AS 726746
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UNCLASSIFIED
DASA 2C57
, r
May 1*71
STATIC UNIAXIAL DEFORMATION OF 15 ROCKS
FINAL REPORT
W.F. BRACE
and
D.K. RILEY
HEADQUARTERS
Defense Atomic Support Agency
Washington, D.C. 20305
^£p£artment of Earth and Planetary Sciences
Massachusetts Institute of Technology
i* '•
Cambridge, Massachusetts 02139
Contract No. DASA01-69-C-Q122
D D G
JUl *7 1971
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Approved for public release; distribution unlimited
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May 1971
DASA 01-69-C-0122
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Program Code 9F10
REPORT ICC UNITY CLARIFICATION
11| • v • N »i
I OmaiNATIN* ACTIVITY (Ci pM tt MAilJ
Massachusetts Institute of Technology
Division of Sponsored Research
UNCLASSIFIED
DASA 2657
May 1971
STATIC UNIAXIAL DEFORMATION OF 15 ROCKS
FINAL REPORT
(This work was supported by the Advanced
Research Projects Agency under ARPA Order No. 862)
W.F. BRACE
HEADQUARTERS
Defense Atomic Support Agency
Washington, D.C. 20305
Department of Earth and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Contract No. DASA01-69-C-0122
Approved for public release; distribution unlimited
UNCLASSIFIED
l
ABSTRACT
Samples of 15 rocks with porosity ranging from nearly
zero to 40 percent were deformed under the constraint of
uniaxial strain by stresses which reached 30 kb. No faults
formed/ although widespread small scale fracturing
accompanied the compaction of the more porous rocks.
Rocks with porosity less than 2 percent recovered from
a cycle of loading with negligible permanent strain.
Calcite twinned extensively in Bedford limestone and
white marble, and in the latter there was indirect
evidence of recoverable flow. Rocks loaded uniaxially
reached very nearly the same stress-strain state as rocks
loaded first hydrostatically and, then, in triaxial
compression. The onset of dilatancy for granite, limestone
and marble was close to the stress in a triaxial experiment
at which strain was uniaxial.
TABLE OF CONTENTS
INTRODUCTION . 1
Previous work . 2
The present investigation . 4
THE ROCKS STUDIED . 5
EXPERIMENTAL PROCEDURE . 9
Jacket . 9
Sample preparation . 10
Strain measurement . 11
Leading procedure . 11
Compressibility . 16
MICROSCOPIC OBSERVATIONS .. 17
Marble . 17
Bedford limestone . 18
Solenhofen limestone . 19
Tuff/ tonalite and sandstones . 19
DISCUSSION . 19
Reproducibility . 19
Recoverable behavior . 20
Permanent strain . 23
Path dependence of the stress-strain relation . 24
Dilation and failure in relation to uniaxial
deformation . 27
Implications for failure theory of rocks . 30
IV
TABLE OF CONTENTS (continued)
ACKNOWLEDGMENTS . 31
REFERENCES . 32
FIGURE CAPTIONS . 37
LIST OF TABLES
1. Rocks studied . 6
2. Stresses, strains and porosity change during
uniaxial strain loading . 12
LIST OF FIGURES
1. Stress-strain relations during uniaxial loading ... 39
2. Stress in the axial direction as a function of
axial strain . 40
3. Photomicrographs of Bedford limestone (a) before
deformation and (b) after one cycle of loading . 41
4. Uniaxial strain behavior of Westerly granite . 42
5. Comparison of permanent volumetric compaction
with initial porosity . 43
6. Comparison of axial strains in uniaxial loading
(a) and in hydrostatic plus triaxial loading (b). 44
7. Dilation stress and uniaxial deformation
compared for Westerly granite . 45
8. Dilation stress and uniaxial deformation
compared for marble . 46
9. Path dependence of uniaxial deformation for
Solenhofen limestone . 47
10. Stress at fracture and dilation compared for
Westerly and Kitashirakawa granites . 48
INTRODUCTION
In a state of uniaxial strain, two of the principal
strains are zero. Strain is usually assumed to be uniaxial
in material loaded by a plane shock wave; this type of loading
is achieved in impact experiments [ 1 ], and approximately in
underground nuclear explosions [ 2 1• The unique strain
direction is perpendicular to the shock front. In tectonically
inactive regions of the Earth's crust where, for example,
vertical compaction in flat-lying rocks is taking place,
strain may also be uniaxial [3 ].
The mechanical behavior of rock loaded in uniaxial strain
is poorly understood. Few experimental studies are available.
