is symmetric in y and that U, = 0 for y = + -i is


(3 q)
S[C--{-/ czJ


The average velocity is given by


X A -) _V (21 (40)
^C(AV) I. 1


The friction stress at the wall Y = \ is

-_Jd I C L_ __r r ^ a (41)



This is one-half of the total friction force.
In order to use the results the following ratio is used:


(4 2)


where ti- is a non-dimensional variable defined by

(43)





-16-


It can be shown that

._ _. = __. (44)
K,, "W 0 1" Av) C:,

which corresponds to the case of Poiseuille flow. If we put
12 K ;
F K.i (45)
^ K,
then F (o) I and

2 ( ) (4G)
U, (Av) 0,

The quantity F (K.) is complex and represents the
deviation from Poiseuille flow for increasing frequency. At high
frequencies or large values of K-

F (K) K + L (47)
I3 r2 3^

which shows that the friction force is 45 degrees out of phase with
the velocity at high frequency. When Eq. (45) is reduced to the real
and complex parts we have,

Re F (K.)= {- ^ .[ (coo F-K + ^V\if2K

-LAi, \C- + ^Fc2, +. ,[ :,+- ^ ,)K
(48)



,K,





-17-


Im F( Q -L I K z"FK,(c + c \fa- K,)


(49)






+ r K(c, K + c K., ) -,K



For values of K I the following approximations may be used
which have a maximum error of one per cent at K = I

Re F,(K,) = 1.00
S2 (501
Im F, (K,) = (
15

The problem is also solved by considering the three dimensional
flow through a circular duct of radius L Equation (38) is replaced
by
a0 + vLA (5+)



By assuming Ux is independent of X and that the flow is
axially symmetric we have

(a + i U, X (s2)
7 (ar2 ar at





-18-


For sinusoidal functions of time we have

d2K u. d u. i = X
+ V
dr r dr 7 V

which is a Bessel's differential equation for U,
boundary conditions that the solution is finite at
U, = 0 at r= a0 we have


iXf


(53)


SIntroducing the
S= 0 and that



(5-4)


The average velocity is given by


X fi a ber'K + L bei'K K
'(A) i6 L K ber K + Z bei K I


K =-(^

JT (K vT-) = ber K

ber' K d ber K
dK

be'K d be; K
dK


-+ be' K


The friction force on the wall is


L oI beF -Kt i be; K
6w / be- K- + bei K


where


(55)


(56)


I





-19-


The ratio of the
velocity is given by

1, (Av)
'(Av)


where


T(K) =


total friction force, 21TTZT to the average


a2rr K T(K)




ber' K -+ bei'K
bet K +- bei K


It can be shown that

&
K- o


2TT T
=rT 8(Avr
SI (Av)


If we introduce a function


F(K) = -



2 frQ -
U(Av)


KT(K)
i F T(K)
K


81Tr F(K).


The function F(K) is exactly similar to F ((K) which
indicates the deviation from Poiseuille flow friction as a function
of the frequency parameter K.
It is shown that for large K,


( o)


F(K) = + K
4V'a 4\Vil


(?7)




(58)


then


(5q)





-20-


The above results are applied to a porous material assuming that
the variation of friction with frequency follows the equations for the
tube of uniform cross-section.

In the equilibrium equations the friction force per unit volume
of material in the X -direction was expressed as

x = b 2t ( u U,) (611


The term (Av) has the same meaning as a/at (Ux Ix)
Thus, the coefficient b is the ratio of the total friction force to
the average fluid velocity and should be multiplied by a frequency
correction factor such as F (C,) or F (K) It should be
recalled that b is the friction coefficient for Poiseuille flow.
The correction factor derived from parallel plates should correspond
to pores shaped like narrow slits while that derived for the tube should
correspond to circular pores.
Biot states that the functions F (K,) and F (K) are
practically indistinguishable when a scale factor is introduced making
the asymptotic directions equal. In this condition

K1- K. ( ra)
3\2 4V2)

Since the correction function derived for the two plates is much
easier to evaluate than the series for the tube, we use with a good
approximation the relationships

Re F(K) ~ Re F(. K)
(63)
Im F(N) = Im F ( K).





-21-


Thus, when pores have the shape of narrow slits its frequency

dependence function may be taken the same as for circular pores with

a radius C- =(4/3) .

This shows that in the extreme cases of slits and circular pores

the frequency correction function is the same except for a scale change

in the frequency parameter. Therefore a universal complex function F(K)

is adopted to represent the frequency effect with a nondimensional

parameter

K = 2 (G4)



where QC is a length which is a characteristic of pore size and

geometry.

In order to establish the relationship between K and f/f

we note that K may be put in the form

Kf = 1 (65)


where



First consider the case of pores with parallel tubes of radius

in the direction of flow. The total friction at K = 0 per tube

and per unit length is

2 = 8 Trv U (Av) (G7)


The value per unit cross section of the tube is

S2T1 8A kL
S- 2 I(Av) ( 6)





-22-


In order to calculate the friction per unit bulk volume of the
material we must consider the porosity n which gives

2Tn 8 n U(', (69)

However,
b
f (70)

Hence,
4 v
= (71r
Thus,
K2 = 8 (72)

By a similar procedure for slit like pores we have


IK = (73)
3 Jc

Since the pore shape will have a sinuosity it is necessary to
multiply the expression for b by a factor t > I which takes
this effect into account. Then we write

b 8 4 (74)


Ka (75)


Thus, in general various geometric factors such as sinuosity and cross-
section shape enter into the frequency correction by a factor multiplying
/ f In general





-23-


F(K) = F ( ] (76)


where c is referred to as a "structural factor." With the sinuosity
factor / > I the expression for varies from

S- (8)2 (77)

for circular pores, to

(G (78)

for slit like pores.
The best value of should be determined by experimental data.
Using the frequency correction function the dynamic equilibrium
equations become

GViL + .'- D+(DG) +Q E] |p(., +9t)2
+ b F(K) (( -a (

[Q + R ] = T6 (Lf +R,2)R -b F(K)( -0) .

The above equations give the following results for the rotational
and dilatational wave properties.
For the rotational waves

2 G a (80)


C = ca B<(--- pr (81)


= 2 1n/OC (82)
B+





-24-


In this case if we let F (K) = F +- i we get





[(fPz acJ b iF)a + bF ,2']
Bc,- b ^[ (P1 P2 2)] +6,P2 2OJFr (4
[(P?2W+bF)2 +b2F,2]

The equation governing the dilatational waves becomes

Za-(z,- +z F, -M F,)M Z +(MF +Z,Z2-cMF,)=O (8s)

where Z, and Zz are the roots to Eq. (18) and

b
M (= a C86)
cop (c.,oc2( i -

Equation (85) has two complex roots Z and Z Let

-Z = L, a F T,
(87)

Then

CC ( o8)




= (90)
(4


Identical relationships exist for fTf ,


and 3. .





-25-


A discussion of the determination of numerical values for a
condition representing that of Ottawa sand is given in the appendix.

Pisarenko
Pisarenko (1962) has evaluated the torsional vibrations of a
circular shaft which has a non-linear stress-strain relationship. He
assumes that the equation for increasing stress is

d a [l- r(EGA, +- E ) (9 )
d GxL

and the equation for decreasing stress is


d G F[i r (e )- e ) j (92)
d xL

where Ex4(MAx) is the amplitude of the shear vibration and F
and A are hysteresis parameters to be determined experimentally.
If we let A = Ad + I and integrate the above equations we have

A A-I A ll
S(93)

=V +e rE + exJ(MAX) A A) I A (q3)C



Equations (93) are integrated over a circular shaft to determine the
expressions for moment. These are substituted into the equilibrium
equation which gives

+= & a .t (94)





-26-


where )(X,t)
parameter,

( <', t)


represents the twist along the shaft, 6 is a small
represents the driving force and (4 t) is given by

S d cP(x) a (x, t))
dx ax
A-1 d ,(d (x ^ 4 rA- '
-2 A (A +- 3)


A solution to Eq. (94) is sought in the form

( X, ) r(X) 9 C(oa u t +- ) E P,(x,l) + j- 2(x,t)+..

