ABSTRACT

EFFECTS OF CONFINING PRESSURE ON POLYCRYSTALLINE ROCK
BEHAVIOR ANALYSED BY RHEOLOGICAL THEORY

by Ezra D. Shoua

The influence of confining pressure upon stress distri-
bution and deformation fields of polycrystalline rock under
load has heretofore been studied theoretically through empiri-
cal relations which have serious shortcomings in describing
accurately characteristic rock behavior. This inadequacy
may be attributed in part to the fact that rock is a hetero-
geneous material whose crystals possess physical prOperties
which differ from the grain boundary properties,

In this investigation, the rock behavior is simulated
by a rheological model composed of two independent systems,
one representing the crystal prOperties and the second the
prOperties of the grain boundaries. This unique arrangement
offers a means whereby in the mathematical analysis the sep—
arate contributions of the crystals and grain boundaries are
correlated to the degree of confinement. This rheological
description of rock behavior not only yields results in good
agreement with existing eXperimental data but furthermore
provides a basis for understanding the variation of physical

constants with changes in confining pressure.

EFFECTS OF CONFINING PRESSURE ON POLYCRYSTALLINE
ROCK BEHAVIOR ANALYSED BY RHEOLOGICAL

THEORY

BY

Ezra David Shoua

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil Engineering

1965

ACKNOWLEDGMENTS

The author would like to express his deep gratitude
for the guidance of Dr. George E. Mase, Professor of Applied
Mechanics and acting Chairman, of the Metallurgy, Mechanics
and Material Science Department, under whose supervision
this study was conducted.

Special thanks are due to Dr. L. E. Malvern, Professor
of Applied Mechanics, for his valuable suggestions during
the formulation of this analysis; to Dr. 0. B. Andersland,
Associate Professor of Civil Engineering, for his interest
and help Whenever requested, to Dr. J. L. Lubkin, Professor
of Civil Engineering and Applied Mechanics for serving on
his committee, and to Dr. S. Serata, Associate Professor of
Civil Engineering for introducing the writer to the field
of rock mechanics.

Thanks go to the National Science Foundation and to
the Division of Engineering Research at Michigan State
University for the financial assistance rendered during
part of the writer's doctoral program.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS O O O O O O O O O 0 O I O O O O O O O O i i
LIST OF TABLES O O O O O O O O O O O O O O O O O O O O V
LIST OF FIGURES O O O O O O O O O O O O O O O O O O O 0 Vi
NOTATION O O 0' O O O O O O O O O O O O O O O O O O O O O ‘ ix
Chapter page
I 0 INTRODUCTION 0 O O O O O O O O O O O O O O O O l
1.1 Statement of the problem . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . 2
1.3 Scope and objectives . . . . . . . . . . 6
II. ANALYSIS OF MATERIAL CHARACTERISTICS . . . . . 8
2.1 Structure of the material . . . . . . . . 8
2.2 Definition and elastic relationships
between differential stress and
differential strain . . . . . . . . . . . 9
2.3 Possible criteria of testing . . . . . . 11
2.4 Behavior of material under different
testing conditions . . . . . . . . . . . 15
III. THEORETICAL ANALYSIS . . . . . . . . . . . . . 20
3.1 Proposed rheological model . . . . . . . 20
3.2 Assumptions . . . . . . . . . . . . . . . 25
3.3 Variation of behavior with confining
pressure . . . . . . . . . . . . . . . . 26
3.3a Variation of the equivalent
elastic response with the degree
of confinement . . . . . . . . . . 31
3.3b Variation of the equivalent visco-
elastic response with the degree
Of confinement O O O O O I O O O O 32
3.3c Variation of the equivalent visco-
plastic response with the degree
of confinement . . . . . . . . . . 41
IV. APPLICATION AND PREDICTIONS OF THEORY . . . . . 45

4.1 Variation of the elastic modulus Eeq
with confining pressure . . . . . . . . . 45

iii

TABLE OF CONTENTS - Continued
Chapter Page
4.2 Variation of the visco-elastic range
with confining pressure . . . . . . . . 51
4.3 Variation of the visco-plastic dif-
ferential strain rate with confining
pressure . . . . . . . . . . . . . . . . 58

V. SUMMARY AND CONCLUSIONS . . . . . . . . . . . 60

5.1 Discussion . . . . . . . . . . . . . . . 60
5.2 Future research . . . . . . . . . . . . 63

BIBLIOGRAPI—IY o o o o o o o o o o o o o o o o o 65

iv

LIST OF TABLES

Table Page

3.1 Calculations of E vs C‘ for different
eq
values Of a O O O O O O O O O O O I O I O O O O O 33

LIST OF FIGURES

Figure

1.1

Schematic diagram of Adams' apparatus for
testing rocks under high confining pressure

(after Adams and Bancroft)3 . . . . . . . . . .
Griggs' early apparatus (after Griggs)15 . . .
Griggs' later apparatus (after Griggs)17 . . .

Polycrystalline rock salt specimen shoWing
randomly oriented crystals . . . . . . . . . .

General triaxial principal stress state . . . .
Hydrostatic confining stress state . . . . . .
Differential stress state . . . . . . . . . . .
Uniaxial differential stress state . . . . . . .
Balanced biaxial differential stress state . . .
General biaxial differential stress state . . .
Variation of Eeq with m . . . . . . . . . . . .
Idealized stress-strain diagram . . . . . . . .
Failure of rock salt specimen tested under
uniaxial differential stress with a confining
pressure equal to one atomsphere . . . . . . . .
Failure of rock salt specimen tested under a
balanced biaxial differential stress with a con-
fining pressure equal to one atmosphere . . . .
Failure of rock salt specimen tested under
uniaxial differential stress with a con-

fining pressure equal to one hundred

atmospheres‘ O D O O O O O O O O O O O O O O O 0

Graph representing the different creep curves
at varying magnitudes of differential stress . .

Typical creep curve at constant differential
stress 0 O O O O O O O O O O O O O O O O 0 O 0

vi

Page

10

10

12

13

14

15

16

17

l7

18

20

21

Figure

3.3

Burgers model showing the relationship
between differential strain rate and dif—
ferential stress for the steady-state

or visco-plastic creep . . . . . . . . .

SVB model showing the relationship between

differential strain rate and differential
stress for the steady-state or visco-
plastic creep . . . . . . . . . . . . .

Proposed mechanical model . . . . . . . .

Differential stress-strain—time relationship .
(a) Deformation.of model due to applied dif-

ferential stress . . . . . . . . . .
(b) History of loading . . . . . . . . .

Typical variation of Eeq with c . . . . .

Variation of Eeq vs. c for various values

Variation of the visco-elastic creep with

Variation of the range with the degree of
confinement . . . . . . . . . . . . . . .

Variation of visco-elastic creep with the
degree of confinement for case (a). . . .

Variation of visco—elastic creep with the
degree of confinement for case (b) . . .

time

Relationship between differential strain rate
and differential stress for the steady state

response 0 O O O O O O O O O I O O O O 0

Relationship between differential strain rate

and degree of confinement . . . . . . . .

Differential stress-strain curves of rock

salt

Specimens deformed in compression (after J.

Handin (1953)18) . . . . . . . . . . . .

Graph of Ee
room temperature 0 O O O O O O O C O O 0

vs. p for rock salt at constant

Differential stress-strain curves,for Solen-

hofen limestone (after H. C.Heard)20. . .

Graph of E vs. p for Solenhofen Limestone at

a constantegoom temperature of 250 C. .

vii

Page

24

24

25

27

27
27

32

34

37

37

39

4O

43

44

46

48

49

50

Figure

4.5

4.6

Creep test results on Solenhofen Limestone
specimens (after B. C. Robertson28) . . . . .

Creep test results on Solenhofen Limestone
(after Robertson (196O)28) . . . . . . . . . .

Normalized visco-elastic creep curves of Solen-
hofen Limestone at different confining pres-
sures and same differential stress . . . . . .

Comparison of the effect of confining
pressure on the range and initial siope of
the visco-elastic curves of Solenhofen
Limestone . . . . . . . . . . . . . . . . . .

