Triaxial Tests on Granular Materials
Transcript of Triaxial Tests on Granular Materials
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Pow der Technology, 60 (1990) 99
-
119
Recent Results of Triaxial Tests with Granular Materials
D KOLYMBAS and W WU
Znst ztut e or Sozl M echanzcs and Rock M echanzcs, Unzv erszty of Karl sruhe, Kazserstr 12, D-7500 Kar lsruhe 1
VRG)
(Recewed December 13,1988, m rewed form June 22,1989)
99
SUMMARY
I n this paper are presented some recent
traaxur l test r esul ts obtamed w h dry sand,
sugar, rape, wheat and synthetic granulates
The device used was a tr laxwl apparatus
specaal ly desrgned to test dry sdo mater s
The resul ts are reported wrth a mew to facll l-
ta mg development and checkmg of appropn-
ate constltu twe equati ons. This IS only pos-
sible if special pr ecauti ons have been taken
to suppr ess error sour ces and guarantee a
homogeneous deformation The resul ts pre-
sented here reveal some character lstlcs of the
sample behavlour , namely (I ) even durang the
ml eal lsotroplc consohda tlon the samples
behave anasotropwal ly, (11) he mhomogene-
ous sample deformati on sets m from the
begmnmg of the traaxl al compression and,
therefore, the test resul ts cannot be evaluated
without a deconvolu tlon technique, and (ur )
with loose sands and granulates constltu ted
f rom soft grams, as well as at high stress
levels, a peak state 1s not obtained and, there-
for e, any reference to a fr iction angle 1s
questionable A simple deconvolutlon tech-
ni que B also presented
INTRODUCTION
Loading h&ones occurrmg m practice are
very complex, and very few can be simulated
by laboratory tests. In general, deformation
occurs together with a rotation of the prm-
clpal stress dlrectlons. Despite several
attempts, e g the simple shear tests described
by Budhu [ 11,
it
has not been possible to
simulate this sort of motion m the laboratory
with a homogeneously deformed sample.
Homogeneity of the deformation is, however,
an mdispensable property of tests which are
supposed to provide the basis for developmg
and checkmg constitutive equations. Thus, a
0032-5910/901 3 50
distmctlon should be drawn between the
laboratory tests which do not fulfil the
requirement of homogeneous deformation
(e g. the shear box test) and those which
allow homogeneous deformation to some
extent.
The mam representative of the latter group
1sthe so-called tr laxzul test, which was mtro-
duced mto soil mechanics m the twenties by
Ehrenberg The pnnciple of this test is as
follows: a cylmdncal sample is compressed m
the axial direction, while the hydrostatically
applied lateral stresses u2 = u3 are kept con-
stant. During the test, the axial and lateral
displacements ui and u3, respectively, are
measured as well as the axial force F,. The
results are evaluated as follows
(1)
f3 =
log
with
A= i (do - 22.~~)~
Of course, this evaluation presupposes that
stresses and strams are homogeneously (1.e
,
constantly) distributed withm the sample,
otherwise the above evaluation is meanmgless.
Although the tnaxlal test appears quite
simple, a series of difficulties and errors has to
be circumvented
TEST DEVICE
A new tnaxial apparatus (see Fig. 1)
has been designed m the Institute of Soil
0 Elsewer Sequola/Prmted m The Netherlands
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1 Loadnlg frame
2 Lorbdmg piston
3 Pressure cell
1 Top cap
5 Bottom cap
I
Sample
7 Load cell
3 Bellows
9 Spoke-wheels
D~splacamnt
transducer
Fig 1 Layout of the trlaxlal test apparatus
Mechanics and Rock Mechamcs of the Karls-
ruhe Umversitya. The apparatus has been
designed for samples with the mitral dunen-
sions h, = 10 cm and d, = 10 cm. The axial
load is exerted by movmg the loadmg piston.