C) = C +

+= + +...

where 9o9 is the amplitude of vibrations, Jc is the natural frequency
of vibrations of the bar and OC is the phase angle between force and
displacement. A solution for the boundary conditions of a bar fixed at
both ends is found and given by

(A) IGG rf 2 o r A-1 r Xd
c)-2 = I + c 3) xo 7
W (A + 3) 1 & le j


(~) e +
X (IQo +)A-a rr3 CJ + 'Xo


(95)


4 r f -P.^
S.(A+ 3cn)^- a


O([|c~ )-


1A I


1 Aede.


The above equations were solved
for Ottawa sand. The values used were
(1)MAx 0.0005
G = 18,000 Ih/i


(9)


for the conditions representative
as follows.






-27-


5 x 104



04






3\

2 -




1-




0.95 1.0

W7


Fig. 2. Theoretical variations of amplitude with frequency for
torsional vibrations of a shaft as given by theory of Pisarenko.






-28-


r 2 cm.

1 50 cm.

A m3

r = 2 x 107 (calculated to give a value of
8 i 0.10)


The resulting resonance curve is shown in Fig. (2). The

variation of the frequency of maximum amplitude with exciting force

is as a second degree parabola with its origin at Cy/CJc 1.0.

The stress-strain relationships used in the derivation are such that

the logarithmic decrement will be independent of the magnitude of

vibrations.



Theories for the Experimental Determination
of Velocity and Damping


Velocity

The measurement of stress waves through a column of soil was

determined by measurement of the resonant frequency of the column for

the type of wave in question. Resonance is found by varying the frequency

of vibration and determining the frequency for maximum amplitude. By

knowing the mode of vibration the wave length may be determined by the

length of the specimen. From these two measurements velocity may be

calculated from the relationship

/ = L (97)

where L represents the wave length of the stress wave.

If the shear wave velocity and the longitudinal wave velocity are

known the dynamic shear modulus and the dynamic modulus of elasticity






-29-


may be computed. Love (1944) showed that in a cylindrical specimen the

velocity of a longitudinal wave is given by





where r is the radius of the specimen. Equation (98) is valid for

small values of r/L The specimens used in this research had an

r/L a 0.02 for the first mode of vibration. Using this value and

a maximum value of Poisson's ratio of 0.5 the first term of Eq. (98)

is 0.999 which is 1.00. Thus, for the specimens used in this

research we can use the relationship

(V (99)


for calculating the dynamic modulus. The dynamic shear modulus may be

found by using the relationship

-,v" (too)


In order to vibrate a specimen some sort of a driving mechanism

must be attached and another mechanism must be attached for measurement

of the response. The addition of a mass to a material in which the

resonant frequency is to be measured results in a slight change in the

resonant frequency. The conditions of the specimen in the present

investigation may be represented by Fig. 3. The solution governing the

natural frequency of such a system under torsional vibrations is given

by


Y CVGr 7(






-30-


Fig. 3. Model representing theoretical
conditions in present research.





where I is the mass polar moment of inertia of the specimen and Io

is the mass polar moment of inertia of the mass attached to the free

end. Equation (101) must either be solved graphically or by trial

and error. It is convenient to put Eq. (101) into the form


'9 rL 8 = -L
I.


(1oa0


Thus,


2- T


(103)


Figures 4, 5 and 6 show graphical solutions to Eq. (102) for the first

three modes of vibration.


Several methods may

waves in a soil specimen.

of the resonance curve for

to vibrate the specimen at


be used to measure the damping of stress

The simplest method is based on the shape

constant driving force. Another method is

resonance and measure the decay of vibrations







-31-


1.7


, I'











1.4 t-- --* -- ...---.4 -.. - -I i4 .









12i -


-1- 7
::; ] ..... 1 I ... .. i : [ . *. j'l -.. : . 1 ; -. *i ; .







































2 5 10 20 50 100 200 500 1000
. .I .. i... .. .. ,






Ii ,...


































Fig. 4 Graphical solution to Eq. (102) for the first mode
of vibration.
of vibration. ~ :






-32-


4.9
: I I -, -i I .


4.8. ..


4.7 - - - --


4.6 1


4.5 -
.I.. .. .I* , I. . -.. ,













S' ' I
. ... '








4.1 - --







4.0 -0


3.9 T -F



S I 3.7 .
, I--- ,t ,--:- -r /- -- i- -- . . ,I-- ,-- ,- ,--- ,






















3.6 --


3.5 I
2 5 10 20 50 100 200 500 1000


Fig. 5.- Graphical solution to Eq. (102) r the second mode
of vibration.
__; .. .. .,... .,.. ...,. ... .. . -._,.. . .[
i i0 2 0 10 2050 10

', 5. Gr/!]ic soluio t, .. (12 ,b ,' ecn
of ra i o-n,1 i ' ,' '







-33-


24 -44Z -- '-

I.,. I.'i







I..
I-c - ~ I .i I -' .


---- ,--* I
4 i ' i 1 i 4 i '








I-






S ''


r 1 t t


r;j- !.,


i i i


-,_ - I-.
: ,


.1-C I


5 10 20


50 100


* I




V T







I~




* I I I


200


Fig. 6. Graphical solution to Eq.
of vibration.


(102) for the third mode


7.4


7.3



7.2



7.1



7.0



6.9



6.8



6.7



6.6



6.5


I I
i-
















I I 1
I, I r



II I





II


I I

I- j


500


1000


r ,


i l


: I I ;


...1..-.1


111~1
,- i
I






-34-


when the driving force is cut off. Another method makes use of the

relationships between force and the magnification factor at resonant

frequency. Various relationships for damping are shown in Fig. 7.

Shape of resonance curve. When the exciting force of the

vibrating system can be written as F = F, sin w t certain

relationships exist for a single degree of freedom system with

viscous damping. Forster (1937) expressed this relationship in general

as


(104)


where Af is the width of the resonance peak

in cycles per second. AMAX is the maximum

resonant frequency ~o

Decay of vibrations. When a specimen is

allowed to decay as a free vibrating system the

may be determined by measuring the amplitude of

vibrations. If these amplitudes are designated

the logarithmic decrement is defined as


at amplitude AX ,

amplitude at the



vibrated and then

logarithmic decrement

two successive

as Anand An+, then


An+


The average value for l cycles can be expressed as


I= j, Ao
n An


(ioS)


(tG06)


a
SAf T f a A- -
fo AMAX V A





-35-


T 9!0o e











- -= e e

EXCiTN FoRCE F F S
'0 2 7X7/ v]'
-t I e tn+i


&Tr



-Nd. t MAa t- A











Sy EXCITING FORE = FY e


Fig. 7. Various relationships for damping.





-36-


Exciting force. For the case of forced vibrations with a constant

exciting force the following relationship exists.