How, for example, do rocks fail in uniaxial strain, and is
behavior prior to failure in any sense predictable from
independent measurement of elastic properties? What is the
role of porosity? One might suspect that rocks of high porosity
will respond to this type of loading quite differently from
those of low porosity. Finally, what is the role of strain
rate? Do rocks deformed in uniaxial strain behave the same
at high strain rates (shock loading) as at low strain rates
(geologic loading)? The present study was designed to throw
light on questions such as these. In addition, experiments in
uniaxial strain provide a means of exploring path dependence
of the stress-strain relation for rocks. This too is a
subject which has received scant attention in rock mechanics.
1
Previous work
SERATA [ 4 ] investigated rock-salt, limestone, and
dolomite under conditions approximating uniaxial strain.
Cylindrical samples were compressed axially while being
restrained laterally by thick-walled steel cylinders. The
lateral strains were not zero in his experiments, but were
quite small. SERATA reported yielding in his materials,
particularly in the rock-salt. Unfortunately, the materials
he used have little application to the problem at hand, and
there is some question as to the exact conditions of strain
in his experiments. HENDRON [ 5 ] and TZUNG [ 6] tested a
variety of sands at quite low pressures using a triaxial
configuration in which -ateral deformation of material was
monitored. Confining pressure was varied so as to maintain
zero lateral strain. BROWN et al [ 7] and SMITH et al [ 8]
studied the behavior of several rocks (granite, tuff, diabase,
rhyolite, and concrete) in uniaxial strain, using HENDRON's
technique. They were capable of applying axial stress to 5 kb
and confining pressure to 2 kb. The sample used was a very
short cylinder. They reported a number of interesting charac¬
teristics of the elastic behavior of their materials, including
maxima in the moduli at around a kilobar stress. No failure of
their rocks was reported. The significance of porosity was not
particularly clear, although they observed some densification
of their more porous materials. BROWN and SWANSON [ 9 ] loaded
Westerly granite, Cedar City tonalite and a quartzitic sandstone
2
(the Nugget sandstone of this study) in uniaxial strain as well
as along other loading paths. Their main object was development
of constitutive relations for rocks, although they investigated
several of the questions posed above. They reported, for
example, that volume contraction during uniaxial strain of
Westerly granite was closely predictable from compressibility.
They found no evidence of failure in the granite for stress as
high as 11 kb. Their technique of loading and strain measure¬
ment was nearly identical to that used here, although they were
limited to confining pressure of about 6 kb. Loading rate was
similar to this study. No microscopic observations were given.
A number of rocks have been subjected to shock loading
in order to determine an equation of state (see, for example,
MCQUEEN et al [10]; LOMBARD [ll]; AHRENS and GREGSON [12]) or
fracture or yield characteristics (for example, PETERSEN et al
[13]; AHRENS and ROSENBERG [14]; GIARDINI et al [15]). There
have been few attempts to correlate shock results with laboratory
experiments; strain rates in the former reach 10 7 sec’“ 1 , in the
latter, range from 10“ 3 to 10” 6 sec" 1 . One noteworthy study is
that of FROULA and JONES [ 1 ] who studied Westerly granite,
Solenhofen limestone. Cedar City tonalite, and Nevada Test Site
tuff. Solenhofen limestone behaved linearly up to crushing at
a stress of 6 kb; the crushing observed at higher stress was
time-dependent. Westerly granite behaved elastically to the
maximum stress of 45 kb applied during their experiment. Based
on a reinterpretation of the granite data, GREGSON, ISBELL, and
3
GREEN [16] reported evidence of yield in the granite at a stress
of about 17 kb.
The present investigation
In this paper attention is focussed in experimental
procedures used in uniaxial strain loading, and on both macro¬
scopic and microscopic aspects of the deformation. In companion
papers a comparison of shock and static deformation of three of
the rocks is given [17], and the recoverable, quasi-elastic
behavior of certain of the rocks is analyzed [18].
Our experimental procedure followed closely that of BROWN
et al [ 7 ] and BROWN and SWANSON [ 9], who used jacketed
cylindrical samples of rock with strain gauges fixed to the
surface to measure axial (ei) and circumferential (£3) strains.
Pressure and axial stress were applied to the sample and varied
independently in such a way that the lateral strain e 3 was
maintained equal to zero. The two stresses, 01 and cr 3 , were
observed during loading as well as the single strain ei, which
equals volume change. Compression here is a positive stress;
volumetric compaction is a positive strain.
A suite of rocks from our previous studies was particularly
chosen for the problems at hand. Porosity ranged from 40 percent
to nearly zero; composition covered typical igneous rocks,
schist, tuff, and sandstone. As many rocks as possible were
included from previous shock studies for our comparison of
shock and static behavior [17].