The velocity of the piston can be regulated m
the range 4 pm/h to 20 mm/mm. In the pres-
ent tests, a downwards piston velocity of 10
mm/h is used. The ram is fixed to the top end
plate of the specunen. The apparatus allows a
maxunum axial load of 100 kN. The maxi-
mum design confmmg pressure u2 = u3 is
1400 kPa. The tnaxial apparatus is character-
ized by the followmg special features
Axzal force measurement
The axial force 1s measured beneath the
pressure chamber by a load cell with a precl-
sion of *30 N The force is transmitted out-
side the pressure chamber by means of a rod
guided by two spoke-wheels (see Fig. 2). A
steel bellows is used to separate the pressurized
cell au from the atmosphere and makes it
possible to transrmt the axial force outside
the pressure chamber, while the two spoke-
wheels (see Fig. 3) guarantee a vertical align-
ment of the transmission rod The influences
due to the stiffnesses of the bellows and the
*A Jomt research project (‘Sonderforschungs-
oerelch’) on ~110shas been estabhshed by several mstl-
tutes of the Umverslty of Karlsruhe with the fmanclal
support of the German Research Community (DFG)
In the framework of this project, the authors mvestl-
gate the mechamcal behavlour of sdo materials
Fig 2 Prmclple of the axial force measurement
SFQKE-
WHEEL
Fig 3 Schematic representation of the spoke-wheels
and the bellows
spoke-wheels are determmed by an appropn-
ate calibration.
Since the axial force is measured beneath
the pressure chamber, the measurement is not
mfluenced either by the fnction between the
loadmg piston and the sealmg or by the
confmmg pressure.
Adjustable cell pressur e
Air 1sused as cell fhud. The cell pressure
can be measured with an accuracy of Au, =
Au, = +0.2 kPa with a pressure transducer. As
already mentioned, m the usual tnaxial tests,
the lateral stress is kept constant. Complex
loadmg histones can be apphed by varymg the
cell pressure. This is achieved by a computer-
controlled motor valve, with which the cell
pressure can be adjusted with an accuracy of
+2 kPa
Lateral stram measurement
Problems and methods related to the lateral
stram measurement are discussed by Tatsuoka
[2]. The use of a proximity transducer is
reported by Dupas
et al [3].
The method
applied by Ueng et al [4] (freezmg) is
mapphcable to dry materials. In the present
mvestigation, the lateral stram of the sample
is measured directly by means of three collars
which contact the sample m the upper,
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Fig Lateral strain collars
middle, and lower parts, respectively These
steel collars are equipped with electric stram
gauges (see Fig. 4) and are pre-stressed m such
a way that they contact the sample with a
gentle pressure. An mcrease m the sample
diameter causes a change m the curvature of
the collars, which results m a local stram
bemg measured. With
d,
bemg the thickness of
the collar and
r
bemg the radius of curvature at
the location of the stram gauges, the stram E
of the collar caused by the displacement u3 is
given by E =
d,u3/r2.
Typical values for the present apparatus are
d, =
0.15 mm, u3 = 10 mm,
r = 50
mm,
resulting m a stram of E = 1 5 X 10P4 The
datalogger fmally allows the determmation of
the lateral stram of the sample with an
accuracy of +0.02 % Calibration shows a
neghgible hysteresis and a satisfactory linear-
ity. For a detailed description of the lateral
stram measurement, the reader 1sreferred to
[5]. Because of the mcompressibmty, the
rubber membrane surroundmg the sample is
not expected to mfluence the measurement of
the lateral deformation of the sample.
End plate lubracatlon
In conventional tnaxial tests, the sample
contacts the filter stone directly. The friction
at the upper and lower end plates hmders the
lateral expansion of the sample, which is a
requirement for the homogeneous deforma-
tion of the sample [6]. To overcome this
effect, tall samples
(ho/d,, = 2.5)
have been
used m the past, and it was expected that the
end plate friction would not mfluence the
middle part of the sample. However, this
method forces the sample to deform mhomo-
geneously and, therefore, lubncated ends have
been used to reduce the friction between the
end plates and the sample [ 71.