Comparison of theoretical and actual test
results for the variation of the range with
confining pressure at constant differential
stress (8:1500 kg/cmz) . . . . . . . . . . . .

viii

Page

52

53

55

56

57

T1

T2

NOTATION

principal compressive stress

principal strain

strain caused by hydrostatic confining stress
differential strain

poisson's ratio
ratio of ty to 5x
differential stress

equivalent elastic modulus

elastic constants of grain boundary bonds
visco-elastic coefficient of boundary bonds
visco-plastic coefficient of boundary bonds
differential yield strength of boundary bonds
elastic constants of crystal grains
visco-elastic coefficient of crystal grains
visco-plastic coefficient of crystal grains
differential yield strength of crystal grains
time

degree of confinement

confining pressure

ratio of E2 to El

. * *
Ratio of E2 to El
. * *
Ratio of E1 to pl
. * *
Ratio of E2 tp n2

ix

ratio of 1/12 to l/Tl
ratio of n2 to n1
ratio of Y2 to Yl

range of visco-elastic creep

hydrostatic elastic constant

CHAPTER I

INTRODUCTION

 

1.1 Statement of the Problem

 

Geologists have long recognized that while most rocks
are capable of sustaining a rather large amount of flow with
little accompanying fracture in highly contorted regions of
the earth's crust, these same rocks exhibit an exceedingly
brittle behavior when deformed under ordinary atmospheric
conditions.

This great increase in ductility within the earth'S‘
crust has been attributed to the environmental conditions
thought to have existed at the time of deformation, and is
ascribed to the effect of any combination of the following
five significant factors:

1. Confining compressive pressure

2. Differential compressive pressure

3. Duration of time

4. High temperatures

5. Presence of solutions

Although these facts have been known for many years,
only since the turn of the century have investigators begun
to study the mechanical prOperties of rocks by attempting to
simulate in the laboratory the complex conditions under which

1

2

they exist in underground formations.

1.2 Literature Survey

 

This transition from brittle fracture to ductile flow
in triaxially confined tested rock samples was first qual-
itatively shown to be dependent upon confining pressure
by Adams and Nicolson (1901)1 and later by Adams (1910)2
and (1912)3 and by Adams and Bancroft (1917)4 when they

compressed small cylinders of marble specimens under a con-

Piston
Specimen

Steel Vessel

 

Fig. 1.1 Schematic Diagram of Adams' Apparatus
for Testing Rocks Under High Confining
Pressure (After Adams and Bancroft)3
fining pressure provided by the walls of a steel vessel.
Their technique, however, had the following disadvantages:
(i) The confining pressure varied during the experiment.
(ii) The confining pressure could not be accurately computed

because of friction between a rock specimen and the steel vessel.

(iii) The specimen was not free to fracture.

3

These findings were later verified quantitatively by Von
Karman (1911)33 and his student B3ker (i915)7. They obtained
accurate stress-strain diagrams by deforming jacketed cylindrical
specimens of Carrara marble and red sandstone in an apparatus
which permitted compressive loading of samples subjected ex-
ternally to liquid pressures of several thousand atmospheres.
Their results indicated that marble under high confining pres-
sure behaved much like a ductile material capable of sustaining
large permanent deformations.

In 1936 Griggs15 published a series of important con-
tributions to the problem of rock deformation utilizing more
precise equipment. His early apparatus (as shown diagram-
matically in Fig. 1.2) was designed for use up to a maximum
pressure of 13,000 atmospheres, or the equivalent of about 28
miles deep in the earth's crust.

Among the significant results of his work on marble and
limestone are the increase in the elastic limit and the ultimate
strength with increase in confining pressure and their dep—
endence upon the duration of the test.

More improved techniques and testing apparatus capable
of handling up to 50,000 atmospheres of pressure were later
used by Griggs (1939)16 (1940)17 to study the effects of time,
temperature and the chemical environment of solutions.

One of the important outcomes of his study was the
presentation of an empirical equation which describes the creep
of rock in compression in which he suggests that the total
strain e(t) at any time t can be represented by the relation-
ship

E(t) = A + B log t + c t (1.2-1)

4

in which A, B, and C are constants of the material.

d:

../*’

’
(

b—V:

\

-4-------- -

it
‘6

I

~

\

\

 

 

 

 

-4---__..-

an“
'0"!('! '0°'l>'l
Gnu-u ll hum. Tums

r l
r—m
III. PACKING

~30»: cannula:
10 unuum
«on '\ ”can“

 

 

 

 

 

 

 

 

THIUST OF
IAIN PIESS

pats:
l LJ

 

 

 

 

 

Fig. 1.2 Griggs' Early Apparatus (After Griggs)15

More organized and systematic tests displaying the
effect of confining pressure were then performed by a number
of Geophysical research workers. Among them is Handin's
work (1953)1§ in which he obtained stress-strain curves of
rock salt deformed in compression, and Handin and Hager
(1957)19 in which stress-strain curves for a variety of poly—
crystalline rocks were presented. More recently, tests were
performed by Heard (1960)20 in which stress-strain curves of
Solenhofen Limestone Specimens (confined up to 5,000 atmos-

pheres) were obtained at different constant temperatures.

5
In addition, tests were performed to evaluate the interstitial
fluid pressure effect within the interstices of a rock sub-

jected to high confining pressures during a triaxial test.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1.3 Griggs' Later Apparatus (After Griggs)l7

Brace (1963)8 tested the ultimate strength of rocks
both in tension and compression, while Baron et al (1963)5
attempted to analyse both the dynamic and static elastic
constants for several types of rocks at various degrees of
confinement. In all of these studies data was compiled
together with analytic observation of microscopic fabric
changes in an attempt to establish the deformation mech-

anism that is involved.

6

Although elastic and fracture prOperties were thoroughly
investigated, very little systematic tests were done on the
creep behavior of polycrystalline rocks. Robertson (1960)28
carried the second and only organized creep testings on Solen-
hofen Limestone specimens at constant room temperatures and
different confining pressures that varied up to 4,000 bars.
Some of his important conclusions were:

1. The creep behavior of polycrystalline rocks exhibit
three distinct stages: transient, steady state and
accelerating.

2. The following empirical relationship was deduced

ta = Kl = K S - K3 (1.2-2)

2
which fits the time t, transient creep strain rate
e and differential stress S of the experiments; K1'

K2, and K3 being constants for a given confining

pressure p.

1.3 Sc0pe and Objectives

 

Geologists and Geomechanic investigators have long
known that the behavior of polycrystalline rocks under load
are influenced by the five factors mentioned previously in
Sec. 1.1, but they have been unable to establish a satisfactory
theory correlating any combination of these factors so that
such a characteristic behavior could be explained or pre-
dicted.

The first attempt by Griggs (l939)l7, Robertson (1960)28
and others was to establish empirical relationships in an

effort to explain some aspects of such a behavior. However,

7
it was found that the empirical relations could neither fit
the whole range of all the results obtained nor account for
the actual mechanism of the stress-strain-time relationship.

More recently, inspired by the application of mechanical
models to simulate viscoelastic creep behavior, several
efforts were made by Emery (l963)12, Price (l964)26, and
others to establish a satisfactory model that would represent
the characteristic mechanism of polycrystalline rock. These
mechanical models mostly simulated uniaxial behavior only and
failed to explain the transitional behavior from the uniaxial
to the triaxial state of stress.

In this presentation, based on observation of the
research work done so far on these polycrystalline rocks, a
new mechanical model is prOposed in which differential strain
and time are related as a function of both the differential
stress and the confining pressure, which accounts for the
change in behavior that is displayed over the range between
the uniaxial and triaxial stress states.

In view of the above, possible testing criteria are
established, thus demonstrating a need for the additional

systematic experiments which are proposed.

CHAPTER'II

ANALYSIS OF MATERIAL CHARACTERISTICS

2.1 Structure of the Material

 

Polycrystalline rocks exist as a compact composition of
an aggregate of crystal grains having various shapes and
orientations. Theory and experiment suggest that between a
crystal grain of one orientation and other neighboring ones,
there exists a grain boundary layer only a few atoms thick in

which randomly arranged atoms take up equilibrium positions

 

Fig. 2.1 Polycrystalline Rock Salt Specimen Showing
Randomly Oriented Crystals

9

and possess a higher free energy than the compact geometrical
arrangement of the atoms within the crystal grains.

The mechanical properties of such a single-phase aggregate
are determined mainly by two factors:
(a) the properties of the crystals of the aggregate, and
(b) the properties of the amorphous grain boundary which
binds these individual crystals together to form the com—
posite mass.