4
We report here the stress-strain relations for these
materials under uniaxial loading to stresses which reached
about 30 kb, a limit set by our ability to generate a
confining pressure and, therefore, a lateral stress of 10 kL.
For approximately half the suite of rocks, strains were
nearly recoverable in our experiments, and for these
compressibility was determined to 10 kb. This served two
purposes; it provided a sensitive test of cracking by
comparison of initial compressibility before and after
loading, and it enabled us to compare volumetric strains in
uniaxial and hydrostatic situations, as in the work of BROWN
and SWANSON [9], Also for these rocks, static Poisson's
ratio was measured as a function of pressure. This provided
a comparison with the value obtained from the relation
between ai and a 3 during uniaxial loading [18],
THE ROCKS STUDIED
Bulk density, total porosity, and modal analysis of
the rocks studied are listed in Table 1. As indicated, most
of the rocks have been investigated before in our studies of
elastic and electrical properties. The Cedar City tonalite
was supplied by S. Blouin of Kirtland Air Force Base. It is
from the same general area as material studied in [1], [22],
and [9]. A detailed petrographic description is given in [22].
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Rhyolite Tuff 40 33 gl, 20 qu, 40 or, 4 an 1( ., 2 ox
Our specimens of Westerly granite and Solenhofen limestone
are from different blocks as those of [9] and [1]. The
Navajo sandstone is from an unknown location. The Nugget
sandstone (quartzitic sandstone of [9]) comes from Parleys
Canyon/ Salt Lake County, Utah. The schist was supplied by
Dr. Larry Schindler, OCE, from an undisclosed site. The
Barre granite is from material currently being quarried at
B-irre, Vermont. Porosity was determined by immersion [21] ,
and for the new materials here has an uncertainty of 0.002.
EXPERIMENTAL PROCEDURE
Jacket
The function of the jacket was twofold, to exclude the
hydrostatic pressure medium from the rock sample, and to
provide a smooth continuous surface for mounting strain
gauges. Inasmuch as circumferential strains were to be
maintained equal to zero during the experiments, circumfer¬
ential strain in the jacket would also be negligible, so
that radial constraint due to the jacket did not have to be
considered. Seamless tubing 1.85 cm ID and 0.033 cm wall
thickness of annealed copper was used; spun caps of copper
were soft soldered to the tubing.
9
Sample preparation
Precisely ground right circular cylinders, 1.85 cm in
diameter and 3.8 cm long, were prepared from rock cores. At
this stage porosity was determined. Then, the rocks of low
porosity were jacketed as described above. The porous
materials (porosity greater than a few percent) were given
special treatment prior to jacketing.
Previous work had shown that porous rocks such as the
tonalite or the Indiana limestone cannot be jacketed and
gauged in the usual manner. Under high pressure the jacket
is forced into surface pores; failure of the jacket often
occurs. Even without failure, the apparent strain reported
by the gauges is often very different from the true strain in
the interior of the rock. To prevent collapse of jacket and
gauges into surface pores, a filled epoxy was applied to the
surface of the rock prior to jacketing. The filler was metal
powder so chosen that the elastic properties of the cured
epoxy approached that of the minerals. In a previous study
of the tonalite [24] , this procedure prevented surface pore
collapse under pressure; strains recorded from measurements
at the curved surface of samples treated in this manner agreed
with those measured externally.
Before strain gauges were mounted, the jacketed samples
were subjected to several hundred bars confining pressure.
This seated the jackets firmly against the surface of the
samples and also revealed jacket leaks. If dimples and other
10
depressions appeared at this stage in the jacKetea suriace,
they were filled with solder and smoothed with a hand grinder.
Strain measurement
Strain gauges were BLH epoxy-backed foil types (FAE-50-
12S6 or FAE-100-12S6) cemented with EPY-150 cement cured
according to manufacturers specification, using the additional
precautions outlined in [25], They were mounted axially and
circumferentially on the samples.
The effect of pressure on strain gauges was taken into
account following 126]. The pressure effect for the present
gauges was +0.60 x 10 -7 bar _1 . The apparent strain in the
axial direction, e*, was corrected for the pressure effect
in the usual way; the corrected quantity is given in Table 2.
The circumferential strain was to be maintained equal
to zero. Because of the pressure effect on the gauge, this
required that the gauge indicate an apparent strain exactly
equal to the pressure effect. The experiments were so
conducted that this condition was satisfied.
Loading procedure
The gauged samples were pressurized (medium was
petroleum ether) and loaded in a large screwdriven press.