AA STATIC(
A. -(P CV T



where in is the undamped natural frequency, Cv is the viscous

damping coefficient, CvcR is the critical damping coefficient and

ASTATIC is the static deflection produced by a force of the same

magnitude as the exciting force. For small values of Cv/CvcR the

difference between the frequency at maximum amplitude and the undamped

natural frequency is very small and we can consider / = I

Thus, at resonance we have


A (0TATI8)
MAX Cv
CycR

From Fig. 7 we have the relationship that

Cv
CVCR

for small values of Cv/CvcR

Finally,

ASTAA ( 09)


for small values of Cv/CvcR. In order to measure AsrTrre and

AMAx the driver must be calibrated for force and the pickup

must be calibrated for amplitude. By measurement of the resonant






-37-


frequency the modulus may be computed for calculating the static

deflection. The static deflection should be based upon a length of

specimen corresponding to L/4 .

In this investigation the values obtained for damping by

measurement of the decay of vibrations were corrected to compensate

for the added mass of the pickup and driver. The effect of the added

mass is approximated by considering a single degree of freedom system

as shown in Fig. 8.

S m





C,

Fig. 8. Single degree of freedom system.


The mass of the specimen is represented by Mn the mass of the driver

and pickup by mo and the spring constant is s First consider

the case without mf We have the relationships



W m




Thus,
Tr Cv
o 1 hv sm ilo)

With the addition of mo we have similarly

T' C= CO
Jas(m+mc ( )






-38-


Finally, +
(112)


In order to use Eq. (112) it is necessary to convert the mass

of the soil specimen into an equivalent concentrated mass. It can

be shown that for the conditions in Fig. 9 the equivalent concentrated

mass is 0.405 m



A" -- 0'



m' = 0.405 m

Fig. 9. Conversion to a concentrated mass system.


This is based on the condition that both systems have the same undamped

natural frequency. Using the above approximation the corrected value of

logarithmic decrement is given by

S= + 3)
0o.405 n

The same correction may be used for torsion by substituting the

analogous torsional inertias.

The decay curves of this investigation are also affected by the

membrane which encloses the specimen. A calculation was made in order

to determine the order of magnitude of such an error. For a given

angle of twist the ratio of the torque resisted by the membrane to the

torque resisted by the specimen is given by





-39-


Tn m GGm
-FS Jr; CS


where T- represents the polar moment of inertia of the membrane or

specimen as designated by the subscript. For a circular specimen with

a thin membrane we have

TM 4 t
Ts r

where tm is the thickness of the membrane and r is the radius
of the specimen. For the specimens and membranes used in the present

investigation tm /r 0.02. In order to determine the shear

modulus of the membrane an extension test was run on a piece of membrane

approximately 0.5 in. long. This gave a value of Em which was

converted to G, Assuming a value of /L = 0.5, Gw, 7.5 x 103

lb./ft.2 Using a value of Gr equal to 1.5 x 106 lb./ft.2


%" 4 x lo-4
Th


We also have the relationship that

2W


where W is the total energy available in one stress cycle and AW

is the energy lost in one stress cycle. If we make a conservative

estimate that AW/W for the rubber material is 0.5 then

-4
10 .






-40-


Since the smallest logarithmic decrement measured was on the

order of 2 x 10-2 the error introduced by the membrane is insignificant.














CHAPTER III


MATERIALS, EQUIPMENT, CALIBRATION AND
PROCEDURE FOR THE PRESENT
INVESTIGATION



Materials


There were four different materials used in this investigation.

Each is described below and the grain size curve for each is shown in

Fig. 10.

Ottawa sand. Standard Ottawa sand which had been sieved for the

fraction between the No. 20 and No. 30 sieves was used for most of the

investigation. This is the material prepared and used by Hardin (1961).

He reported that the minimum void ratio was 0.50 corresponding to a unit

weight of 110.5 lb./ft.3 and the maximum void ratio was 0.77 correspond-

ing to a unit weight of 93.6 lb./ft.3

Glass beads No. 2847. Glass beads, all of which lie between the

No. 16 and No. 20 sieve, were obtained from the Prismo Safety Corporation,

Huntingdon, Pennsylvania. These beads are essentially perfect spheres

as viewed from a microscope. They have a specific gravity of 2.499.

The minimum void ratio was 0.57 and the maximum void ratio was 0.75.

Glass beads No. 0017. This material was also obtained from the

Prismo Safety Corporation. These are very fine beads and near silt

size. Ninety-five per cent pass the No. 200 sieve and 96 per cent are

retained on the No. 400 sieve. They have a specific gravity of 4.31


-41-


















Gai baAldi


----
4---,




-I-c


.4 -




* ft


10 0.5


4847

- p--


I._6



: f
L_




4- -....


..

-V.-


L


I--..,


...4,.. -














i-
G. -
F2._


- .-I -


ass
. OC


100


pi1-







^r




If I
1- 1- 1 --I
f-. I
VTr.-


-4-




- I







* I-


- V


SAI
I------ I I I


-


---- ; fJ -f --
------------- -I- .-.L_ __

S....... ; T cul te -
: \ 1150


Il-i:






Ii






-4----i-
4- -4


I


4- -.4
i-'I-*'
I. ; 2.
I .L


-i -I--
,
r-t-


2 - _~L 4_I r L- _


---.- I--
I- i -


S -A I I I I I I


- j-1;.


I --


SFC-


I- -- --i


..P.I


0.05


. -


-- 4-I-I- ---


(17

I- --


*-. :4:--i---~


I -.


xx:-:: r ,j

a-'"t' .i-;


0.02


,... L.11. .. -

' .- -e -:
1~ '--- -i~-------i*--~ ---- --
I I---II .-.l. I ;
- r -4-.- --- i--L L
-.- -. :-----j --


0.01 0.005


0.002 0.001


Grain Diameter in mm.


Fig. 10 Grain size curves for the materials used in the present research.


~1. ~. -.... - -4 i -I
11 I- '-
I I.. -

124 .xj .- - -
I-~ I---
...
S*-*- I- 2 r-i---
- I... --I--. .1 -
I i i
i -r -- t v -


"


_-4


am


r a :


i r- rl- I -


-- i


'


~- --


--A.






-43-


which is very high. The reason for the high specific gravity is due to

the fact that a high index of refraction is desired in their commercial

use. The minimum void ratio for this material is 0.57 and the maximum

void ratio is 0.76.

Novaculite No. 1250. This is a very fine quartz powder obtained

from the American Graded Sand Co., 189-203 East Seventh Street, Paterson

4, New Jersey. This material was considered to be a silt as shown by

the grain size curve in Fig. 10.



Tests


Three groups of tests were run and are summarized in Table 1 and

as follows.

Group I. These tests were run with Ottawa sand to obtain data

on the effects of amplitude, pore fluid (air, water and dilute glycerin),

mode of vibration and density on velocity and damping for both torsion

and compression.

Group II. After the tests of Group I were completed it was

decided to run tests with the two sizes of glass beads described above

in the dense condition both dry and saturated.

Group III Triaxial CFS-tests as developed by Professor

J. H. Schmertmann of the University of Florida and described by Schmert-

mann and Osterberg (1960) were run by Mr. George Kao to obtain cohesion

and friction characteristics of the Novaculite No. 1250 in the dry con-

dition. A torsional vibration test, as well as damping characteristics,

was run on Novaculite No. 1250 in the dry condition.