2.2 Definition and Elastic Relationships Between Differential
Stress and Differential Strain

 

 

In order to simulate the behavior of the above poly-
crystalline material under load, it is important to define
the stress state that governs its deformation.

Consider a general state of triaxial principal stress

condition with ax, oy, and'oz acting in the x, y, 2 dir-

ections respectively, with Ox > 0y > 02. (For simplicity
oz will always be considered as the smallest principal stress.)

Compressive stress is considered positive.

X

 

 

0y

2— fl

 

 

 

 

 

Fig. 2.2 General Triaxial Principal Stress State

10
This general state of triaxial stress is equivalent to a
hydrostatic confining stress tensor, oz, acting in the x, y, 2
directions, plus a differential stress state tensor, Si, defined
as
Si = (oi — oz) (2.2-1)

acting in the x, y directions only.

 

/ °2
02/

Fig. 2.3 Hydrostatic Confining Stress State

Fig. 2.4 Differential Stress State

Corresponding to the above defined stress conditions, the

state of triaxial elastic strain is given by

Ee'x = Ox"- u(oy + oz) in the x-direction
Eg'y = 0y - p(OX + 02) in the y-direction (2.2-2)
Ee'z = oz - u(ox + oy) in the z-direction

11

which can be resolved into a strain, 8? caused by the hydro-
1
static confining stress, namely

E8". = o - ”(Oz + o

1 z in the x, y, z—directions (2.2-3)

2)
and a strain caused by the differential stress tensor, defined

as differential strain tensor, 8i which is given by

 

Eex = E(e'X - e"X) = (0X - oz) - u[(oy - oz) + (oz - 02)]
or Eex = S - uS in the x-direction
X Y
Eey = Sy - uSX in the y-direction (2.2-3)
Eez = - u(Sx + Sy) in the z-direction
from which
62 " Ll
= (2.2—5)
EX + 5y 1 - u
if we define
E
m = y (2.2-6)
6x

then we get the relationship

8 - (l + m)u
= (2.2-7)
1 - u

 

 

2.3 Possible Criteria of Testing

 

With the above definitions, there are three possible
cases of loading: (i) uniaxial differential stress state, (ii)
balanced biaxial differential stress state, and, (iii) general
biaxial differential stress state. Cases (i) and (ii) rep—
resent the extreme conditionSqthe general testing pattern is

represented by case (iii).

12

Case (i) Uniaxial Differential Stress State

 

 

 

in which
ox>0y=oz>0
giving
Sx = (0X - oz) - - T
Y
_ ‘ - S2 = 0-
0x
4'

02

Fig. 2.5 Uniaxial Differential Stress State

and Bax = SX
Eey = - uSX
Eez = - uSX
whence
e
m ziz —p
6x

(2.3-l)

(2.3-2)

(2.2-3)

Case (ii) Balanced Biaxial Differential Stress State

in which

13

Fig. 2.6 Balanced Biaxial Differential Stress State

 

 

giving
-SX = (oX - oz) - - 1
- Sy = (0y - oz) - (2.3-4)
_ - - S2 = 04
and Bax = SX - uSy = Sx(l - u)
Eey = Sy - uSX = Sx(1 - u) (2.5-3)
Eez = - p(Sx + Sy) = - ZuSX
hence
e
m _____Y a 1 (2.3-6)
6x

Case (iii) General Biaxial Differential Stress State

 

in which
OX > 0y > Oz & 0
giving
-Sx = (0X - oz) - - -

- S = (0y - oz) - (2.3-7)

 

 

l4

OX>Oy

Fig. 2.7 General Biaxial Differential Stress State

and Eex = Sx - uSy
E6 = Sy - qu
Eez = - p(Sx + Sy)
hence
E
S =-—————%€ + us )
X 1 _ U2 X \y

\

but by equation 2.2-6

6 = max

Y
and therefore

S = = E 5x
X 1-112 eq

 

 

Equation 2.3—ll gives the
differential stress Si and the

a general biaxial differential

the equivalent elastic constant Ee

(2.3-8)

(2.3-9)

(2.3-10)

(2.3-ll)

relationship between the
differential strain £1 for
stress state. The change in

q with m is shown in Fig. 2.8.

15

 

 

 

 

 

 

 

 

 

I?”

Uniaxial Differéntial Balanced Biaxial
Stress State (m-= - uh 23 A/"“ = 1/2.' differential stress
V r / em:

/ 11:
IE'
1 3..
-l/2 -l/3 0 1/2 1 m

Fig. 2.8 Variation of Eeq with m

2.4 Behavior of MaterialyUnder Different Testing Conditions

 

It has been experimentally observed and verified (esp-
ecially by Brace (1963)8 who ran tests on polished sections
under dark field illumination) that for a certain constant
rate of uniaxial loading an idealized stress-strain curve
shows four characteristic stages as illustrated in Fig. 2.9.

In stages I and II the behavior is elastic; nearly all
the strain is recoverable. The degree of curvature in stage
I varies for different rocks, depending on their degree of
porosity or compactness. In general, compact rocks have a
straight stress—strain curve in this region whereas the loose

rocks show rather pronounced curvature which disappears when

l6

1 IV

III

II

Differential Stress

 

Differential Strain
Fig. 2.9 Idealized Stress—Strain Diagram

and if a small hydrostatic pressure is applied to the Specimen.
Starting with stage III, important permanent changes
in the microsc0pic character of the compact rock occur. The
rock takes a somewhat lighter color which was traced at high
magnification to reflection of light at grain boundary surfaces.
It was therefore apparent, that the crystal grains are becoming
detached at their boundaries; and, when this occurs, the
boundary becomes totally reflecting and therefore easily
visible.
These reflecting surfaces become more numerous as fracture
is approached in stage IV. However, this characteristic be-
havior in uniaxial deformation completely changes as soon as

a confining pressure is applied; rock specimens cease to fail

l7

 

Fig. 2.10 Failure of Rock Salt Specimen Tested Under
Uniaxial Differential Stress with a Confining
Pressure Equal to One Atmosphere

 

Fig. 2.11 Failure of Rock Salt Specimen Tested Under a
Balanced Biaxial Differential Stress with a Con-
fining pressure Equal to One Atmosphere

18
along the grain boundaries, and fracture takes place both in

the crystals and their boundaries.

 

Fig. 2.12 Failure of Rock Salt Specimen Tested Under
Uniaxial Differential Stress With,a Con-
fining Pressure Equal to One Hundred
Atmospheres

These facts would indicate that:

(a) In a uniaxial differential stress test where the confining
pressure is negligible, failure will occur when the differently
oriented crystals tend to re-orient themselves in an effort to
resist the applied differential stress; thus transmitting part
of the load to the grain boundary layer or bond, and failure
will ultimately occur when these bonds are destroyed.

(b) When a confining pressure is applied, the ability of the
grains to reorient themselves will be restricted and hence a
greater proportion of the imposed differential stress will be

carried by the grains themselves rather than their being

partially transmitted to the boundary.

19
The above physical characteristic behavior suggests
that the inhomogeneous polycrystalline rocks are made up
of two independent mechanical systems: (1) the grain boundary
system and, (2) the composite crystal grain system, each of
which will share the applied differential load in a pro-

portion depending upon the degree of confinement.

CHAPTER III
THEORETICAL ANALYSIS

3.1 Proposed Rheological Model

Griggs (1939)16, Robertson (1960)30 and other investigators
have shown that the time-strain pattern exhibited by almost all
polycrystalline rocks under a state of constant differential

stress is similar to the curves shown in Fig. 3.1.

Fracture

Differential Stress
ei(t) (>> Elastic Limit)

High Differential Stress
(> Elastic Limit)

 

 

 

 

Initial '11——
Eigsgigstic Low Differential Stress
Deformation (Without Elastic Limit)
.112 1
Time

Fig. 3.1 Graph Representing the Different Creep Curves
at Varying Magnitudes of Differential Stress

These curves exhibit an instantaneous elastic strain
response which takes place upon the immediate application of

the differential stress and is represented by CA in Fig. 3.2.