In the present tests, the followmg standard-
ized method of lubrication is apphed: A 0.05-
mm thick film of the grease UNISILKON,
TK44 N3RECA is applied to the surfaces of
the end plates, which are made of glass. The
grease film is then covered by a 0.3-mm thick
rubber disk. This method has been found to
successfully suppress the friction at the end
plates. The thickness of the lubncation layer
is kept constant from test to test
ERROR SOURCES AND CORRECTIONS
Fr lctl on between the end plates and the
sample
The use of lubricated ends reduces the
friction between the end plates and the sam-
ple considerably and the deformation of the
sample becomes more uniform However, it is
generally acknowledged that the friction
cannot be ehmmated completely by using the
lubncated ends. Besides, the effect of the
friction at the end plates on the test results is
difficult to assess In the direct shear test, the
fnction angle between the lubncated end and
the sand (fine to medium) was found to be
smaller than 0.25” [8]. This fmdmg is m
accordance with that of Goto and Tatsuoka
[9], accordmg to which the friction angle was
reduced to 0 14” .
0 16” by the use of lubn-
cated ends. Fnctlon reduction without
bedding error can possibly be achieved by
using extremely hard and smooth endplates.
For this purpose, we have examined end
plates which were ground, lapped, pohshed
and covered with a thm film of tltamum-
alummum mtnte. However, the friction
between sand and end plate could not be
suppressed below 2”. This fmdmg 1sm accor-
dance with the observations of Lmton
et al
[lo]
and Ueng
et al
[4]
Corr ectzon for the beddmg err or
A problem associated with the use of the
lubncated ends is that the axial deformation
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measured mcludes not only the deformation
of the sample but also the deformation of the
lubrication layers the so-called
beddzng
error).
There are two mam approaches to correc-
tion of the beddmg error. The first approach
is theoretical or semi-theoretical, whereas the
second approach 1s experimental. For a
thorough exposition of the first approach, the
reader is referred to [ll]. Because of the
many sunphfications mvolved, an exact cor-
rection cannot be achieved through the
theoretical approach. In the present study,
the experimental approach wilI be discussed.
There are two experimental methods
proposed by Newland and Alley [12] and
Roscoe et al [13] respectively. In [12], the
beddmg error is corrected by evaluatmg an
isotropic compression test. The difference
between the axial and the radial stram gives
the correction for the beddmg error. This
method seems to be quite simple at first
glance. Isotropic compression tests, as will be
described m the sequel, show, however, that
the samples behave amsotropically This
renders the method by Newland and Alley
mapphcable.
The method by Roscoe et al
was origmally
proposed to deal with lateral membrane
penetration and the same pnnciple was used
to correct the beddmg error by Sarsby et al
[ 141. In our mstitute, a test senes has been
carried out by Goldscheider [ 151 swnmg at an
exact determmation of the beddmg error.
Figure 5 shows the results under monoto-
mcalIy increased normal stress for dense
Karlsruhe medium sand. A large scatter m the
test data can be readily seen. The bedding
error can be roughly accounted for by the
followmg empirical equation [ 151:
u IkN/m*l
E
E to=03mm
zi 02
Fig 5 Beddmg error us normal stress after Gold-
schelder [ 15 ]
At
- = al[l - exp(-a20)]
1
co
where
At
results from the compression of the
rubber membrane and from the mdentation
of grams mto it;
to 1s
the mitral thickness of
the rubber membrane; (TIS the normal stress,
ul = 0.3 and a2 = 0.0037 m’/kN are con-
stants dependmg on the material tested.
This fmdmg can be compared with that of
Mochlzuki et al [
171. The bedding error
correction accordmg to eqn. 5) has been
apphed to treat the data presented m Figs. 7
and 9. This correction does not take mto
account the compression of the grease layer.
Neglecting the correction, however, appears
to be Justifiable since the thickness of the
grease layer amounts only 0.05 mm. Accord-
ing to Sarsby
et al [
141,
the untial
density
of the sample has minor influence on the
beddmg error, so that eqn. 5) can be apphed
with equal force to loose Karlsruhe medium
sand. For materials other than Karlsruhe sand,
the correction for the beddmg error 1s made
by assummg that the rubber membrane is
totally compressed at u = 1000 kPa, i.e. At =
tw
Obviously, this correction overestimates
the bedding error. However, it offers an upper
bound for the bedding error.