-44-


TABLE 1

Summary of Tests


Group Test No. Material Void Ratio Pore Fluid Type

I 10 Ottawa sand 0.52 Air Torsion
11 Ottawa sand 0.67 Air Torsion
14 Ottawa sand 0.52 Water Torsion
21 Ottawa sand 0.64 Water Torsion
12 Ottawa sand 0.52 Air Compression
16 Ottawa sand 0.66 Air Compression
13 Ottawa sand 0.51 Water Compression
15 Ottawa sand 0.66 Water Compression
20 Ottawa sand 0.50 Dil. glycerin Compression
19 Ottawa sand 0.64 Dil. glycerin Compression

II 25 Beads #2847 0.59 Air Torsion
Water Torsion
26 Beads #0017 0.58 Air Torsion
Water Torsion
23 Beads #2847 0.58 Air Compression
Water Compression
24 Beads #0017 0.58 Air Compression

III 28 Novaculite 0.80 0.83 Air Torsion






-45-


Equipment of Previous Investigators


Several methods have been used to measure the stress wave ve-

locities in a cylindrical specimen. The methods differ mainly in the

end conditions which are imposed or are assumed to exist.

Wilson and Dietrich (1960). The apparatus developed at the

laboratory of Shannon and Wilson is shown in a schematic diagram in

Fig. 11. An amplified audio-frequency signal is supplied to a driver

unit adapted from a loudspeaker. For longitudinal vibrations the driver

is directly connected by an aluminum rod to a clamped rim diaphragm of

aluminum having a natural frequency several times greater than that of

the soil specimen. For torsional vibrations the driver is directly con-

nected to an aluminum clamp to provide a torsional twist to the specimen.

The specimen rests on a brass plate and is enclosed with a lightweight

cap and a rubber membrane. A standard phonograph crystal is suspended

by rubber thread from a tie rod or stand and records the motion of the

top cap on a cathode ray oscilloscope.

It is stated that the restraint of the specimen has been verified

to correspond to the lower end clamp and the upper end free.

Hardin (1961). Hardin developed apparatus to measure shear wave

velocities and compressive wave velocities for a soil specimen which was

considered free at each end. Figure 12 shows a diagram of the driving

equipment for each apparatus. A permanent magnet was attached to each

end of the specimen by means of Plexiglas caps. For the compression

wave the bottom magnet rested on rubber pads between electromagnets. The

other end of the specimen was placed between two sets of coils which were

identical to the driver, and these produced a signal which indicated the















110 v.
A.C.


crystal pickup


vacuum line


110 v.
A.C.


detail -



rubber suspension

to oscilloscope

cap)
) ,specimen


stand for -
pickup


aluminum vacuum
liaphragm line

see
speaker detail
driver system


'"- driving rod

brass stem

aluminum clamp


Lgnal lead base- to amplifier
generator


Logitudinal vibration apparatus Torsional vibration apparatus


Fig. 11. Schematic diagram of apparatus used by Wilson and Dietrich.













specimen


electromagnet


frame --14 ' I -
-J ii permanent magnet
Side view frame
p.
rubber









Top view Top view

(a) (b)


Fig. 12. Drawing of vibration mechanisms used by Hardin (a) shear wave apparatus.
(b) compression wave apparatus.





-48-


motion of the top of the specimen. The apparatus for torsional vibrations

was identical except that the electromagnets were placed in a position

that would produce torsional motion. In this case the specimen rested

on a pivot.

Both types of equipment considered above have end conditions

which are not completely specified. The equipment used by Wilson and

Dietrich is considered to be fixed at the driving end. If this were

true then there would be no vibrations in the specimen. The equipment

relies on the fact that at the natural frequency of the specimen very

little motion is needed at the base in order to maintain steady state

vibrations. The relative amount of motion between the top and bottom

of the specimen for small values of damping is such that the node is

very close to the base.




F= SINwc t SPECIMEN





Fig. 13.- MEdel representing apparatus used by
Hardin.


A model of the conditions for the apparatus built by Hardin may

be considered as shown in Fig. 13. The torsional case is shown. The

rotational inertia of the end caps is represented by the discs at each

end of the specimen and the rubber pads are represented by springs.

The magnitude of the effect of the rubber pads is essentially indetermi-

nate. However,all of Hardin's tests were run at such small amplitudes

that this effect would be negligible.






-49-


Equipment for the Present Investigation


Several practical considerations must be made when deciding upon

the equipment for measuring the dynamic properties. The most important

thing to keep in mind is that the design must be as simple as possible,

otherwise, considerable time will be spent on refinement. In choosing

between a free-free and a fixed-free type of test condition, the fixed-

free condition has two main advantages. First of all, the end conditions

are more easily determined, and second, the first mode of vibration will

occur at one-half of the frequency for the free-free condition. Exper-

ience has shown that it is much easier electrically and mechanically to

work with lower frequencies.

Two pieces of equipment were specially designed and built to vi-

brate the specimen at relatively large amplitudes with longitudinal and

torsional vibrations. Each was based on the fixed-free condition. The

equipment was different than that described above in that the base of

the specimen was fixed to a large mass and the vibrations and displace-

ments were applied and measured at the top of the specimen.

Compression apparatus. The frame of the apparatus was built from

a piece of 4" standard steel pipe. The base for bolting the aluminum

bottom cap of the specimen was made out of a piece of steel 1/2" thick

and 1 1/4" wide. The ends of the base were drilled and tapped with one

hole on each end. Screws which fit through the sides of the pipe frame

hold the base in place. The holes in the frame were drilled so that the

base position may be adjusted in order to align the specimen vertically

in the apparatus. In order to "fix" the base of the specimen the inside

of the frame was lined with four sheets of lead 0.10" thick. This makes






-50-


the mass of the specimen very small compared to the mass of the frame to

which the specimen is bolted. The total weight of the above apparatus was

29 Ib. as compared to 1.2 lb. for a specimen.

The driver and pickup used is shown in Fig. 14. The design of

both the driver and pickup is similar to that of an ordinary loudspeaker.

The coils were made by placing a piece of paper around a tube of the

desired diameter. This paper is glued so that it will not unroll and

also be loose enough to be removed from the tube after the coil has been

wound upon it. After each layer of wire has been wound upon the paper

tube a layer of glue was applied to hold the coil together. If the coil

was to be used as a pickup a clear spray lacquer was used as a glue, but

for the driving coils it is necessary to use a glue that will withstand

high temperature. For this purpose "2-ton" brand epoxy glue was used.

It is absolutely necessary to insure that the loops of the coils are

rigidly held together,otherwise erratic wave forms will be recorded. It

is also essential to have a rigid connection of the coils to the top cap

of the specimen, i. e., the natural frequency of the connection must be

several times greater than the frequency range used in the experiments.

The driver and pickup must also be made as light as possible. The total

weight of the driver and pickup for the compression apparatus was 30.6 gm.

The permanent magnets were made from cylindrical Alnico V magnets.

Soft steel was machined and placed around the cylindrical permanent mag-

net so that the lines of flux would be concentrated in an annular opening

through which the coils would fit. The Alnico V is very hard and

brittle and cannot be machined easily. Therefore, it is necessary to

either glue or clamp the soft steel to the Alnico V. Non-magnetic ma-

terials should be used for clamping so that the magnetic flux is not









brass rod



Ii (


soft steel


frame


permanent
m a g n e t ---

driving
coil


1/16 in. brass
rod frame for
holding coils


pickup
coil


permanent
magnet


pickup
coil


permanent
magnet


top cap of specimen


Top view

(a)


Side view

(b)


Fig. 14. Vibration mechanisms used in present research (a) shear wave apparatus (b) compression
wave apparatus.






-52-


disturbed after the magnet has been constructed. The magnetism can be

increased several times by placing the finished magnet in a strong field.

If the soft steel is later removed it will be necessary to remagnetize

the magnet when it is put back together.

The magnets are fastened to the top of the frame by means of

threaded brass rods as shown in Fig. 14. These provide a means of ad-

justing the vertical position of the magnets. The holes through which

the threaded rods pass are much larger than the rods themselves. This

allows a limited amount of movement so that the magnets may also be

positioned with respect to the horizontal plane.