'20

21

There follows a period of primary creep AB in which the

 

rate of deformation decreases with time. Primary creep is

sometimes referred to as delayed elastic deformation or visco—

 

elastic flow, for if at any time T1 the specimen is unloaded,

 

there is first an elastic recovery BC followed by a time-elastic

recovery represented by the curve CD.

i

 

 

 

 

 

 

 

 

Visco-elastic Visco-plastic Accelera ing
or or or
Primary _ Secondary ‘ Tertiary
Creep"V Creep '7 Creepvl
Failure
. t
61()
E _____ €i(t-)
l
A C '
| Permanent
, 1 D ! Jj‘Deformatigp
0 T1 T2 T3
time

Fig. 3.2 Typical Creep Curve at Constant Differential

Stress

However, if the load is not removed at time T

1, and if

the differential stress is greater than the elastic limit,

the specimen begins to exhibit secondary creep, a phase of

 

deformation in which the rate of strain is constant, whence

the secondary creep is often called steady state creep.

 

22

Because the rate of strain is constant and because the spec-
imen has undergone permanent deformation (as could be shown if
the specimen was unloaded at time T2) this deformation is also

sometimes termed pseudo-viscous or visco-plastic creep.

 

 

Many empirical equations were introduced (as shown
earlier by equations 1.2-l and 1.2-2) in an effort to estblish
a relationship between the differential stress-differential
strain-time elements which would fit experimental results;
however, these equations do neither succeed in describing
the full range of the experimental results nor do they
account forthe deformation mechanism of the material.

As a result, many investigators [e.g. Zener (1948)33,
Eirich (1956)11 and Bland (1960)6] find it convenient to
express the observed behavior of the material in terms of a

mechanical model that is comprised of a number of Simple

 

units (for which the usual assumptions are made regarding
isotropy and homogenity).

One such model which can be used to interpret time-
strain curves of this type is known as the visco-elastic or

Burgers Model, (see Fig. 3.3). It mainly consists of a Max—

 

well unit coupled in series with a Voigt unit, in which the
spring E represents the component of the body which gives
rise to the instantaneous elastic strain.

The Voigt unit consisting of a Spring and a dashpot
coupled in parallel (E*, n*) represents the component res-

ponsible for primary or visco-elastic creep. The component

23

of secondary creep or visco—plastic flow is contributed
by the dashpot n. However, this model will eventually
exhibit permanent deformation even when the stress becomes
exceedingly small, and therefore represents a viscous
liquid.

In order to provide for a yield strength Y of the
material which exists before attaining a steady state creep
rate, a Bingham body is used to replace the dashpot of the
Maxwell unit and is used in series with the Spring and
Voigt units. This new composite model is called The SVB

(spring, Voigt, Bingham) Model, since there is no Specific

 

name given to it yet. AS in the visco-elastic model, the
spring E accounts for the instantaneous elastic deformation
and the coupled spring and dashpot (E*, n*) accounts for

the visco-elastic flow. However, before the visco-plastic
deformation of the dashpot n can take place, the frictional
resistance of the Slider Y must be overcome. This frictional
resistance in the model simulates the yield strength of the
original substance.

Since polycrystalline rocks are inhomogeneous and have
rather displayed the existence of two separate systems, the
following mechanical model (Fig. 3.5) is introduced not only
to relate the strain-time relationship as a function of the
differential stress but also as a function of the confining
pressure.

The model consists of a grain boundary system and a

crystal grain system in parallel; each system is represented

24

‘ Viscous
Liquid

E
0* E*
Tl

Fig. 3.3 Burgers Model showing the relationship
between differential strain rate and
differential stress for the steady-state
or Visco-plastic creep.

A Solid
E
n. :IEE.
Yield
Strength
n Y . i
’-

Differential Stress Si

 

Differential Strain Rate 64“)

 

)—
Differential Stress Si

Differential Strain Rate 3 l(t)

 

 

 

Fig. 3.4 SVB Model showing the relationship
between differential strain rate and
differential stress for the steady-state
or visco—plastic creep.

 

 

 

E
2
Grain Crystal
Boundar * ~* Grain
.__—_X ”1 E2 _
System System
n1 Y2

 

 

 

77/ln/lll7/li7/TF/l ///

Fig. 3.5 Proposed Mechanical Model

as an SVB model fixed at the bottom to a rigid base. At the
top, the two systems are coupled together by a rigid bar
hinged to each system, while the position of the applied
differential stress Si along the hinged bar relative to the
two systems varies according to the degree of confining pres-

sure C.

3.2 Assumptions

 

(i) The element constants shown in the prOposed model
representing the prOperties of the aggregate material are
statistical averages of the properties of the crystals and
their boundaries taken over all orientations.

(ii) Specimens sufficiently large to contain enough

crystal grains are assumed to be statistically isotropic.

26

(iii) The crystal grain elements' constants are always
the same for a certain type of polycrystalline material;
however, the grain boundary elements' constants might differ
for the same material, depending on the previous temperature
and stress history to which the Specimens have been subjected.

(iv) The effect.of a hydrostatic principal stress tensor
(0x = 0y = Oz) on the deformation of polycrystalline rock is
assumed to be linearly elastic and to have negligible

variation with time. Thus it can be represented by the

equation

 

0- = G6. = e; (3.2-l)

where G is the hydrostatic elastic constant of the material.

3.3 Variation of Behavior.with Confining Pressure

The relationship between differential stress Si and
differential strain Ei and the degree of confinement c is
determined by utilizing the proposed rheological model (Fig.

3.6-a with a history of loading as shown in Fig. 3.6-b) in

which
E1, E: = Elastic constants of grain boundary bonds
ni, nl = Visco-Elastic and visco-plastic coefficients
of bonds

Y1 = Differential yield strength of boundary bonds
SE = Applied differential stress on boundary system
a? ‘= Differential strain of boundary system

E2, E3 = Elastic constants of crystal grains

27

 

S.
I i
(l-C) Si |.._- c_. l—c) _,:CS.'L
—.—+---- ,e—c
b I I EL -’-€i
‘ 8i I T
——i——+ E2
E1 IIIIE
* * * - E2
_ n
n. I .1 2
211.. Y1 n2 - Y2

 

/7//7/7////////////////////

(a) Deformation of Model due to applied
differential stresS

 

Differential Stress

 

 

 

 

0 Time t
(b) History of Loading

Fig. 3.6 Differential Stress-Strain-Time
Relationship

28

= Visco-elastic and visco-plastic coefficients
of grains

Y2 = Differential Yield strength of grains

S§'= Applied differential stress on grains
ti = Differential strain of grains
t = Time

From the equilibrium of the rigid bar,

S? = c S-
1 1 (3.3-l)

b _ _
Si - (l c) Si

and from the compatibility condition of the two systems

5i = CE + (1 - C) (a? - CE) = (l - c) a? + cs: (3.3-2)
but a? = [Bil] elastic + [£52] visco-elastic
+ [5:3] visco-plastic (3.3-3)

and by considering the constitutive relationship of the three

elements, it follows that:

 

 

 

- c _ 1 _
(i) Eil - c Si (_L) (3.3 4)
32
(") * deiz + E* C c s
11 T12 -4——h E- — .
dt 2 i2 1
degz E; C
or . + ‘T 5:2 = * Sl *
dt n2 n2 E2
'1'
DZ -
and by using the integration factor e , we get
E2 t E5 t
3'3" 7"
,c b 2
d (ti2 e ) = c Si e
dt n;
'k
E t E t
t 2
11* C 11*
c 2 _ 2
or 512 e - _* Si e dt

 

 

 

 

 

assuming that 5E2 = 0 at t = 0.
*
-E2 t
Therefore *
l n
c _ _ 2 _
€12 — C Si[ _* (1 e )] (3.3 5)
Ez
9833
(iii) c Si = Y2 + n2 ( if c Si > Y2)
dt
d 8‘53 _ 1 .
or _ (cSi - Y2)
dt 712
t
or £9 = 1 (l - __ ) dt
13 CS
D2 1 O
. C = = .
assuming €i3 0 at t 0
Therefore
Y t
c 2
= S. 1 - _ 3.3-6
613 C l ( ) ( )

Hence, by substituting equations 3.3-4, 3.