It can be seen that no matter how the cor-
rection for beddmg error is made, theoreti-
cally or expenmentally, an exact correction
can never be expected. Without proper pre-
cautions, the
correction could even brmg
about a greater error than no correction at all
Resides, the beddmg error may only mfluence
the deformation behaviour It does not have
any influence upon the strength charactens-
tics. Certamly, this does not mean that we
should simply overlook the bedding error
Rather, the difficulty as well as the necessity
for the correction should be appreciated.
In the haste to obtam corrections for
bedding error, experimental results are also
presented m the hterature without cor-
rection for the bedding error, e.g. [18]. We
are of the opmion that the significance of the
bedding error should be studied for certam
typical tests. The total test results, however,
should be presented without any correction.
Sufficient data, e g. thickness of the rubber
membrane and of the grease layer, the elastic
modulus and the Poisson ratio of the rubber
membrane, the density of the sample and the
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103
mean diameter of the grams should be pre-
sented m case such corrections are required.
In the present paper, the beddmg error is
corrected for several typical tests m order to
show its influence on the stress-&ram behav-
iour, see Figs. 7 and 9. For the total tests,
however, the beddmg error is left uncor-
rected.
Correctaon for t he effect s of t he l at eral
membrane
The corrections
to
account for the effect
of the lateral membrane on the stress stram
behavlour should consider.
(1)
the axial load carried by the lateral mem-
brane;
(u) the lateral confinement caused by the
expansion of the lateral membrane durmg
compression.
A correction for the axial load carried by
the membrane has been discussed by Bishop
and Henkel [19]. There, the membrane was
assumed to have the form of a right cyhnder
durmg compression. This correction is negh-
gibly small. Moreover, it becomes meanmgless
as soon as the specimen bulges.
The second correction can be made usmg
the followmg equation:
(6)
In denvmg eqn. (6), the membrane 1s
assumed to have the form of a nght cylmder.
In the case of bulgmg, a mean value of the
lateral stram can be used.
In the present tests, the rubber membrane
placed around the sample has a Young modu-
lus of
E =
1400 kPa and a Poisson ratio of 0.5
[20]. In the unstretched state, the diameter
and thickness of the rubber membrane
amount 94.0 mm and 0.3 mm, respectively.
Accordmg to eqn. (6), at a lateral &am
e3 = 10% (which corresponds - roughly - to
the peak state for a sample of dense Karlsruhe
medium sand), the rubber membrane exerts a
lateral compression of
ca
1.26 kPa on the
sample. If we do not take thus effect mto
account, we overestimate cp by the amount
shown m Table 1.
MATERIALS TESTED
The materials tested are Karlsruhe sand,
sugar, wheat, rape and synthetic granulates.
TABLE 1
CorrectIons for the frlctlon angle due to lateral mem-
brane confmement
FiPa)
cp= 20”
cp= 40”
50
0 59”
0 48”
100
0 30”
0 24”
200
0 15”
0 12”
500
0 06”
0 05”
1000
0 03”
0 02”
The gram size distribution curves of the mate-
rials are given m Fig 6. In Table 2, the
extreme densities, the mean diameters of the
grams and the specific gravities are summarized.
(The maxmum and mmimum densities are
expenmentally determined by convention
according to the German Standard DIN
18126 )
SAMPLE PREPARATION AND TESTING
PROCEDURE
Sample preparation
The specimens are prepared by pluviation.
The setup for the preparation procedure con-
s&s of a silo with a central outlet setting
on a distnbutmg cylinder. Three sieves are
mounted m the cylinder. The particles flow-
mg through the opening are distributed by
the sieves and fall homogeneously into an
auxiliary mould. Durmg pluviation, the mould
IS moved downwards with a velocity of 12
mm/mm to keep the falhng height constant.