Torsion apparatus. The frame of the torsion apparatus is very

similar to the frame of the compression apparatus except for the design

of the driver and pickup, which is shown in Fig. 14. The coils were

mounted in a frame constructed of brass rods 1/16" in diameter. The

frame was made as rigid as possible while keeping the torsional inertia

to a minimum. The permanent magnets were circular bar magnets which

were mounted on the frame and projected into the center of the coil.

Positioning of the magnet was accomplished by soldering a threaded brass

rod in an off center position to the magnet. The threaded rod provided

in and out movement and horizontal movement was obtained by rotating the

eccentrically located rod. Vertical movement was accomplished by cutting

a vertical hole in the frame of the apparatus.

The equipment described above could be used to measure the

resonant fequencies for torsion and compression and also the decay curves

after vibration in a steady state condition by cutting off the power to

the driving coil. The decay curves were recorded photographically on an

oscilloscope.





-53-


Commercial apparatus. An MB Electronics Type P 11, Model T 135234

power supply was used for driving the coils in the vibration apparatus

and also for driving an MB Electronics Model C 31 pickup calibrator. The

range of the Model C 31 calibrator is 5 to 1000 cycles per second with

a maximum force vector of 25 lb. A probe type pickup, Model 115, also

manufactured by MB Electronics was used for measurement of vibration

amplitudes and for calibration. A Tektronix Model 502 dual beam oscillo-

scope was used for the measurement of output from the pickups and drivers.

Decay curves for damping measurements were recorded with a Dumont Type

450 oscilloscope camera.

The apparatus for measuring vibrations was placed in a triaxial

cell for testing. The triaxial cell was manufactured by Geonor A/S,

Oslo-Blindern, Norway. In order for the testing equipment to fit inside

the cell a longer Plexiglas tube and fastening rods were made. Air

was used for the confining pressure and pressures from 0 to 50 lb./in.2

were measured by a mercury manometer.


Calibration

Compression pickup. The pickup for the compression apparatus was

calibrated using the C 31 pickup calibrator. The C 31 calibrator was

bolted to th concrete floor and the pickup coil was mounted on the

calibrator. The magnets for both the driver and pickup were held in

position by a stand which was resting on the floor. A 100 lb. weight

was placed on the stand to hold it in position. When the coil was vi-

brated no vibrations could be detected by touching the permanent mag-

nets. Since the amplitude of the coil motion was on the order of 0.01 in.

the movement of the magnets is negligible. The pickup was calibrated in

the range of 100 to 1000 cycles per second for different positions in








relation to the coil and the magnets as shown in Fig. 15. It was found

that the calibration was insensitive to the horizontal position or

centering of the coil. There was a slight amount of variation with the

vertical position, but for the ranges indicated in Fig. 15 the variation

was within + 3 per cent.

The actual calibration was carried out by displaying the output

from the C 31 calibrator and the output from the pickup simultaneously

on the dual beam oscilloscope. Both outputs were directly proportional

to velocity and for sinusoidal vibrations, amplitude and acceleration

may be found by dividing or multiplying respectively the velocity by the

circular frequency.

Torsion pickup. The pickup used for the torsional vibrations was

not completely satisfactory in that the calibration was sensitive to the

relative position between the coil and the magnet. This required that

the pickup be calibrated before each test. After the specimen was placed

in the testing apparatus and held in position by a vacuum it was vibrated

at the first and second modes of vibration. The amplitude was measured

using the MB Type 115 probe pickup and correlated with the output from

the pickup on the specimen. A typical calibration is shown in Fig. 16.

For a velocity type pickup the plot of the calibration factor in inches

per millivolt vs. frequency will be a straight line with a slope of

minus one on log log paper. This is the case for the compression pickup

but not for the torsion pickup, due to a decrease in sensitivity with

increasing frequency. Since only two points are obtained for the cali-

bration curve a straight line was plotted and assumed to be relatively

accurate for the frequency range actually used in the experiments. The

errors will be small near the two calibration points, and the actual

portion of the curve used in the experiment is shown in Fig. 16.






-55-


1 x 10-4







5


500


1000


200


Frequency, cycles/sec.
Fig. 15. Calibration curve for the compression pickup.