equation 3.3-3, we obtain

3-5 and 3.3—6 into

1
53 = c Si {[._ 1 -+
. E2
*
-32 t
"_T—
n2
. Y2 t
+ [._: (l — e )] + [(1 --——-) - ]} (3.3—7a)
E2 CSi r12
*
-E1 t
Similarily, it can be shown that :;__
b 1 l l
8i = (1 - c) Si [ _ ] + [ _} (1 - e )]
El E1
Y1 t
+ [(l - ._______ ) _]} (3.3—7b)
(1 - C)Si n1

30
By substituting equations 3.3-7a and 3.3-7b into equation

3.3-2, we get the general relationship

 

-El t
——7-
n1
2 1 l (l - e )
El El
-E3 t
——7-
n2
Y t l l ,
+ (l - 1 ) _ ]+ c2[ _, + _} (l ' e )
(l - C)Si n1 E2 E2
Y2 t
+ (l - )._ ]} (3.3-8)

C Si 02
which can be written as
(1 - c)2 c2
e- = S. {[ ________ + __ ]
1
El E2

representing the equivalent elastic response

 

 

 

* *
-El t -E2 t
"T‘ —~lr"'

(l - c)2 n1 c2 n2

+ [ ____1r___ (1 - e ) +-.1; (l - e )1
El E2

representing the equivalent viSCO—elaStic (3.3-8a)
response

(1 - c)2 t Y1 c2 t Y2
+ [ ———————— (l - ei)+ (1 - )1

representing the equivalent visco-plastic reSponse

31

3.3a Variation of the equivalent elastic response with
the degree of confinement

From the derived general relationship (equation 3.3-8),

it was shown that

 

1 (1 - c)2 c2
.. = _ + _._
Eeq El E2
If we let E2 = a El (3.3-9)
1 (1 - c)2 c2
we get _ = ________.+
Eeq El a El
OLEl
or E = (3.3-10)

 

eq 397+ a(1 - c)2

Maximizing equation 3.3-10 with respect to c, we obtain

d(Eeq) -aEl [2c - 2a(l - c)]

d c [c2 + a(1 - c)2]2

 

from which we find that (Eeq) max. occurs at

 

5 = (3.3-ll)
l + a
Hence (Eeq)max = (1 + a) E1 = E1 + E2 (3.3-12)
Also when Eeq = E2 ='a El' a quadratic equation of c is

obtained giving

c2 +a(l - c)2 - 1 = 0 (3.3-13)
which could be re-arranged as

[(a + l) c - (a - 1)] [c - 1] = 0

0:71
with two roots: c = 1 and c = '

 

a +11

degree of confinement c is shown in Fig.

Equivalent Elastic Modulus Eeq

32

A typical curve showing the variation of Be with the

q
3.7 for the particular

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

than:1 . E1+E7
/,
Ez
./
a-l
a+l '7
E 1 u
1 R 1:1— '
0 0.2 0.4 0.6 0.8 1.0
Degree of Confinement, c
Fig. Typical Variation of Ee with c

q

value of a = 4; while in the next two pages, the calculations

(Table 3.1) and a graph (Fig.

for various values of a are presented.

3.8) showing this same variation

3.3b Variation of the equivalent visco-elastic response
with the degree of confinement

The visco-elastic creep response as obtained by the

derived general expression (equation 3.3-8) was shown to be:

 

-E§ t

 

11*
(l - e 2 )1

33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a
1 1.22E2 1.47E2 1.72E2 1.92E2 2.0 E2 1.92E2 1.72E2 1.47E2 1.22E2 E2
2 0.61E2 o.76E2 0.93E2 1.14E2 1.33E2 1.47E2 1.49E2 1.39E2 1920E2 E2
3 0.41E2 0.52E2 0.64E2 0.81E2 1.00E2 1.19E2 1.32E2 1.32E2 1.19E2 E2
4 0.31E2 0.38E2 0.49E2 0.63E2 0.80E2 1.00E2 1.18E2 1.25E2 1.18E2 E3
5 d0.25E2 0.31E2 0.39E2 0.51E2 0.67E2 0.86E2 1.06E2 1.19E2 1.16E2 E2
6 0.21E2 0.26E2 0.33E2 0.43E2 0.57E2 0.76E2 0.97E2 1.14E2 1.15E2 E2
a
7 0.18E2 0.22E2 0.28E2 0.37E2 0.50E2 0.68E2 0.89E2 1.09E2 1.14E2 E2
8 0.15E2 0.19E2 0.25E2 0.33E2 0.44ES 0.61E2 0.83E2 1.04E2 1.12E2 E2
\
9 0.14E2 0.17E2 0.22E2 0.29E2 0.40E2 0.56E2 0.77E2 1.00E2 1.11E23E2
50 0.12E2 0.16E2 0.20E2 0.27E2 0.36E2 0.51E2 0.72E2 0.96E2 1.10E2 E2
Table 3.1 Calculations of Ee vs c for different values of a

q

 

Equivalent Elastic Modulus, Eeq

34

 

Degree of Confinement, c

Fig. 3. 8. Variation of Eeq vs. c for various values of at.

35

 

 

If we let E; = d* E: (3.3—14)
*
a*(1 - c)2 c2 a (1 - c)2 n;
then 6i (t) = Sl {[ * + ‘7 ] - [ 1.1, e
E2 E2 E2
*
-E2 t
2 “T‘
C D2 7
+ _w e ]} (3.3-15)
32

By denoting

E; 1

T = —

n2 T2

(3.3-l6)

E: 3*
and _*.=.__

01 T2

n*
then _§.= a* 8* (3.3-16a)

n1

By substituting equations 3.3-16 into equations 3.3-15, we

obtain
-t
Si T2
6i (t) - _?.{[c2 + a* (1 - c)2] + [c2 e
E2

 

36

which can be rewritten as

 

 

 

 

 

 

 

 

 

 

 

 

 

t
_ (8* + 1)
S' 2T2
Ei (t) =_: [c2+a*(l-c)2] +(\L* c2 (l-c)2e
E;
t t
e (s*-l) ** 2 _ (e*-1)
C2 212 a (l - C) 2T2
[ * e + e 1}
0 (l - c)2 c2
t
or _ (8* + 1)
S- 2T2
El (t) = _;.{[c2 + a (l - c)2] +~’a* c (1 - c) e
32
1 c2 1 t *
_ 10g * ‘— —— (8 - l)
2 a (1 - c)2 2 12
[e e
1 c2 t .*
-- log * 2 - (B - 1)
2 a (1 - C) 2T2
+ e e ]
or
t
_ (3* + 1)
8- 2T
5. (t) = l {[c2 + 0*.(1 - c)2] + ZA/a* c (1_- c) e 2
1 '1'
E2
1 * t c2 V
cosh — [(8 - l) _. + log * 2 ] (3.3-17)
2 T2 a (l - C)

This relationship between ei(t) and t is illustrated by

the curve shown in Fig. 3.9, in which the expected creep range

37
t

 

 

*
- 2(8 +1)
e-(t) l 21 1 t
1 a _T{[c2+a*(l-C)2 +21/a* c(1-c)e cosh -3521) _
. + log _____1
a*(1-C)2

 
    

-—==—- woo

 

 

 

 

time, t

Variation of the Visco-elastic Creep with time

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3.9
*
r-t \ [[b a = 5 *
N ,
:6 _.r _X (r. 0.82
C. V‘\ a + 1
* - \
N: .‘N If \ {1* I
L... _.a = 3 ~—= 0. f5
H|*CE \ * '7' i
& \\“-
,..a* = \
‘\L\ f- O.* = 0.2 ¥
I
- L-

 

 

 

O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Degree of confinement, c

Fig. 3.10 Variation of the Range with the degree of
confinement

1.0

38

R is the value approached by ei(t)/Si as t +w, given by

5- (w) l

—-—-——_ = R = [C
S. E*

l 2

2

 

+ a* (l-c)2] (3.3-18)

The variations of the range R with the degree of confine-
ment c is shown.in Fig. 3.10, and is dependent on the value of
a*. It is worthwhilesto note'that thecrange is always an in-
creasing function.of c whence is sufficiently*near to 1.