The auxiliary mould consists of the lower end
plate and a supportmg lateral wall composed
of three removable pieces.
TABLE 2
Extreme densltles, mean duuneters of the grams and
speclfx gravities of the mvestlgated matwals
Material
rm1n
7max
dso
Ys
(kN/m3) ( kN/m3) (mm)
Karlsruhe medmm
14 10 17 00
0.33 2 65
sand
sugar
8 46 9 49
0 43
Wheat
7 14 8
15 300 125
Rape 6 45 6 99 1 54 1 04
Luran
6 38 6 75
2 45 1 18
Lupolen
5 53 5 88
2 88 1.01
Polystyrol
5 96 7 03
2 52 088
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006 02
06 2
6 20
gram size [mm
1
Fig 6 Gram size dlstrlbutlon curves of the materials tested
symbol
A
.
Cl
v
.
0
materlal
k%Fuhe
Sugar
Rape
Polystyrol
Luran
Lupolen
Wheat
Accuracy of the measur ement of the m al
densz y
The accuracy for the mltlal density can be
estimated by conadermg the total differential
of the density
VAW+ WAV
AT<
V2
(7)
where V and W are mltlal volume and weight
of the granular mass, Ay, AV and AW are the
vanatlon of the mltlal density, of the mlt1a.l
volume and of the weight of the granular mass
respectively.
In the present tests, the mltlal diameter of
the sample 1smeasured at the upper, nuddle and
with an accuracy of 0 1 mm and the sample is
weighed with an accuracy of 0.5 g. The initial
volume
V =
785 cm3 and weight
W =
1354 g
have been obtamed for dense samples of
Karlsruhe medium sand. Substltutmg these
quantities m eqn. (4), we obtam AT < 0 05
kN/m3
Scatter of the m al denslty
The mltlal density depends on the fallmg
height and the pourmg mtenslty. For a gwen
pourmg mtenaty, the density 1sproportional
to the falling height, while for a @ven falling
height, the density decreases with the mcrease
of pourmg intensity, see also [21]. It was
found that a constant falhng height of
25 cm produces dense sand samples with a
speclflc gravity of y = 17 kN/m3 Vanatlon of
the fallmg height
h
results m different denw-
ties accordmg to the followmg emplrlcal
relation.
y = y. -a
exp(--bh) (8)
where y0 = 17.0 kN/m3,
a =
2 5 kN/m3 and
b = 15/m
In order to enunciate the vanatlon of the
initial density, 30 tests with the same falling
height were carried out. With the afore-
mentioned samphng set-up, a fanly good
reproduclblhty of the mltlal density was
achieved: The mean value of the mltlal den-
sity was 7 = 16.92 kN/m3 with a standard
deviation of 0 12 kN/m3.
Test pr ocedur e
After obtammg the final sample height, the
sample surface 1s equahzed by sucking off all
roughness aspenties with vacuum. The mould
is then gently placed on the pedestal m the
tnaxlal apparatus. The three collars are
mounted on the auxiliary mould. The piston
1s moved downwards until contact between
the upper end plate (which 1smounted on the
piston) and the side walls of the auxiliary
mould 1s estabhshed Subsequently, the
rubber membrane 1s ixed to the upper plate
and a vacuum of 15 kPa 1s apphed to the
sample mtenor. As soon as the vacuum 1s
apphed, the external atmosphenc pressure
acts upon the sample and makes it stiff (I e
,
self-sustammg) so that the auxiliary wall
becomes dispensable. After removmg the
auxlhary mould, the collars are mounted on
the sample m the upper (1 cm from the top
end plate), mtermedlate (m the middle of the
sample) and lower (1 cm from the bottom
end plate) height (see Fig. 4). The pressure
cell is closed and sealed by lowenng the
chamber, which 1s made from reinforced
perspex The cell pressure 1s then mcreased
step by step followed by regulation of the
axial force This computer-controlled process
1s performed m such a way that a nearly
hydrostatic stress path 1s apphed. The vacuum
1s released as soon as the value of the cell
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105
pressure reaches 15 kPa. Subsequently, the
sample is compressed m the axial direction by
movmg the piston downwards.