Pressure was generated externally, and recorded together with
total axial force exerted by the press and the two strains as
described above. Procedure was somewhat different for low and
11
TABLE 2 Stresses, Strains and Porosity Change During Uniaxial Strain
Stresses are in kilobars, strains in 10 '
Compressive stress and compressive strain are positive
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14
high porosity rocks. For the latter, application of pressure
or axial compression generally caused permanent compressive
strain, whereas for the former, strains were typically recoverable.
For the low porosity rocks only, compressibility was determined
before and after uniaxial strain loading. As noted above, the
purpose was to detect possible cracking during uniaxial strain
loading. A pressure of about 1 kb was applied for two or three
cycles.
During an actual experiment, procedure was as follows.
The sample was placed inside the pressure vessel and leads
were connected to the strain gauges. The motor driven screw
was then advanced at a rate equivalent to a strain of about
10 -5 sec _I . A continuous record to axial force vs confining
pressure was made, as well as a record of pressure vs both
strains- As soon as the axial piston contacted the sample, load
began to increase; pressure was then manually raised so as to
maintain the circumferential strain equal to zero. As the
piston advanced, continuous plots were made until the fluid
pressure reached 10 kb, which was the limit of our pumping
system. Axial load and then pressure were dropped, and in the
case of the low porosity samples, compressibility to 1 kb re¬
measured. For the high porosity rocks, final external
dimensions were measured with a micrometer.
Axial load was measured with an external force cell
which had been calibrated against a proving ring. Accuracy
15
of force measurement was about 1 percent. A correction for
Q-ring friction at the pressure vessel seals was applied to
the measured force during data reduction.
Pressure was measured by a manganin coil which also,
through a bridge, provided an electrical signal suitable for
recording. Aocuracy was about 0.5 percent.
Strains were accurate to no better than 1 percent of the
measured value, the uncertainty in the gauge factor. The
condition of no circumferential strain could be maintained to
about ±25 x 10 -6 . It is not known how strain gauge character¬
istics change 'or strains as large as those recorded for the
more porous samples (up to 17 percent). Considerable uncertainty,
perhaps as high as 10 percent, must be attached to the values
of ei given below which exceed a few percent.
The data are collected in Table 2 and plotted in Figs.
1 and 2 for the fifteen rocks. Duplicate samples of Westerly
granite were run to test reproducibility so that two sets of
data appear for that entry in Table 2.
Compressibility
Measurement of linear strain as a function of hydrostatic
pressure was carried out for the low porosity rocks for two
reasons. First, increase in crack porosity during uniaxial
strain loading could be estimated using the procedure outlined
in [25]. Second, change in volume as a function of
pressure could be compared with volume changes during uniaxial
16
loading [18]. In Table 2 the nonrecoverable strain, or new
crack porosity, remaining after one cycle of uniaxial strain
loading is given as rip. This is given for the calcite rocks
(marble and limestones) even though it was likely that plastic
flow has occurred; this is known [27] to cause anomalous length
changes upon release of pressure that may have nothing to do
with cracks.
MICROSCOPIC OBSERVATIONS
The rocks in Table 2 are listed in order of increasing
initial porosity. This is also very nearly in order of
increasing compaction as given by n • Homogeneous compaction
P
in the axial direction was in fact the only obvious mode of
deformation. The samples contained no faults, or fractures
larger than the grain sise. All of the rocks below the Nugget
sandstone in Table 2 were sectioned to obtain further details
of the deformation. A section of the marble was also prepared
when we observed [18] that the effective Poisson's ratio of
this rock had reached a value close to 0.50.
Marble
A thin section of the marble cut parallel with the sample
axis revealed extensive twinning compared with untested material.
Traces of the twins were typically 10 to 45° to the axis, and
therefore to the Oi axis. Twin spacing was compared in tested
and untested marble. The average for 50 grains in untested
was 3.6 twins/mm compared with 11.4 twins/mm for the tested
sample, as measured on the flat stage. Twins in more than
one direction in a single grain were common in tested
material.
Bedford limestone
Our measurements, Table 2, revealed that much of the
original porosity of the Bedford limestone had been eliminated
during uniaxial deformation. This was borne out by study of
the thin sections, which also revealed interesting details of
the mechanics of compaction. Elimination of porosity is clearly
seen in the pair of photomicrographs (Fig. 3) made of untested
and tested material. Considerable plastic deformation of the
individual grains is evident in the deformed sample. Twinning
is quite abundant in the calcite which formed the cement between
the shell fragments and other debris in the original rock. Some
of the fossil fragments, nearly circular in cross section
originally, became elliptical during uniaxial strain deformation.
Measurement of axial lengths as seen in thin section show that
apparent strains in the Oj direction ranged from 5 to 15 percent;
owing to original ellipticity of the fossil fragments, actual
strains were probably closer to the smaller value. This number
may be compared with the ej of 9.8 percent (Table 2) caused by
uniaxial deformation.