Thus, by finding:the appropriate values of 8* and T,
almost any visco—elastic creep behavior can be approximated,
after determining the.approximate-value of a* from the pre-
ceeding range analysis. To illustrate this characteristic
change in behavior, consider for simplicity, the variation

of the visco-elastic creep curves with the degree of con-

finement c in the following two cases:
*

Case-(a) a = 1
T = 2
B*=1
*

Case (b) a = l
T = 2
B*=5

By examining the two plots of cases (a) and (b) shown in
figures 3.11 and 3.12 respectively, we see that because 0* in
both cases was chosen to be the-same and équal to 1, the ranges
of the visco-elastic curves were equal for the same-c. However,
by changing-8*, the ratio of B*/T has beenChanged causing a

change in the dependence of the initial slope on c.

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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41
In case (a) the initial slope increased as c increased, while
in case (b) the initial slope first decreased and then in-
creased as c approached l.
3.3c Variation of the equivalent visco-plastic response
with the degree of confinement
The visco—plastic response derived from the general

solution of the proposed model (equation 3.3-8) was given as

provided that
c S. > Y2
and (l - c) Si > Y1

now if we let

n2
_.=B
n
1 (3.3-19)
Y
_i.= A
Y1

and substitute equations 3.3-l9 into the above equation, we

 

 

 

 

 

.obtain
Si t Y Y2
ei(t)= [B(l-C)(l-c- )+C(c-__)]
"2 A Si Si
8. t Y B (l—c)
or ei(t)= l [B(l-c)2+c2-__2.(c+ )]
’02 Si. A
S. t Y B B
or s. (t) =: 1' B (l - c)2 + c2 -._E.[c (l - _) + _J1(3.3-20)
1
”2 Si A AI

42
Again to illustrate the change in the steady state lepe
of the creep curve with the degree of confinement, we choose
for simplicity an example in which B/A = l, and therefore, the
visco-plastic response becomes

8- t Y
(t) = l [s (1 - c)2 + c2 - .3] (3.3-21)

from which the steady state slope of the creep curve is

 

obtained by differentiating 6i (t)with respect to t, thus

getting
Y2
- ——] (3.3-22)

Si

n2 21 (t) = si [s (1 - c)2 + c2

which when plotted for a constant c and varying Si will give
a straight line relationship as shown in figure 3.13.

In order to plot the variation of the differential strain
rate with the degree of confinement c, it is noticed that the
behavior will depend on the value of B, which is expected to
be much less than 1, since it is experimentally observed that
the grain boundary is much more viscous than the actual crystal
grains themselves. This fact has been observed even in metals
and verified by KS (1947)22 in a set of experiments on poly-
crystalline and single aluminum crystals, in which aluminum
wire was used in a torsion pendulum oscillating at a constant
frequency. K3 found a relaxation peak in the internal friction
when measured at different temperatures for polycrystalline
aluminum which was not observed when single aluminum crystals
were tested, and concluded that the phenomenon was primarily
due to the high viscosity of the grain boundaries.

Thus if we assume an arbitrary value for Y2/Si and a

43
value of 8 less than 1, a decrease in the initial part of the
curve showing strain rate with the degree of confinement (Fig.

3.14) is observed, followed by an increase as c tends to ap-

proach 1.

n2 2(t)

Slope = 8(1 - c)2 + c2

(for a constant c)

Yield
Strength

 

 

‘7

Y2
Differential Stress Si

Fig. 3.13 Relationship Between Differential Strain
Rate and Differential Stress for the
Steady State Response

The value of c corresponding to the minimum value of the

differential strain rate is obtained by differentiating 8i (t)
with respect to c

i =_i_[-2B(1—c)+2c]=o

dc rh

44

Hence

2c (8 + l) = 28

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B
or (c) . = (3.3-23)
mm B + 1
for the case when 8 = A.
J /
.8 //
7 for B = 1/5 J
B A = 1
.5 /
n2 81 (t) 4 11/,
S
l .3 B i l
.2 ' A / 1
.lts‘
-~
3
O 0.2 0.4 0.5 0.8 1.0

Degree of Confinement, c.

Fig. 3.14 Relationship Between Differential Strain Rate
and Degree of Confinement

CHAPTER IV
APPLICATION AND PREDICTIONS OF THEORY

From most of the work that has been done to relate the
five factors affecting the behavior of polycrystalline rock,
all that can be concluded is that there exists some sort of
variation in the behavior; since, up to the present time, no
theoretical basis to guide investigators has been established.

As a result, only a very few regular and systematic
tests have been performed using a relatively wide range of
experimental variation to demonstrate the influence of con-
fining pressure and temperature. These will be utilized
here to show how the experimental results could be compared
with the theoretical solution obtained by utilizing the
proposed mechanical model if the prOper values of the dif-

ferent elements were known.

4.1 Variation of the Elastic Modulus BA with Confining

Pressure

 

q

After experiments done by Griggs (1936)15, Handin (195318
and 1957 19), Heard (1960)20, Brace (1963)8 and Baron (1963)5,
it has become an established fact that an increase in con-
fining pressure tends to increase the elastic modulus.
This could be explained by the fact that confining pressure
raises the activation energy of the crystal grains that con-
stitute the skeleton of the rock by restricting their dis-
placement or re-orientation.

.45

46

Nevertheless, Heard (1960)20 and Baron (1963)5 working
with confining pressure up to 5,000 Kg/cm2 were surprised to
get an increase in the elastic modulus with an initial in-
crease in the confining pressure followed by a decrease of
the elastic modulus with a subsequent increase in the con-
fining pressure.

Such a behavior can easily be explained by examining
the solution of the proposed model as demonstrated by the
following two examples:

Example 1: Results of differential stress versus dif-

 

ferential strain of rock salt specimens at different con-
stant confining pressures were presented by Handin (1953)18
(Fig. 4.1). These results demonstrate the increase of the

elastic modulus with the degree of confining pressure. Upon

X’MNOVIS FINN)"

CWINING “550'! IN
lYUOS’NE'KS GWEN ON
EACH CURVE

uncut-1m. Stats: I: uczcu‘

 

570“" W P! ICINY

Fig. 4.1 Differential Stress-Strain Curves of Rock
Salt Specimens Deformed in Compression
(After J. Handin (1953)18)

47

measuring the initial elastic modulus, one finds the following
values E25 = 4,500 Kg/cm2

3100 = 24,000 Kg/cm2

E2000 = 42,000 Kg/cm2

32800 = 63,000 Kg/cm2

A fitting method was used to determine the best possible

values of the Springs El and E2 that could represent the same
variation. To do this, one must first assume what the ex—
pected maximum value of Be would have been had the test been

q

run for the full range. This value of (Eeq) is equal to

max
the sum of E1 and E2 as was shown previously. Again if E2

is not known, a feasible value of E2 is guessed and hence the
ratio of the two springs a is determined from these assumed
values. By knowing the value of a, a curve is plotted for
the experimental values of Eeq versus the degree of confine-
ment c, utilizing table 3.1. The actual points (Eeq' p) are
then plotted over this theoretically determined curve in an
effort to establish the relationship between the relative
degree of confinement, c.and the actual confining pressure,
p. If no fit is obtained between the two curves, a new set
of spring values should be assumed, and the same procedure

is repeated until a close fit between the theoretical and
actual variation is obtained.

For this particular example, it is found that by using

9,000 kg/cm2

El

E2 63,000 kg/cm2

a = 7

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49
a close agreement between most of the theoretical and actual
values is obtained (Fig. 4.2).

Example 2: Heard (1960)20 presented results of dif-

 

ferential stress versus strain curves at constant confining
pressures for Solenhofen Limestone specimens, at a constant

room temperature of 25°C as shown in Fig. 4.3.

 

, ///]......

l

 

 

 

 

 

 

 

 

    

 

 

unnumuIn-I-awflhn'
O

. “"'"‘"“'“‘ll

 

 

 

 

 

 

 

 

 

 

 

Fig. 4.3 Differential Stress-Strain Curves fox?0901-
enhofen Limestone (After H. C. Heard

From these curves, the following initial elastic moduli
were obtained.

E = 3.25 x 105 kg/cm2

_ 5 2
E3000 — 4.65 x 10 kg/cm

2
E5000 = 4.20 x 105 kg/cm

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Again, the same method of fitting is applied and by
assuming a value of E2 = 4.20 x 105 kg/cm2
E1 = 0.70 x 105 kg/cm2
0 = 6
a close fit is obtained as shown in Fig. 4.4.
It is worthwhile to note the linear variation of the
degree of confinement c with the confining pressure p in
each of the above two examples.