OBSERVATIONS DURING HYDROSTATIC
COMPRESSION
Although there is enormous experimental
research concemmg tnaxlal tests m the hter-
ature, most of the references are centered on
the material behaviour under devlatonc load-
mg. Only a few references describe the mate-
rial behaviour under hydrostatic loadmg
[22,23]. The reasons are as follows. firstly,
the deformation developed at this stage is
usually small compared with that durmg shear
and 1s considered to be neghgible, secondly,
exact measurement of the deformation durmg
the hydrostatic loadmg 1smore difficult than
durmg the subsequent compression
In the present tests, the axial and lateral
deformations dunng hydrostatic loadmg are
measured by a commercial displacement
transducer mounted between the two end
plates and the three collars described m the
section on
L. era1 strazn measur ement
The displacement transducer permits mea-
surement of axial stram with an accuracy of
+O 02% (by absence of the beddmg error)
Illustrated m Fig. 7 are the test results with
different materials evaluated with and with-
out correction for the beddmg error
It can be seen that the small magnitude of
deformation durmg hydrostatic loadmg can
only be expected for dense sand. For loose
sand, however, especially for granular mate-
rials consistmg of compressible particles, e.g.
rape and wheat, the deformations resulting
from hydrostatic loadmg can be as large as
those during the subsequent shear The values
of the maximum stram max ei , es)) at the
end of the isotropic loadmg are given m
Table 3
The beddmg error has a stnkmg mfluence
on the deformation behavlour durmg iso-
tropic compression This is especially the case
when the resultmg strams are small, see for
mstance Fig. 7(a) and (b).
An mterestmg observation is that the axial
stram is usually not equal to the lateral stram
although the loadmg path apphed is hydro-
static The mitral amsotropy 1s found to
depend on the mitral density of the sample.
TABLE 3
Maxlmum &rams under hydrostatic loadmg
Material
Dense Karlsruhe medium sand 0 267
Loose Karslruhe medium sand 0 496
sugar 1 241
Wheat 1840
Rape 5 467
Lupolen 4 143
Dense sand behaves nearly isotropically,
whereas loose sand seems to be stiffer m the
axial direction than m the circumferential
direction, see Fig. 7(a) and (b). This mitral
amsotropy of sand under hydrostatic loading
has also been reported by other mvestlgators
[22, 231. The tests m Fig. 7 with dlffer-
ent materials and mltial densities show a
great diversity of the deformation behaviour
under hydrostatic loadmg, both quantitatively
and qualitatively, dependmg on the materials
and densities concerned The mitral amso-
tropy has been found to persist dunng the
subsequent shear and has a remarkable mflu-
ence upon the strength and deformation
dunng shear [ 241
RESULTS OF TRIAXIAL COMPRESSION
Figure 8 shows some of the typical test
results on dense and loose Karlsruhe medium
sand Cauchy’s stress and logarithmic stram
are used for the evaluation. No corrections
are made m the evaluation either for the
bedding error or for the membrane effects
The symbols 0, C and A stand for the corre-
spondmg quantities denved with reference to
the upper, middle and lower part of the sam-
ple It can be seen from Fig. 8 that three
stress stram and volumetnc stram curves are
obtamed as a consequence of the mhomoge-
neous deformation
Quan
tztatzve descrzptzon of the tests and
determznatzon of the parameters
Quan z a zve descrzp tzon of the tests
The followmg parameters are used to
describe the stress stram and volumetnc strain
curves quantitatively.
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106
0 3
0 2
0 1
0
(4
E,C%I
/
Dr =
90
0 /
8
corrected /
0
uncorrected
J
?I
0
0
J
‘3
0
CI
7’ q
&,[ I
0
0 1 0 2
0 3
E,C%I
/
.
D
/
. q
.
/
.
. D
o”
/’
.
q
/’
D, =
18 2
. D
/
’
e corrected
0
uncorrected
E,[%l
0
0 5
1 0 1 5
(cl
(d)
6
E,[ l
t
Dr = 62 4
,’
8
0
F”
E,C l
0
0 2
4
6
(e)
0 4
E,C%l
.
o/
.