18
Solenhofen limestone
An especially thin section was prepared of this fine¬
grained rock to see if plastic deformation of calcite had
occurred here too. No twins were observed, althouqh the 20
micron grains could be clearly resolved.
Tuff, tonalite and sandstones
All of these rocks became less porous (Table 2) by
amounts which ranged up to 13 percent of total volume. In
thin section the deformed material appeared minutely fractured,
but unfortunately in a way which could not be easily
distinguished from untested material. Perhaps the difficulty
lay with thin section preparation, which, particularly in the
case of the tonalite, may have introduced microfractures about
the same size as those produced during deformation. In any
event, details of the compaction mechanism in these rocks
were not obvious in thin section. Clearly, there is need
for further work in this area.
DISCUSSION
Reproducibility
Results for Westerly granite are plotted in Fig. 4 as
oj vs 03 and Oi vs ej. The two samples studied here were
virtually identical in the 01-03 plot (see also Table 2) and
19
very close to that of [9] , whose values are also shown in
Fig. 4. In the ai-Ci plot, our two samples differed by about
two percent and were within a few percent of the BROWN and
SWANSON values. Data from two different shock experiments
are also given in Fig. 4 for comparison with our static
values. The differences, which are seen to be small, are
discussed in [17]. One of the shock studies was done on the
so-called Bradford granite [28], which is said to come from
a quarry adjacent to that of Westerly granite.
Agreement in Oi-a 3 for granite is probably as good as
can be expected for two different laboratories, particularly
when the samples are not taken from the same block of rock.
The agreement in the Oi-ei for Westerly is also quite
satisfactory.
Recoverable behavior
Recovery as opposed to yield is defined in terms of r^.
A sample is said to recover if, after a cycle of uniaxial
strain loading, n had a magnitude less than 0.5 x 10~ 4 .
P
Much smaller strains than this can be detected when strain
gauges are used in more conventional applications, but in
view of the large strains imposed here this is quite a
satisfactory limit.
n (Table 2) is small and typically negative (denoting
a permanent increase in volume) for all the rocks through
20
Nugget sandstone; for the more porous rocks making up the
balance of the Table, rip ranges up to 13.1 percent. Behavior
of the first group, which recovered, is discussed in [18];
that of the second, in which permanent volumetric compaction
took place, is considered in the next section.
The small extensions shown by many of the rocks may be
a manifestation of the effect first noted by PATERSON [27] and
more recently studied in detail by EDMOND [29]. For a wide
variety of ductile rocks (limestone, marble, soapstone, poly¬
crystalline alkali halides, and talc, and serpentinites) they
observed a permanent increase in volume during the release of
pressure following triaxial deformation to large permanent
axial strains. The volume increase was particularly marked
for the calcite rocks. It is of interest that in our study
marble and Oak Hall limestone increased in volume 50 to 60 x 10“ 4 .
This might imply, according to [27], that some plastic
deformation of these materials took place during uniaxial
strain. As we noted above, this was substantiated by our
microscopic study of the marble. A small increase in volume
for schist, felsite and both granites was also noted (Table 2),
although plastic flow of these materials in our experiments
seems unlikely.
The volumetric strain, ei, for all the rocks are compared
as a function of Oi in Fig. 2. The curves for the rocks which
recovered are seen to be very nearly linear, whereas, with the
21
exception of Pottsville and Nugget sandstones, all of the others
are strongly curved. The marked linearity and small permanent
strains of the low porosity rocks suggests that the recoverable
behavior was largely elastic. Elsewhere we analyzed this
behavior [18] and found that volumetric compression during
uniaxial strain was closely predictable from independent
measurements of compressibility for diabase, gabbro, schist,
marble, and two granites. However, the Poisson's ratio in
uniaxial strain was appreciably higher than given by direct
measurement. The difference could be explained by sliding
motion on closed cracks, combined, in the case of marble, with
flow of calcite.
Behavior of the marble in uniaxial strain was unusually
interesting, as it may be an example of recoverable plastic
flow. As noted in Table 2, recovery from the strain of about
2 percent was nearly complete, although there seemed compelling
microscopic evidence of flow in the calcite grains, and the
apparent Poisson's ratio during uniaxial deformation was nearly
0.50 [18]. The question immediately arises, did some of the
calcite untwin during unloading, or did one set of grains twin
during loading and another, differently oriented set twin during
unloading? This question could probably be resolved by careful
petrofabric analysis, which was, unfortunately, beyond the scope
of the present study.
