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AS 726746 


r* 


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 

5EUu ibti 
c 


Approved for public release; distribution unlimited 


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II. •nOMMUIN* Ml LI T An V ACTIVITY 

The Director 

Advanced Research Projects Agency 
Washington, D.C, 20301 




NOtlt (Any •*« imnMw DM may A* M»l|nM 


May 1971 


DASA 01-69-C-0122 

A nnojcc t mo. 

ARP A Order 862 
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]. 


5 





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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 
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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 


REFERENCES 


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2. BUTKOVICH, T.R. Calculation of the shock waves from an 
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3. HUBBERT, M. KING, and WILLIS, David G. Mechanics of 
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4. SERATA, Shosei. Transition from elastic to plastic states 
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5. HENDRON, A.J. The behavior of sand in one-dimensional 
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6 . TZUNG, Fu-Kong. Sand in one-dimensional compression. 
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7. BROWN, W.S., DE VRIES, K.L., and SMITH, J.L. Properties of 
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32 


SMITH, J.L., DE VRIES, K.L., BUSHNELL, D.J., and BROWN, W.S. 
Fracture data and stress-strain behavior of rocks in triaxial 
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BROWN, W.S., and SWANSON, S.R. Constitutive equations for 
Westerly granite and Cedar City tonalite for a variety of 
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MCQUEEN, R.G., MARSH, S.P., and FRITZ, J.N. Huguniot 
equation of state of twelve rocks. J. Geophys. Res. 72(20) , 
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LOMBARD, D.B. The Hugoniot equation of state of rocks. 
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AHRENS, T.J., and GREGSON, V.G., Jr. Shock compression of 
crustal rocks; data for quartz calcite and plagioclase rocks. 
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PETERSEN, C.R., MURRI, W.J., and COWPERTHWAITE, M. Hugoniot 
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AHRENS, T.J., and ROSENBERG, J.N. "Shock metamorphism; 
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Book Corp., Baltimore (1968). 


33 



15. GIARDINI, A.A., LAKNER, J.F., STEPHENS, D.R., and STROMBERG, 

H.D. Triaxlal compression data on nuclear explosion shocked, 
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Westerly granite under shock loading. Trans. Amer. Geophys. 
Union 51(4), 423 (1970). 

17. BRACE, W.F., and JONES, A.H. Comparison of uniaxial deformatior 
in shock and static loading of three rocks. J. Geophys. Res. , 
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1^. WALSH, J.B., and BRACE, W.F. Elasticity of rock in uniaxial 
strain. Int. J. Rk. Mech. Min. Sci. , in press (1971). 

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crystalline rocks. J. Geophys. Res. 70 , 5669-5678 (1965). 


34 



2. PERKINS, R.D., GREEN, S.J., and FRIEDMAN, M. Uniar'ial 
stress behavior of porphyritic tonalite at strain rates to 
10 3 /second. lnt. J. Rock Mech. Min. Sci . 1_, 527-535 (1970). 

3. GREEN, S.J., and PERKINS, R.D. Uniaxial compression tests 
at strain rates from 10 _ Vsec to 10 “/see on three geologic 
materials. DASA-2199, Final Report , 44 pp. (1969). 

!4. WALSH, J.B., BRACE, W.F., and WAWERSIK, W.R. Attenuation 
of stress waves in Cedar City diorite. Tech. Rept. No. 
AFWL-TR-70-8 , 76 pp. (1970). 

25. BRACE, W.F. Some new measurements of linear compressibility 
of rocks. J. Geophys. Res. 70 , 391-398 (1965). 

26. BRACE, W.F. Effect of pressure on electric-resistance strain 
gages. Experimental Mechanics 4(7), 212-216 (1964). 

27. PATERSON, M.S. Secondary changes of length with pressure in 
experimentally deformed rocks. Proc. Roy. Soc, London, A , 271 , 
57-87 (1963). 

28. GRINE, D.R. Progress Letter No. 7, SRI Project PGU 7852 , 
Stanford Research Inst,, Menlo Park, Celif .(1970). 

29. EDMOND, J.M. Effects of pressure during rock deformation, 

PhD thesis, Australian National University (1969) . 

30. BRACE, W.F., PAULDING, B.W., Jr., and SCHOLZ, C. Dilatancy 

in the fracture of crystalline rocks. J. Geophys. Res. 7_1(16) , 
3939-3953 (1966). 


35 



31. HEARD, H.C. "Transition from brittle fracture to ductile 
flow in Solenhofen limestone as a function of temperature, 
confining pressure, and interstitial fluid pressure", in 

R ock Deformation (D. Griggs and J. Handin, Eds.), GSA Memoir 79, 
Chap. 7, 193-226 (1960). 

32. SCHOLZ, C.H. Microfracturing and inelastic deformation of 
rock in compression. J. Geophy. Res. 73 , 1417-1432 (1968). 

33. MATSUSHIMA, S. On the deformation and fracture of granite 
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Bull. No. 36, Kyoto Univ. (1960). 

34. PAULDING, B.W., Jr. Crack growth during brittle fracture in 
compression. PhD. thesis, M.I.T. (1965). 

35. BRACE, W.F., ^nd BYERLEE, J.D. "Recent experimental studies of 
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Amer. Inst, of Mining, New York (1966). 

36. WAWERSIK, W. , and BRACE, W.F. Post-failure behavior in 
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37. KOIDE, H. and HOSHINO, K. Development of microfractures in 
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85-97 (1967). 

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(1967) . 

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