4.2 Variation of the Visco-Elastic Range with Confining
Pressure

 

Creep experiments using polycrystalline rock were first
started by Griggs (1939)16 in an effort to investigate the
variation that takes place for various combinations of the
five influencing factors, and as a result some empirical
relationships were suggested.

Robertson (1960)28 has presented the only systematic
creep work; his work was on Solenhofen Limestone at different
levels of constant differential stress, different confining
pressures, and at constant room temperature. Some of his
plots are shown in Fig. 4.5. Several empirical relationships
were suggested in an effort to establish a correlation between
the differential stress-differential strain-time elements,'
and it was concluded that a more theoretical approach is
certainly needed to relate these variables and improve the
planning of creep experiments.

Price (1964)26 and others have recently applied model

analysis to some creep tests on rocks. However, since they

52
used a single system model, it was only applied satisfactorily

to the behavior in the uniaxial state of stress.

 

 

  

 

  
  
    
  
   

 

 
 

 

 

 

 

 

 

 

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Fig. 4.5 Creep Test Results on Solenhofen Limestone
Specimens (After E.C. Robertsonze)

In analysing the work of Robertson which is replotted
in Fig. 4.6, the following two assumptions are made:
(i) The visco-plastic creep is negligible, or has not begun
during the whole analysis of the visco-elastic response.
(ii) The visco-elastic response is negligible (or has been
completed) during the steady-state creep.
Curves (i), (ii) and (iii) in Fig. 4.6, representing
the visco-elastic creep reSponse (since their differential

stress level is within the elastic limit) are normalized

Differential Strain si(t) (x10—3cm/cm)

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1
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Fig. 4. 6. Creep Test Results on Solenhofen.Limestone [ after Robertson

(1960)28]

 

 

54
to the same differential stress level of curve (i) by reducing
their respective differential strain ordinate in the ratio
of the differential stress of curve (i) to the differential
stress of the respective curve. The normalized curves are
plotted in Fig. 4.7, and for comparison, they are again plot—
ted in Fig. 4.8 after being reduced to the same origin.
The range of each of these three curves is then est-
imated by making a reasonable assumption for the ordinate
approached as t-w-oo. The estimated ranges are plotted versus
the degree of confinement c (since the relationship of the
actual confining pressure p to the degree of confinement c
has already been established from the earlier elastic analysis)
as shown in Fig. 4.9.
By assuming what the maximum range would have been at
c = l, the usual method of fitting is applied to determine
(by trial and error) the best value ofa* to give the closest
fit. In this particular example, it has been found that if
a=l
and the maximum range = 2.2 x 10'3 cm/cm
(at c = 1)

a close fit is obtained.

Thus, it is worthwhile to note that by observing the
variation in behavior as obtained from experimental results
of systematic tests, the values of the elements of the grain
boundary system can be roughly determined if the exact values
of the elements of the crystal grain system have already

been established.

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58

4.3 Variation of the Visco-Plastic Differential Strain Rate
with Confining Pressure

 

 

From Robertson's creep curves (iv), (v) and (vi) in
Fig. 4.6, it can be observed that they exhibit visco-plastic
or steady state differential strain rates (their differential
stress level being greater than or equal to the elastic
limit) with the following values:

Curve (iv) Confining Pressure = 2000 kg/cm2

 

Differential Stress = 5800 kg/cm2
Differential Strain Rate = 3.8 x 10'7 cm/cm/sec

Curve (v) Confining Pressure = 3000 kg/cm2

 

Differential Stress = 4700 kg/cm2
Differential Strain Rate = 8.8 x 10'7 cm/cm/sec

Curve (vi) Confining Pressure = 3000 kg/cm2

 

Differential Stress = 5700 k'g/cm2

Differential Strain Rate 12.5 x 10'7 cm/cm/sec

It is observed from curves (iv) and (vi) that with both
curves being almost at the same differential stress level,
an increase in the differential strain rate is observed
merely as a result of the increase of the confining pres-
sure. Also from curves (v) and (vi) which are at the same

2, an increase in the dif-

confining pressure of 3,000 kg/cm
ferential strain rate takes place again with increase in
the differential stress level.

This variation is found to be qualitatively in good
agreement with the theoretically expected variation; how-

ever, due to thexundetermined values of the elements in

the Bingham units of both systems, it is difficult to show

59
any fit between theoretical and actual values without at
least knowing the approximate values of Y2 and n2.

The actual values of the crystal grain system constants
can be determined from the elastic analysis and from creep
tests that should be conducted at a confining pressure cor-
responding to c = l and for different levels of constant

differential stress.

CHAPTER V

SUMMARY AND CONCLUSIONS

5.1 Discussion

 

As a result of the interest that has developed at the
beginning of this century, a study of the deformation of
rocks under load was begun and several experiments were
performed in an effort to establish the factors that affect
the behavior of such materials.

One of the important factors that influences such a
characteristic behavior, is the effect of confining pres-
sure. Several empirical relationships incorporating this
effect were presented. However, such relationships had
serious shortcomings in describing accurately such a char—
acteristic behavior, and have failed to account for the
actual deformation mechanism that is responsible for such
behavior.

The slow progress in this field of study is due to the
following two reasons:

(i) It is a very expensive and tedious process to run
experiments on rock at extremely high pressures, and to
develop the necessary equipment by which such experiments
can be performed.

(ii) The current theories of elasticity and viscoelasticity
are inadequate to describe such characteristic behavior.

This inadequacy may be attributed in part to the fact that

60

61
rock is a heterogeneous material whose crystals possess
physical properties which differ from the grain boundary
prOperties.

In this investigation, the rock behavior is simulated
by a rheological model composed of two independent systems,
one representing the crystal properties and the second the
prOperties of the grain boundaries. However, in order to be
able to use this new model to describe the transitional be-
havior from uniaxial to triaxial stress state, it was nec-
essary to redefine the effect of a general loading condition
as a superimposed sum of a hydrostatic confining stress
tensor and a differential stress tensor. As a result of
this new formulation, three testing criteria for the usual
case of a general principal compressive stress state were
established in Chapter II, namely:

(i) The uniaxial differential stress test
(ii) The general biaxial differential stress test
(iii) The balanced biaxial differential stress test

By utilizing this model, the separate contributions of
the crystals and grain boundaries are correlated to the
applied confining pressure by varying the position of the
applied differential stress relative to the two systems
according to the degree of confinement. This unique ar-
rangement offers a basis for the mathematical analysis shown
in Chapter III, where several variations in the characteristic
behavior are displayed, depending on the ratio of the physical

constants of one system to the other. The main characteristic

 

62
features that were obtained from the mathematical analysis
can be summarized as follows:
(i) In the elastic analysis, the equivalent elastic modulus
increases with initial increase in confining pressure and
decreases with subsequent increase in confining pressure
depending on the value of a, which is the ratio of E2 to El°
(ii) In the visco-elastic analysis, the creep range de-
creases with initial increase in the degree of confinement,
c, and increases with subsequent increase of c depending on
the value of a*, which is the ratio cf E; to E1. In addition,
the initial slope of the visco-elastic creep curves either
increases with increase in confining pressure or first
decreases and then increases with subsequent increase in
confining pressure, depending on the value of 8* which is
the ratio of l/I2 to l/Tl.
(iii) In the visco-plastic analysis, the steady state creep
rate first decreases and then increases with increase in
the degree of confinement depending on the value of B, which
is the ratio of n2 to n1; and it constantly increases with
an increase in the differential stress.

This rheological description has been compared with
existing data in Chapter IV yielding the following results:
(a) by means of graphical fitting, it was possible to deter-
mine approximately the value of a, and the relationship be-
tween the degree of confinement, c, and the.actual confining
pressure, p. For the two particular cases that were pre-
sented, it was found that the relationship is linear, and

that the minimum confining pressure needed to isolate the

63

effect of the crystals (corresponding to c = l) is approx-
imately 5,000 atmospheres.

(b) The range of the visco-elastic creep was found to increase
with confining pressure, and from the variation obtained it
was possible to approximately determine the value of a*.

In this case, the initial s10pe of the visco-elastic response
first decreased and then increased with increase in confining
pressure.

(c) The steady state creep rate of the visco-plastic response
was found to increase both with increase in confining pres-
sure and differential stress; however, there was not enough
data to obtain quantitative results.