22
Permanent strain
Appreciable permanent or nonrecoverable strain, n , was
observed for all of the rocks having porosity greater than
about 2 percent. This took the form of an apparently
homogeneous one-dimensional compaction. As noted above, no
faults, fractures or offsets larger than the grain diameter
were observed.
The magnitude of the permanent strain, rip# correlates
fairly well with initial porosity {Fig. 5). The 45° line in
this figure represents the maximum value of rip* Pottsville
sandstone and Bedford limestone are apparently quite close to
this limit; the others are within about 40 percent of complete
compaction. It is interesting that the degree of compaction
does not always improve with the rocks which have the lowest
strength, as might be expected. (Compare the stronger tonalite
and weaker rhyolite, for example.) Probably a great many factors
affect the degree of compaction at any given pressure, including
grain size, shape of the pores, mineralogy, degree of alteration,
and abundance and continuity of cracks.
In Fig. 2, the shapes of the opEi curves for the high
porosity rocks may be compared. The curves are of two types,
those initially concave upward (Pottsville sandstone and
tonalite) and those initially concave downward. This difference
is probably due to differences in crack porosity; from electrical
«
studies [19], rhyolite, Bedford limestone and Solenhofen lime¬
stones are known to have little or no crack porosity, whereas
Pottsville sandstone does.
23
The stress at which total compaction might occur can be
roughly estimated from Fig. 2. The dotted lines give curves
which would be followed for the different rocks if porosity
were zero. They are obtained from the known elastic properties
of rocks of the same composition; they intersect the abscissa
at e i equal the porosity. For example, the dotted line for
Solenhofen limestone has about the same slope as the solid
line for marble; it intersects the strain axis at 4.8 percent,
the value of the porosity of the limestone. The stress at which
the measured curves intersect these dotted lines would be the
stress at which porosity reaches zero. For the two limestones,
this stress appears to be 15 to 20 kb. For the Navajo sandstone
this stress probably exceeds 30 kb. For the tonalite it may be
a great deal higher.
From the shape of the curves in Fig. 3, the stress at
which pore closure began for tonalite, rhyolite and Bedford
limestone was apparently very low; data were not recorded at
very low stresses, so that the actual magnitude cannot be
definitely established. Finally, it is also of interest that
for the two high porosity limestones, the stress at which pore
collapse began seems to correlate inversely with the initial
porosity (compare Figs. 2 and 5), as might be anticipated.
Path dependence of the stress-strain relation
The uniaxial strain experiments provide an opportunity to
test the dependence of stress-strain behavior of certain of the
24
rocks upon loading path. This interesting theoretical question
has, to our knowledge, only been considered previously by BROWN
and SWANSON [ 9]. They found that stresses at faulting in
Westerly granite and the tonalite did not differ by more than
5 or 10 percent for several stress paths. Here, we do not
compare stresses at faulting but rather the stresses and strains
when strain is uniaxial. This is done two ways. Referring to
Fig. 6 , we first compare the strains in the g) direction when
strain in the 03 direction is zero. In Fig. 6 a the strain,
is just the value found above during uniaxial strain loading
and referred to in Table 2 as ei. We compare e y with sum of
the strains e H and e T from hydrostatic and triaxial loading
respectively. e H is just the linear compression due to a
hydrostatic pressure equal to 03 . We obtain e T from a triaxial
compression experiment at constant confining pressure equal to
a 3 at that point on the stress-strain curve when the lateral
strain, an extension, just equals the linear compression due
to confining pressure 03 . In other words, it is the point on
the stress-strain curve where the lateral shortening due to
confining pressure is just balanced by the lateral extension
due to axial stress.
A second test of path dependence is given by comparison
of stresses in the Oi direction . Our question here is, does
the total axial stress in a triaxial experiment equal o 1 in the
uniaxial strain experiment, when total lateral strain is zero
in both?
25
Triaxial experiments in which both axial and radial
strains were measured were available from a previous study of
dilatancy [30] for Westerly granite and marble. Although
Solenhofen limestone has been widely studied, complete strain
data from triaxial experiments are unknown to us. Eight
experiments were carried out to provide the required information,
using jackets and strain gauges identical to those used in
uniaxial loading as described above.
Comparison following the two schemes outlined above is
shown in Fig. 7 for Westerly granite, Fig. 8 for marble, and
Fig. 9 for Solenhofen limestone. In each, we show all or portions
of the curves of a i vs aj and Oi V£ £i given above and tabulated
in Table 2, for comparison with points from triaxial experiments
at which total lateral strain was zero.
Comparison of the uniaxial with the combined hydrostatic-
triaxial data reveals very close agreement in the strains for
granite and limestone except at high pressures, and a small
(10 percent) but consistent difference for the marble.
