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 .. ...
CFSTest . . . .
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
CFSTests . . . . . . . .. 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 TwoDimensional 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 WaterSaturated
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 WaterSaturated
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 WaterSaturated
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 WaterSaturated Ottawa
Given by the Blot Theory . .
82. Variation of Damping and Velocity with
Pressure for WaterSaturated 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 GlycerinSaturated 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 = nondimensional elastic properties
OC phase angle between force and displacement
xvi
jc= nondimensional 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 nondimensional 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 CFStest
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 \Zr
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
twelveinch specimen fixed at one end were 5 x 10"4 in. longitudinally
and 1.5 x 103 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 stressstrain 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 nonlinear stressstrain 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 fluidfilled 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 fluidsaturated 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 facecentered 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 nonlinear 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 stressstrain
relation for a medium composed of a facecentered 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 stressstrain 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 stressstrain 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 (43/)(877)_______ 3 &2 3 ]
(' o) ((43) (87, )() 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 onethird power variation of loga
rithmic decrement with pressure.
Biot
Biot (1956a) presented a theory dealing with the propagation of
elastic waves in a fluidsaturated 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
solidfluid 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 fluidsaturated 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)
Ka 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 IdI ) 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 fluidsaturated 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 nondimensional 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 LT 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 twodimensional 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 twodimensional 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 onehalf of the total friction force.
In order to use the results the following ratio is used:
(4 2)
where ti is a nondimensional 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 FK + ^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
(54)
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 crosssection.
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,Z2cMF,)=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 nonlinear stressstrain 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 AI 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
A1 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 A1 r Xd
c)2 = I + c 3) xo 7
W (A + 3) 1 & le j
(~) e +
X (IQo +)Aa 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 stressstrain 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 on,1 i ' ,' '
33
24 44Z  '
I.,. I.'i
I..
Ic  ~ I .i I ' .
 ,* I
4 i ' i 1 i 4 i '
I
S ''
r 1 t t
r;j !.,
i i i
,_  I.
: ,
.1C 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 lo4
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 102 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. Ninetyfive 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,
Ic
.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
Ili:
Ii
4i
4 4
I
4 .4
i'I*'
I. ; 2.
I .L
i I
,
rt
2  _~L 4_I r L _
. I
I i 
S A I I I I I I
 j1;.
I 
SFC
I  i
..P.I
0.05
. 

4II 
(17
I 
*. :4:i~
I .
xx::: r ,j
a'"t' .i;
0.02
,... L.11. .. 
' . e :
1~ ' i~i*~  
I III ..l. I ;
 r 4.  iL 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 ri
 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., 189203 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 CFStests 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 audiofrequency 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 freefree and a fixedfree 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 onehalf of the frequency for the freefree 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 fixedfree 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 "2ton" 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. Nonmagnetic 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,
OsloBlindern, 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 104
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 105
5
1 x 105
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 ,
~TTTi 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 104 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 lV10 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
3N10, 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 0rings were used to hold the mem
brane against the cap. Dow Corning silicone stopcock 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
^4l
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 0rings 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 twentyeight 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 waterglycerin 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 103 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 103 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 /VC 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 104 in. for
compression and 1.5 x 103 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
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j 1 I 7
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Fig. 25. Variation of velocity with amplitude for Ottawa sand
dry and saturated with water.
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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
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Fig. 27. Variation of velocity with amplitude for Ottawa sand
dry and saturated with water.
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Fig. 28. Variation of velocity with amplitude for Ottawa
sand dry and saturated with water.
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