____I 2---







______~~~~7 v..-
I Iii
I,.










4-
~~~ :~~ I I



*'? 0.40 in K :

0.7In < < .80 in I I I1 1:1~: '

__________________ K J ~ i < U


1 x 10"5






-56-


7 x 10-5




5


1 x 10-5


100


200


500


Frequency, cycles/sec.

Fig. 16. Typical calibration curve for the torsion pickup.


7T



t- f
I 4-




144it Wit cu:
it 7


_ _- '-1 1it
j I -




T
-- 5 *- M- --,
~T-TT-i --r __



~~r iVIIV j4


1000






-57-


The torsional mass moment of inertia of the driver and pickup

along with the top cap of the specimen was measured using a torsional

pendulum. The pendulum was made from a 1/16" diameter, 11.6" long brass

rod which had two small steel plates soldered onto each end. One end

was bolted to a steel frame and the pickup and top cap were fastened

to the opposite end. The pendulum was calibrated by measuring the angle

of twist per unit torsional moment and also from the natural frequency

of vibration of an object of known torsional mass moment of inertia. The

two methods were used as a check and agreed within 2 per cent. The

natural frequency of the pendulum with the pickup and top cap attached

was measured by observing the frequency of maximum amplitude when a

variable frequency current was passed through the coil. The spring

constant for the pendulum was found to be 0.64 in. lb./rad. and the

torsional mass moment of inertia of the pickup and cap was calculated

to be 2.14 x 10-4 in. lb. sec.2 or 0.247 gm. cm. sec.2

Camera. The lens for the Dumont Type 450 oscilloscope camera

could be adjusted to give an object to image ratio between 1:1 and

1:0.85. The adjustment was made so that a 1:1 object to image ratio

was obtained with the cathode ray tube. Since the cathode ray tube

and the graticule are not the same distance from the camera they will

be photographed at different scales. Therefore, the graticule only

serves as a guide for alignment of the picture when measurements are

made on the photograph.

Polaroid Type 42 film was used in the camera to record the decay

curves.

Wide range oscillator. The oscillator was calibrated for frequency

with an electronic counter over the frequency range to be used in the






-58-


experiments. The maximum error was found to be 0.7 per cent and the

average error was about 0.3 per cent.

Oscilloscope. The calibration of the oscilloscope was checked

periodically and corrected if necessary. When properly adjusted the

accuracy of the instrument is within 3 per cent of the indicated readings.


Electrical measurements

A schematic wiring diagram for the testing apparatus is shown in

Fig. 17. Details of the high pass filter, attenuater and phase shifter,

and time delay and triggering mechanism are shown in Figs. 18, 19 and 20.

Measurements were made on the oscilloscope which showed the input

voltage to the driver and the output voltage of the pickup. However, due

to the mutual inductance between the driving coil and the pickup coil, the

output voltage of the pickup does not represent the motion of the pickup.

It is necessary to compensate for this induction by applying a signal of

equal magnitude and phase relationship to the differential input connection

of the upper beam. The attenuater and the phase shifter is adjusted to

give a correction of the correct amplitude and phase relationship. The

frequency characteristics of the compression apparatus are such that the

high pass filter is used with the phase shifter and attenuater. For the

torsion apparatus it was not necessary to use the high pass filter. The

adjustment of the induction correction was made by setting the frequency

so that it was not near resonance and adjusting the attenuater and phase

shifter so that there was no signal from the pickup. If the adjustment

is correct then there should be practically no amplitude at each side of

the resonant frequency. The adjustment is good for a limited frequency

range and must be readjusted for each mode of vibration.
















upper beam

-o to input A

-o to input B



Soscilloscope
connections



-o to lower beam




-o external trigger


Note: All grounds are to oscilloscope.


110 v.
AXC,


Fig. 17.- Schematic diagram of electrical equipment.


110 v.
A.C.






-60-


input
0-

two layers of
A. W. G. 36 copper
wire


output
---

three layers of
A. W. G. 40 copper
wire


Fig. 18. Detail of high pass filter.


510





input


:put


Phase shift and attenuation


Attenuation


Fig. 19. Detail of attenuater and phase shifter.





-61-


The time delay switch and triggering mechanism was used in the

measurement of the decay curves. Its function was to trigger the sweep

on the oscilloscope and then cut off the power to the driver after a small


power
input -"


to external
trigger on
scope


Fig. 20. Detail of triggering and time delay switch.




time delay. The oscilloscope was triggered by an electric pulse from

three small flashlight batteries connected in series. A hinged arm was

dropped which would momentarily close a contact to trigger the oscillo-

scope and then fall a short distance and cut off the power to the

driving coil. The distance that the arm fell after triggering the scope

determined the time delay. This distance was adjustable. When only

steady state vibrations were being made the triggering apparatus was dis-

connected.



Procedure


Preparation of membranes

The membranes for the test specimens were made at first with a

liquid latex compound obtained from Testlab Corporation, Chicago, Illinois.





-62-


When this material was used up, the membranes were made from a liquid latex

Type Vultex l-V-10 from the General Latex and Chemical Corporation, 665

Main Street, Cambridge 39, Massachusetts. The latex used first had to be

diluted to three parts latex and one part water, while the other was di-

luted to seven parts latex to one part water. Molds were made from 38 mm.

glass tubing 14 in. long with rubber stoppers in each end. These were

dipped into the liquid latex and allowed to dry a minimum of four hours

between dips. Around eight to ten dips were used for each membrane. It

was found that the latex was affected by absorption of water and to prevent

this, the latex membranes were dipped in a liquid neoprene Type Vultex

3-N-10, which was also obtained from the General Latex and Chemical Cor-

poration. The layer of neoprene was then placed on the inside next to

the specimen.

The membrane was removed from the mold by first dusting the outside

with talcum powder and then cutting off the ends with a razor blade.

Next, the membrane was peeled from the mold and the inside was dusted

with talcum powder.


Preparation of the specimen

The soil specimens were approximately 1.5 in. in diameter and 11 in.

long. The top cap was made of Plexiglas and the bottom cap was made of

aluminum as shown in Fig. 21. Rubber 0-rings were used to hold the mem-

brane against the cap. Dow Corning silicone stop-cock grease was used

between the membrane and caps to provide a good seal.

When dealing with granular materials it is necessary to use a mold

in forming the specimen as shown in Fig. 22. The mold was made from a





-63-


rubber

Plexiglas


I I : aluminum
studs


pore pressure
lines


-1 n- 11
II Ii
D ^ n (b)


studs
(a)


Fig. 21. End pieces used with specimens (a) bottom cap (b) top
cap.




piece of PVC tubing which was cut to form two halves. Tubes were placed

on each half for vacuum line connections and filter paper strips were

placed on the inside to allow a good distribution of the applied vacuum.

The bottom cap was greased and the membrane was placed around it.

The mold was then clamped around the cap and membrane with a hose clamp.

The inside diameter of the mold was such that it fit snugly against the

membrane, sealing off the end of the tube. The opposite end of the

membrane was stretched over the end of the mold and a vacuum was applied

which held the membrane securely against the inside of the mold.

Several methods were used for placing the soil into the mold, de-

pending upon the soil type and density desired. For the Ottawa sand and





-64-


joint
\,


-I







I


Fig. 22. Mold used for preparation of the specimen.


plastic
tubing-




filter
paper


I oI I
^4--l






I I



I I
, 11






-65-


the glass beads the dense condition was obtained by pouring in approx-

imately 50 ml. layers and vibrating each layer with a 1/8 in. brass rod

attached to a small vibrator. This resulted in a condition close to

100 per cent relative density. The loose condition for the Ottawa sand

was obtained by pouring the sand through a funnel attached to a 3/16 in.

internal diameter glass rod which extended to the bottom of the mold. The

rod was kept full of sand and slowly retracted from the mold allowing the

sand to be deposited in a loose condition. The specimens prepared with

the Novaculite No. 1250 were compacted. Since the material is very fine

a special procedure had to be followed. A vacuum was applied to the

bottom pore pressure line during the compaction to prevent the material

from blowing out of the mold and also as an aid to compaction. A teaspoon

of material was added and pressed five times with a No. 7 rubber stopper

attached to the end of a standard Proctor miniature compactor. Prior to

construction of the specimen the Novaculite was dried in the oven at

2200 C. for a period of several days. Dry nitrogen was used as a pore

fluid in the CFS tests on Novaculite in which the pore pressure had to be

controlled.

After the specimen had been placed in the mold the top was leveled

and the cap was placed in position. While firmly holding the cap on top

of the specimen with one hand, the membrane was pulled up around the cap

with the other hand. A vacuum was then applied to the pore pressure line

at the bottom cap. The pore pressure line at the top cap was plugged.

The hose clamp was loosened and the mold was removed from the specimen.

The 0-rings were placed on the top and bottom caps with the aid of a ring

stretcher and the specimen was placed under a vacuum for measurement. The

diameter was measured at seven positions vertically and four positions






-66-


circumferentially making a total of twenty-eight measurements for the

diameter. The length of the specimen was measured in three positions.

The specimen was then placed in the vibration apparatus and bolted securely

to the frame. In order to connect the pore pressure line to the base of

the triaxial cell the tubing was closed by using needle nose pliers. The

vacuum line could then be transferred to the cell connections. The driver

and pickup were attached to the top of the specimen, the magnets placed

in position and the electrical connections made. The pore line for the

top cap was attached to the top connection on the cell and sealed with

a neoprene bonding cement. The triaxial cell was clamped together and the

confining pressure applied after which the vacuum line to the pore

pressure was disconnected. The pore pressure was at atmospheric pressure

during the entire test.

For the saturated tests a vacuum line was attached to the top

line of the specimen and a full vacuum was applied for several minutes

before the bottom valve was open to admit the pore fluid. By making

sure that the pore space was evacuated before allowing the fluid to

enter the possibility of entrapped air is reduced. The water or water-

glycerin mixture was boiled and placed under a vacuum to reduce the

amount of dissolved air. The water-glycerin mixture was composed of

3 parts water to 1 part glycerin because pure glycerin is so viscous

that it is impractical to saturate the specimen with it. On the basis

of the length of time required for saturing a specimen of Ottawa sand

with water it would take approximately 1,500 minutes to saturate the

same specimen with glycerin.






-67-


Recording of data

A sample data sheet is shown in Tables 2, 3 and 4. Table 2

shows the data taken for computation of void ratio and the relation-

ships for velocity and modulus in terms of frequency. These are

calculated from the relationships described by Figs. 4 through 6.

The volume of the specimen was corrected for the membrane by weighing

the membrane and computing its volume on the basis of a unit weight

of 0.92 gm. per cm.3

The measurements recorded for the velocity and damping were

taken after the electrical equipment had warmed up a minimum of 30

minutes. There is considerable drift in calibration as the equipment

warms up and accurate measurements cannot be made during this period.

The variation of velocity with amplitude was recorded for the

first mode of vibration and then for the second mode. Each amplitude

was approximately half of the proceeding one which corresponds to the