Comparison of the stresses for the three rocks (the left hand
curve in each figure) gives agreement within about 5 percent
for all three of the rocks, except granite at high pressure.
For the marble, triaxial values are consistently below; for
the granite, above; and for the limestone alternating above
and below the values obtained in uniaxial loading.
Without further work it is not known whether the differences
cited above reflect small but real differences due to path, or
26
whether they are experimental. Of the three, the most careful
comparison was made in the case of Selenhofen; for the other
rocks, the samples used came from different blocks or from
different orientations in the same block. This last factor
may be particularly critical for the marble, which is
elastically anisotropic [25] to a degree consistent with the
differences noted above.
We conclude that path dependence in our three rocks is
of the same order as the path dependence of the stress at
faulting reported in [9]; that is, variations due to path are
10 percent or less. This result is of some interest owing to
the wide range of mechanical properties of the three rocks.
Under the conditions of the experiments, granite was brittle,
and marble was ductile except near room pressure. The limestone
was brittle at low pressure and ductile at high pressure if we
adopt from HEARD [31] the mean stress (2.7 kb) at which he
observed the brittle-ductile transition. In addition, pore
collapse occurs at the higher stresses in the limestone.
Apparently, then, relative insensitivity to loading path is
a common feature of both stress-strain behavior (our result)
and ultimate strength [9] of rocks.
Dilation and failure in relation to uniaxial deformation
The stress paths during uniaxial deformation are shown
for all of the rocks in Fig. 1. Clearly stresses were non¬
hydrostatic even for the weakest rocks, and it is of interest
27
to determine why a particular rock followed a particular path.
From consideration of recoverable behavior [18] and from
examination of the curves in Fig. 1, the determining factors
seem to be: intrinsic elastic properties, crack characteristics,
and the tendency to fail locally by pore collapse or by intra¬
crystalline flow. Of all the rocks, only the diabase followed
a stress path (Fig. 1) which could have been predicted purely
from elastic properties of the minerals [18]. Several of the
other low porosity rocks, for example, the granites, the marble,
and the gabbro were truly elastic only in regard to nondeviatoric
strain; their curves in Fig. 1 reflect a combination of elastic
strain and relaxation due to sliding on closed cracks. The
relative importance of sliding on cracks may be found by
comparison of rocks of similar mineralogy for which, therefore,
elastic strains would be identical. Apparently motion on cracks
played a greater role for gabbro than for diabase, and for Barre
than for Westerly granite. Unfortunately no more quantitative
explanation of these differences is possible.
Certain of the rocks were clearly weakened by high
porosity; for example, Navajo compared with Nugget sandstone,
and rhyolite compared with the granites. However, it is
curious that the stress paths of the calcite rocks (marble,
Bedford and Solenhofen limestones) do not reflect differences
in porosity, particularly at high stress levels. Perhaps the
low shear strength of calcite dominated here.
28
Rocks typically increase in volume nonelastically during
axial compression (termed dilatancy or dilation ) at a stress
■ference which is about half to two thirds the fracture
rength (30, 29]. Where does dilation begin relative to the
.nt in a triaxial experiment at which strain is uniaxial?
i stress at which dilation is first detected, ..he dilation
ress , was previously determined for two of the rocks studied
re, Westerly granite and marble. In Fig. 7 the dilation
ress (heavy bar) is compared for a number of experiments at
fferent confining pressure, o 3 , with the value of ai (open
x) at which total lateral strain in the triaxial experiments
s zero. As discussed in the last section, we also give in
g. 7 the curve (dotted) obtained in the present study by
iaxial loading. Comparison of the three sets of data shows
,ir agreement except at a 3 = 0. The dotted curve, representing
ie stresses at uniaxial strain obtained here, appears to be
.thin about 10 percent of the dilation stress.
A similar comparison for marble is given in Fig. 8 with
riaxial data from the same source. For this rock dilation was
lly observed below a 3 of about 4 kb [32, 29). The dilation
tress (solid bar) is very close to the uniaxial strain point
square) from the triaxial experiment; it is systematically
slow the curve through the values measured here.
The dilation stress is known for another granite from the
ork of MATSUSHIMA [33], This rock, the Kitashirakawa granite,
29
has a grain size two to three times that of Westerly granite,
but has quite similar mineralogy, containing 80 percent feldspar,
10 percent quartz, and 8 percent biotite. The dilation and
fracture stresses are compared with those of Westerly granite
in Fig. 10; the dilation stresses for these rocks are quite
similar, although Westerly is about 25 percent stronger than
the Kitashirakawa granite.