Thus, with this rheological description of rock be-
havior it was possible to obtain results in excellent
agreement with existing data and to provide a basis for under-
standing the variation of physical constants with changes

in confining pressure.

5.2 Future Research

 

Based on this presentation, it was found that very few
systematic tests were conducted on any particular kind of
polycrystalline rock that would enable one to determine the
ten physical constants of the model. This indicates the
need for performing further experiments from which a com-
plete analysis could be obtained.

In order to be able to fully determine the ten physical
constants of the proposed model, the following experiments

should be performed:

 

64

l. The first set of experiments is to obtain differential
stress—strain curves of specimens which are subjected to
various confining pressures. In this first type of exper-
iments it is important to use an apparatus that is capable
of furnishing the needed high pressure for determining the
isolated physical property of theycrystal grains. By
analyzing the results of this first set of experiments, it
will be possible to determine (i) the relationship between
the degree of confinement c, and the actual confining pres-
sure p; and (ii) the approximate values of El and E2.

2. The second set of experiments should be conducted to
obtain, at various differential stress levels, creep data
from specimens that are confined to a hydrostatic pressure
which correlates to a degree of confinment of unity (C: 1).
From the results obtained it will be possible to determine
E3, n3, n2 and Y2.
3. The third set of experiments is conducted with the aim
of obtaining, at a particular constant differential stress
level, creep data from specimens that are subjected to

various confining hydrostatic pressures. From the variation

in the behavior due to the changes in confining pressures,

*

it will be possible to roughly determine El’

”I'"l and Y1
by fitting methods.
Other useful applications of this rheological rep-
resentation are:
(a) investigating the change in these physical constants with

temperature by repeating the above three sets of experiments

at Various degrees of temperatures.

65

(b) studying the effect of temperature and pressure history
on specimens by subjecting them in the laboratory to various
predetermined history conditions.

(c) determining the value of Poissons ratio of each system
from the elastic analysis of differential stress-strain
curves by performing first a set of uniaxial differential
stress tests at various constant confining pressures fol—
lowed by a set of balanced biaxial differential stress tests
at the same constant confining pressures.

(d) although this analysis is only valid for single phase
polycrystalline materials, yet it may serve as a first

step towards analyzing more complicated mechanisms of de-
formation that are exhibited in the behavior of multi-phase

polycrystalline materials such as ice.

 

1.

10.

11.

12.

13.

BIBLIOGRAPHY

Adams, F. D., and Nicholson, J. T., "An Experimental
Investigation into the Flow of Marble," Proc. Roy.
Soc. London, Ser. V., Vol. 195, 1901, pp. 363-401.

. "An Experimental Investigation into the Action
of Differential Pressure on Certain Minerals and
Rocks, Employing the Process Suggested by Prof. Kick,
"Jour. Geol., Vol. 18, 1910, pp. 485-525.

. "An Experimental Contribution to the Question
of the Depth of the Zone of Flow in the Earth's
Crust," Jour. Geol., Vol. 20, 1912, pp. 97-118.

. and Bancroft, J. A., "On the Amount of Internal
Friction Developed in Rocks During Deformation and
on the Relative Plasticity of Different Types of
Rocks," Jonr. Geol., Vol. 25, pp. 597~637.

Baron, G., Habib, P., and Morlier, P., "Deformation of
Rocks under Constraint," Review de 1'institute
Francais du Petrole, June 1963, pp. 97-130. (in French)

Bland, D. R., The Theory of Linear Visco-elasticity,
New York, Pergamon Press, 1960, pp. 125:

 

BBker, R., "Die Mechanik der bleibenden Formanderungen
in Kristallinisch aufgebauten Korpern," Verdeutsch,
Ingenieure Mitt. Forschungsarbeiten, Vol. 175,
1915, pp. 1-51.

Brace, W. F., "Brittle Fracture of Rocks," International
Conference on State of Stress in Earth Crust, May 1963.

Cottrell, A. H., "Dislocation and Plastic Flow in Crystals,"
Oxford, Clarendon Press, 1952.

Cullity, B. D., "Elements of X-Ray Diffraction," Addison
Wesley Publishing Company, 1956.

Eirich, F. R., editor, Rheology, theory and applications,
Vol. 1, New York, Academic Press, Inc., 1956.

 

Emery, C. L., "Strain Energy in Rocks," International
Conference on State of Stress in Earth Crust, May 1963.

. "Testing Rock in Compression," Mine and Quarry
Engineering, April and May, 1960.

66

 

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26

27.

Finnie, 1., and Heller, W. R., Creep Engineering
Materials, McGraw-Hill Book Company, Inc., New
York, 1959.

 

 

Griggs, D. T., "Deformation of Rocks under High Confining
Pressure," J. Geology, Vol. 44, 1936, pp. 541-577.

"Creep of Rocks," J. Geology, Vol. 47, 1938,

p. 225.
. "Experimental Flow of Rocks under Conditions
Favoring Re-crystallization," Bulletin of Geological

Society of America, Vol. 51, July 1940, pp. 1001-1022.

Handin, J., "An Application of High Pressure in Geo-
physics," Trans. A.S.M.E., Vol. 75, 1953, pp. 315-324.

. and Hager, R. V., "Experimental Deformation of
Sedimentary Rocks under Confining Pressure," Am. Assoc,
Petrol. Geol. Bulletin, Vol. 41, 1957, pp. 1-50.

Heard, H. C., "Transition from Brittle Fracture to
Ductile Flow in Solenhofen limestone as Function of
Temperature, Confining Pressure, and Interstitial
Fluid Pressure," Geol. Soc. Am. Memoirs, Vol. 79,
pp. 193-226, 1960.

Hill, R., "Mathematical Theory of Plasticity," Oxford,
Clarendon Press, 1956.

Ke, T. 5., "Experimental Evidence of the Viscous Behavior
of Grain Boundaries in Metals," Physics Review,
Vol. 71, No. 8, 1947, pp. 533-546.

Kvapil, R., "Tectonic Experiments on Natural Rocks,"
Int. Jour. of Rock Mech. and Mining Sciences, Vol. 1,
Jan., 1964.

Murrell, S. A. F., and Misra, A. K., "Time-dependent
Strain or 'Creep' in rocks and Similar non-metallic
Materials," Inst. Mining Metall. Bu11., Vol. 71,
pt. 7., 1962, pp. 353-378.

Nadai, A., Theory of Flow and Fracture of Solids., Vol. 1,
McGraw-Hill Book Co., New York, 1950, pp. 175-228.

 

Price, N. J., "A Study of Time-Strain Behavior of Coal-
Measure Rocks," International of Rock Mechanics
and Mining Sciences, Vol. 1, No. 2, March 1964.

Read, W. T., Dislocations in Crystals, New York, McGraw-
Hill Book Co., Inc., 1953.

 

67

 

28.

29.

30.

31.

32.

33.

34.

35.

36.

68

Robertson, E. C., "Experimental Study of the Strength
of Rocks," Bull. Geol. Soc. of America, Vol. 66,
Oct. 1955, pp. 1275—1314.

. "Visco-elasticity of Rocks,” International
Conference on State of Stress in the Earth's
Crust, May 1963.

. ”Creep of Solenhofen Limestone'under Moderate
Hydrostatic Pressure," Geol. Soc. Amer. Menoir,
Vol. 79, 1960, pp.227—244.

Schwartz, A. E., "Failure of Rock in Triaxial Shear
Test," Proceedings of the Sixth Symposium on
Rock Mechanics, 1964.

Seligman, G., "The growth of glacier crystals,”
Journal of Glaciology, pp. 254, 1949.

Steineman, S., I‘Flow and Recrystallization of ice,"
Association Internationale d'Hydrologie Scientifique,
Publication no. 39, Congres de Rome, 1954.

Tomaschevskaya, I. S., "A Study of Shear Modulus or
Rocks under High Confining Pressure," Izv. Akad.
Nauk, Vol. 18, Geophysics Series, 1961,(in Russian).

Von Karman, T., "Festigkeitsversuche unter allseitigem
Druck," Zeitschr. Ver. deutsch. Ingenieure, Vol.
55, 1911, pp. 1749—1757, (in German).

Zener, C., Elasticity and anelasticity of metals,
Chicago, University of Chicago Press, 1948,
pp. 140-163.

 

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