successive scales on the oscilloscope. With a maximum of 6 volts peak

to peak applied to the torsional driver the double amplitude of

vibrations was approximately 2 x 10-3 radians depending upon the shear

modulus, damping and density of the specimen. Since the torsion

driver consists of a force applied to a lever arm from the top of the

specimen, both bending and torsional modes of vibration will be measured.

This is shown in Fig. 23 which shows the second mode for bending and

the first mode for torsion. For small values of damping the amount

of bending at the natural frequency for torsion will be insignificant.

The bending and torsion may be distinguished by the phase relationship

between the driver and the pickup. For bending the motion of the driver

and pickup is in phase but for torsion they are in opposite phase.

This is easily seen on the oscilloscope.










TABLE 2


Typical Data Sheet


Test No. 23
Material: glass beads #2847
Specific gravity: 2.499

Beaker + specimen = 854.22 gm.
Beaker specimen = 325.39 gm.
Dry weight of specimen = 528.83 gm.
Weight of membrane = 12.65 gm.
Initial diameter in cm.
4.03 4.04 4.04 4.03 4.03
4.06 4.08 4.07 4.06 4.06
4.05 4.06 4.05 4.04 4.04
4.00 4.00 4.00 4.00 4.01


Type of test: longitudinal wave
Driver and pickup dimensions:
A 0.37" B 0.83"
Weight = 30.6 gm.

Relationships for velocity:


4.03
4.05
4.04
4.00


4.03
4.05
4.03
4.00


Wt. of specimen
Wt. of driver

1st mode
2nd mode


Dry
= 17.3,


= 3.791 x f,
= 1.262 x f,


Saturated
21.3


3.753 x f ft./sec.
1.250 x f ft./sec. a


Average diameter 4.035 cm.
Initial height 10.75", 10.76", 10.74"
Average height = 10.75 in., 27.30 cm., 0.896 ft.


Total volume of specimen
Volume of membrane
Volume of specimen
Volume of beads
Volume of voids
Void ratio 0.585


- 349.90 cm.3
= 13.75 cm.3
= 335.34 cm.
= 211.62 cm.3
= 123.72 cm.3


Unit weights:

dry = 1.577 gm./cm.3
sat = 1.946 gm./cm.3















TABLE 3


Typical Data for Velocity


Velocity Data

Frequency Output Frequency Output Frequency Output Frequency Output
c.p.s c. c.p.s. cm. c.p.s. cm. c.p.s. cm.


5.0 psi

5 mv/cm
3.28

2 mv/cm
5.24
7.22

1 mv/cm
7.20
7.08
4.72
0.5 mv/cm
5.30
6.06


- 10.2 psi

5 mv/cm
5.50
3.22
2 mv/cm
6.66
4.18
4.16
1 mv/cm
5.30
4.78

0.5 mv/cm
4.90
6.36


Press. -


186
541

188
549
189

190
550


191
551


24.9 psi

5 mv/cm
5.18
4.60
2 mv/cm
6.42
5.26
3.96
1 mv/cm
5.10
5.98

0.5 mv/cm
4.78
6.94


Press. -


218


220
643


222
643
642

223
641
642

224
641


50.6 psi

5 mv/cm
6.70

2 mv/cm
5.16
3.06

1 mv/cm
5.44
6.10
3.89
0.5 mv/cm
5.44
6.10
3.89
0.2 mv/cm
3.6
2.0


Press. =


138


142
426


144
436
145

148
441


Press.


150
462

154
469
156

159
472


160
476






-70-


TABLE 4

Typical Data for Damping


Damping Data

Pressure Picture Frequency Sensitivity Sweep
psi No. c.p.s. mv./cm. ms./cm.

5.0 499 135 5 10
500 142 2 10
501 145 1 10
502 148 0.5 20
503 150 0.2 20
504 429 2 5
505 439 1 5
506 442 0.5 5
507 448 0.2 5

10.2 508 151 5 10
509 156 2 10
510 159 1 10





-71-


1st mode
torsion


10

8
S2nd mode
6 bending

S4





120 150 200

Frequency, cycles/sec.


Fig. 23.- Resonance curves obtained with torsion apparatus.






With a maximum of 8 volts peak to peak applied to the compression

driver a double amplitude of approximately 1 x 10-3 in. could be obtained.

It was noted that at some pressures the bending mode of vibration and the

compression mode were near the same frequency. This showed up as two

resonances very close together. A change in position of the driving

coil did not seen to correct this condition but it was found that a

slight change in confining pressure would help remove the condition. The

maximum effect of this condition seemed to occur near a confining pressure

of 25 lb./in.2






-72-


After the velocity measurements were made the decay curves were

recorded. Amplitudes were chosen which would give full scale deflections

on the photographs for each sensitivity setting on the oscilloscope.

The frequency and amplitude as well as the time delay mechanism

were adjusted and the graticule illumination was turned off. The cathode

ray was adjusted for moderate intensity. A camera lens setting of f/4

at shutter speed B corresponds to a sweep rate of 10 millisec. per cm.

The picture was taken by holding the lens shutter open while the

triggering and time delay switch was operated causing the decay curve

to be displayed on the cathode ray tube. The shutter was closed and

reset to f/11 at 1/10 of a second exposure for the graticule at full

brightness. The frequency, vertical sensitivity, sweep rate, confining

pressure and picture number were recorded. The picture number was

also written on the picture for later identification. Figure 24 shows

typical decay curves.

The measurement of logarithmic decrement from the decay curves

was done at first by plotting each amplitude vs. wave number on semi-

log graph paper. A straight line was drawn through the points and

logarithmic decrement was computed from Eq. (106). This proved to be

very time consuming since each picture represented only one data point.

A much faster method was developed based on the fact that the points

plotted as straight lines. A set of curves for the function e

for various values of /V-C were drawn and photographed to make a

4 x 5 in. slide which fits into a photo enlarger. The points of

maximum amplitude for each decay curve were traced onto a piece of

tracing paper and a smooth curve was drawn through them. These traced

curves were placed under a photo enlarger and the magnification was






-73-


Fig. 24. Typical decay curves.






-74-


adjusted so that the curves coincided with each other. Knowing the

coefficient //- OC the time scale of the projected curve, the

time scale of the photograph and the frequency of vibration the logarithmic

decrement was computed from the relationship

"-- /-CX MS./CM. PROJCTE0o CURVE
SMS./CM. PHOTOGRAPH















CHAPTER IV


PREVIOUS WORK OF OTHER INVESTIGATORS



Theoretical solutions have been given by many persons concerned

with wave propagation and damping in porous or granular materials. Hara

(1935) and Gassman (1951) have obtained solutions based on a model of

elastic spheres. Brandt (1955) assumed a model of randomly packed

elastic spheres for his solution. However, Duffy and Mindlin (1957) have

given the most complete theory on the basis of a packing of elastic

spheres. Kosten and Zwikker (1949) have based their solution on a porous

material the solution of which was later extended by Morse (1952) and

Paterson (1956). Biot (1956) has given the most complete solution which

is based upon a porous material.

Experimental studies have been undertaken by many investigators

starting as early as 1935. Factors affecting the propagation of waves

have been reported by Birch (1938), Born (1941), Hughes and Maurette (1956),

Bruckshaw and Mahanta (1954), Collins and Lee (1956) and Peselnick and

Zietz (1959). Extensive vibration studies of soils have been made by

lida (1938 and 1940), Taylor and Whitman (1954), Matsukawa and Hunter (1956),

Shannon, Yamane and Dietrich (1959), Wilson and Dietrich (1960) and

Paterson (1956). Duffy and Mindlin (1957) have performed experiments with

low tolerance steel spheres.


-75-






-76-


Hardin (1961) has discussed the above theoretical and

experimental work in detail and this will not be repeated here.















CHAPTER V


PRESENTATION OF RESULTS



Stress Wave Velocities


Group I

Figures 25 through 34 show the results for the velocity of

torsional and longitudinal waves calculated from the tests of Group I

on Ottawa sand. In these tests the variation of velocity with confining

pressure, density, pore fluid (air, water and dilute glycerin) mode of

vibration and amplitude of vibration was determined. The confining

pressures chosen for each test were approximately 5, 10, 25 and 50

lb./in.2 and the results at each pressure are plotted in the same

figure. Tests were run at both loose and dense conditions corresponding

to a void ratio of approximately 0.65 and 0.51 respectively. Only the

first and second modes of vibration were measured due to the fact that

the higher modes of vibration occur at frequency ranges where calibration

of the equipment is questionable. The amplitude of vibrations was varied

over the range of the smallest measurable vibration to the maximum

attainable with the equipment which was approximately 5 x 10-4 in. for

compression and 1.5 x 10-3 rad. for torsion. The ratio of magnitudes

of any two successive amplitude measurements is approximately 2:1 as

this corresponds to the ratio of two successive sensitivity settings of

the oscilloscope.
-77-






-78-


Figures 25 through 28 show comparisons of velocity between dry

and saturated specimens at the first mode of vibration. Figures 25

and 26 are torsion tests with loose and dense specimens respectively.

Similar figures for the compression tests are shown by Figs. 27 and 28.

In addition the tests in which the first and second mode velocities

were measured are plotted individually. The second mode velocity could

not be measured in the early compression tests since the apparatus was

not fully developed until after test No. 15.


Group II

The velocity test results for the glass beads are shown in Figs.

35 through 38. These tests included two sizes of beads one of which

corresponds to a grain size near Ottawa sand and the other a silt grain

size. The tests were performed the same as those in Group I except that

only the dense condition was tested. This corresponded to a void ratio

of around 0.58 for both sizes of beads. The large beads had a specific

gravity of 2.499 which is slightly less than Ottawa sand but the silt

size beads had a specific gravity of 4.31 which is much greater than

most soils. Each specimen except that for test 24 was tested in the

dry condition and then in the saturated condition. The results are

plotted to show the comparison of dry and saturated conditions of each

test.


Group III

Velocity tests were run on a compacted specimen of Novaculite No.

1250 with the torsion apparatus. This is a crushed quartz material with

ten per cent of the particles less than 0.002 mm. The behavior of this

material is quite different than for that of Ottawa sand and glass beads







-79-


i~-:fti e. mi0ri52jF


1PrJ.:'
10.52


1:I


1300





1200





1100



4)
0
4J



0















700





600






500




4 900
400


j 1 I 7


16- 6-


I 77-


3. 20: 1i


:-' :-
!: I: : I
---
L ! i___.._


_,.__L_ -

:: : jl-
i I
i i


a-.fiti *Trtde-4
-sattteate4 -


.0 I


5b30 lb./ft.2


- itii
V .:j ::

* ., i 71


./ ft
..i : ;.


21ii ;ii-


I--t =-~ -4- ET-*-- -fil~ + -I --- -----


-1 1
~-~* 5+

K iLA.


_-77--

: 0-:


5 ji l


I-


aag :1-1 I.tf ~ 2--~!i

. .
2290;l\/t


7 7---7


_..i ftL


-7 7






_2I9 10. -:



2.0 x 10-3


Double Amplitude, Radians


Fig. 25. Variation of velocity with amplitude for Ottawa sand
dry and saturated with water.


-K-i


aq~


-r~~,


: i
--Y----
-u; I-


i


I


b


7t7 717-


--


-"


i


--


I


--L------' -


-I-----r-----c-----r


r


T


.. .~.. ;:--1:7


r-t--I-_LLI-L_ ~-_


I


[


I


__ _T


1


i .-,_







- - ~
i:


I


---c-L-- '




















*"! .U -!' *i "
4-"






1'^ :; -


- LI

I 9 lb./ft.
S.7-n-i HI- 7i


a____1: ti




t I
I I
-- illb-ff. i- :ii~ I :


10
+ ii I





SI I- ;- I-
L^ i
F i

1.0


692

I- I


t- i-.-


Ib.


2

a'-


-:ttilrf- --i-
1


i

j:ll f

_1 _i
i. I



J. i. i. ... 1;.L, _


2.0


-! IS~ Ib
IL87 1b../


''


*i~ I:


.2:


3.0 x 10-3


Double Amplitude, Radians

Fig. 26. Variation of velocity with amplitude for Ottawa sand dry and saturated
with water.


1000




900


800




700




600


500




400






-81-


1800




1700




1600




1500




4 1400

4J
U
u
0
1300




S1200



1100
1100


K. : I 1:7 21~


T.


;: !: !


"!


_ __ -___ .:1.-



'626 lb./


I '... : 'I

605, lb ft



Et I'
i- I it:I.


0.5 1.0 x 10-3


Double Amplitude, Inches

Fig. 27. Variation of velocity with amplitude for Ottawa sand
dry and saturated with water.


7080 Thid/ft-.2] I t
0 iTt o -7.




Ma a )
'----FIrt:jm~

-113t IMD'e

.J~j1 ~ atU ate 7 7


-7 ~ 2 -
4 -t I ;;It :it li: Til



3710 Lb.(ft.2
_3i3O1 i'2IT 1 I I'~'i' i



-3-0,... 2 .) i '-



77!1


-7
~ 1_ . :T-.. 7. _7 7 .... ...... -.i~iiii~








-774
.1 1.. i-








__1 1i11~ 1 .1.
jr I:: :: ::;: ,:: I~: i: : ::,:i.:t
.i ii t : I i~iii~ii jI-I~ I~:J4:
::_: .7
1 i l
1,: i..i-~i.. lll~ .-. i --; ii - I ii; 1.; 111:1.7,

71 : ; i .


1000




900




800


)' It
Q4bi -
ii
:


i






-82-


1700




1600





1500




S1400
0


I,''


3- 7
37::.


I i


-I ,


S.. ..... I'-'
--a6. /

l 2:2J :kl.!i ..