Implications for failure theory of rocks
In previous studies, PAULDING [34] and BRACE and BYERLEE
[35] tested the applicability of Griffith theory of fracture,
modified to include friction on closed cracks, to brittle
fracture of rocks under pressure. Both from theoretical
considerations and from observation of the way elastic
properties changed as stress increased to fracture, it seemed
more likely that Griffith the^- y predicts the dilation stress
rather than the stress at wuich fracture by faulting occurs.
However, in the light of present results, even this modified
view may have to be rejected. For one thing, the dilation
stress does not appear to be very structure-sensitive, as was
shown by comparison of data for the two granites (Fig. 10).
For Griffith theory to apply, dilation stress should vary with
initial crack length, which is, presumably, closely related to
grain size. For another thing, agreement of the dilation stress
with the point of zero lateral strain for three very different
rocks suggests that the dilation stress may be controlled more
by geometrical than by microstructural or mineralogic factors.
30
luch more work is needed here for a fuller understanding of the
Iilation stress, and a particularly interesting area of study is
iehavior under more general stress states. Experiments with
:hree unlike compressions will provide an important test of the
:onsistency of our observations.
The results near atmospheric pressure (Figs. 7, 10) raise
l number of questions. The dilation stress, from the previous
leasurements, is of the order of half the fracture stress, which
.s obviously different from the point of uniaxial strain. For
in experiment at 03 = 0 , strain is uniaxial only at j] = 0 .
:iearly the situation at room pressure is different, then, from
rhat in a confined compression test at a pressure above a few
mndred bars. Does this mean, on the one hand, that the detailed
:racture process is different in the two cases? There seems to
>e some evidence for this in recent crack studies [36, 37]. Or does
.t mean, on the other hand, that the dilation stress has been
.ncorrectly measured in experiments at low pressure? Does
iilation actually begin close to zero stress, as suggested by
:he present correlation? Clearly these questions need to be
resolved.
Acknowledgments - This research was supported by Advanced
Research Projects Agency (ARPA) and the Defense Atomic Support
Agency (DASA) and was monitored by DASA under Contract No. DASA
)l-69-C-0122. Discussion with Clifton McFarland, J.B. Walsh,
and S.P. Green was particularly helpful. Prof. S. Matsushima of
Cyoto University assisted by providing a thin section of the
Citashirakawa granite.
31
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36
CAPTIONS
. 1 Stress-strain relations during uniaxial loading. Along the
line marked HYDROSTATIC, Oi equals 03 .
. 2 Stress in the axial direction as a function of axial strain.
Curves are identified by abbreviated rock name. The small
number on some of the curves is porosity in percent. The
dotted lines are stress-strain curves which would be followed
if porosity were zero.
[. 3 Photomicrographs of Bedford limestone (a) before deformation
and (b) after one cycle of loading in uniaxial strain along
the path shown in Fig. 1.
I. 4 Uniaxial strain behavior of Westerly granite. The open
circles and triangles are static experiments, the closed
squares for shock loading. GRINE 1 s data pertain to Bradford
granite.
I. 5 Comparison of permanent volumetric compaction with initial
porosity for high porosity rocks. Rock names are abbreviated.
Size of boxes indicates uncertainty.
|. 6 Comparison of axial strains in uniaxial loading (a) and in
hydrostatic plus triaxial loading (b). The end state in each
case is the same. The dotted figure in (b) is the position
after hydrostatic loading, the dot-and-dash figure; the
position after triaxial plus hydrostatic loading.
17
Fig. 7 Dilation stress and uniaxial deformation compared for
Westerly granite. The dotted curve of 0 | vs Oj is from
Fig. 4. The dotted area in the plot of Oj V£ ej includes
all of the data points for the granite, from Fig. 4. The
boxes are points in triaxial experiments in which total
lateral strain was zero from [30]. The bars give the
approximate values of the dilation stress, also from [30].
Fig. 8 Dilation stress and uniaxial deformation compared for
marLle. Symbols same as Fig. 7.
Fig. 9 Path dependence of uniaxial deformation for Solenhofen
limestone. Symbols same as Fig. 7.
Fig. 10 Stress at fracture and dilation compared for Westerly and
Kitashirakawa granites. The dotted band includes the values
of stress at dilation from Fig. 7. Fracture stresses for
Westerly are from [30, 38].
38
oru crographs of Bedford limestone (a) before deformation
(: ) after or.e cede of loadina
100
Fig.
O
% ‘Ajjsojod |D|J!U|
Comparison of permanent volumetric compaction
with initial porosity
43
Fig. 6 Comparison of axial strains in uniaxial
loading (a) and in hydrostatic plus
triaxial loading (b)
(a) Uniaxial (b) Hydrostatic + triaxial
Fig. 9 Fath dcpondcn
Solenhofen li