~~~~ .! .: i!


I I

Ia4;







I;II
:12 P I
K .

I 7Lii.


! :. I -.


5 1,/E


.__ __. ...












0.5


'7


.1',

- a


7>1


360(


YTT-1
Tr ii


-i.
-- :-


t .2'


2i~~ii1! I 4'bit2

I J


I I


4 I7 .


I I !
I
I
i_
'

LT'L;,~
iii t:-:

i'i '


i ..



V-


I.


I i i


SI: i:

I I.--


1.0 x 10-3


Double Amplitude, Inches

Fig. 28. Variation of velocity with amplitude for Ottawa
sand dry and saturated with water.


-t
. : i ;


0.'~'


i . .,-'I /I .

2 -- - Dr- .
T I- .. .L... ..f.- 4 ,:',., . ... '- :_- .- t: -.";. ,**



I i i 4 i i _
._._._ 0 b f--- -- -- 0


-- ----- Tet Nd.-lt .

... ..... ..... L .......... .t.. 4- ... ... Ri . .. ..
-- 7200 b/. i I au ted

7200 I b./ Saturlted


1300


.nn I ".l;::


1100


1000





900





800





700


_____ __r_ Cf^r~__P_~


` '! '


: I


~'':''

iit-


77i


'_ "T[ -


"i r:. .


4 4l


'1:
: i. : :


i I
I



I


ii iitil;:




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