.
0 7’
.
,J
. /
02
.
/”
D, = 12 2
’ ” corrected
./.
0
uncorrected
4
E,t%l
0
’ 02 04
@I
2 0
1 5
1 0
0 5
-
E,C%l
/
. /,O
:/ CI
.‘A m
./‘,
D. = 78 6
zy’ ’ corrected
;/D • I uncorrected
0
0
5
1 0
1 5
E,C%l
D, ~00
/
corrected
0
uncorrected
/
/.‘,
q
/‘. o
/
. m
. 0
/-
E,C%l
II
2
4
(f)
Fig 7 Deformations under hydrostatic loading for (a) dense Karlsruhe medium sand, (b) loose Karlsruhe medmm
sand, (c) sugar, (d) wheat, (e) rape and (f) lupolen
- the mtral slope of the stress stram curve,
E, -
the ml&l cldatancy angle, Go
.
cQ-c
E,= v
(9)
Jlo= iA
(10)
61
El= 0 El e,=o
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107
(a)
Fig
5
10
AXIAL STRAIN [Xl
-21
--l
-4
t
15
0
D, = I L 2
r
U3 -2OOkPa
5
10
15
20
AXIAL TRAIN
[Xl
(b)
8 Typical trlaxlal tests on (a) dense Karlsruhe medium sand and (b) loose Karlsruhe medmm sand
-the fn&on angle at the limit state, cp
9=
u1-u3
alTSlil
i 1
l+ (73 max
(11)
-the axial stram at the hmlt state, elf
- the dllatancy angle at the hmlt state, 9
i
= arctan; (12)
El E,=Cf
In the above equations, ilf 1s the axial stram
rate at fdure.
De ermma l on of the parameters
It can be seen that the parameters
E,, J/0
and
are defined by stress and stram rates at a gwen
stress or stram state The rate quantltles are
difficult to evaluate exactly from the test
data. In the present paper, these parameters
are obtamed by a numerical denvatlon pro-
cedure, m which the denvatlve at the stress
state elk (the stress state of the kth reading)
is obtamed by calculatmg the slope of the
straght hne passmg through four nelgh-
bounng pomts usmg the least mean square
method
Effect of the bedding error on the test resul t
As has been shown m the section on
Observations durmg hydrostatic compression,
the beddmg error has a stnkmg influence on
the results durmg lsotroplc compression In
order to demonstrate the effect of the
bedding error on the subsequent tnaxlal
0.4
0 5 10 15
AXIAL STRAIN IX1
Fig 9 affect of the beddmg error on the test result
compression, a typical test with Karlsruhe
medium sand evaluated with and wlthout
correction for the beddmg error 1sshown m
Fig. 9. It can be seen that the mfluence of the
beddmg error on the result 1svery small.
The parameters gwen m Table 4 serve to
appreciate the beddmg error quantltatmely.
It can be seen that the beddmg error has a
remarkable mfluence on the mltlal slope of
the stress stram curve
E,
and the mltlal
d&& ncy
angle O. This fact makes the evalu-
ation of these parameters even more difficult
Whereas the beddmg error has still quite a
small mfluence on the dllatancy angle and the
axial stram at the lmut state, and elf, It has
no influence on the fnctlon angle cp.
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108
TABLE 4
TABLE 5
Parameter of the trlaxlal test m Fig 9 evaluated wlth-
out and with correction for the bedding error
Obtamed scatter of the parameters
Parameter Uncorrected Corrected
cp
43 53”
43 53”
Eolu3
560
720
26 57”
26 57”
-35 10”
-31 63”
6 27%
6 13%
ReproducMl t y of t he t ests
The test results are subjected to systematic
and stochastic errors. The stochastic error can
only be appreciated when a number of tests
are performed. This demands that repeated
tests under the same condltlons should be
conducted to confirm the vahdlty of the tests.
Despite the unportance of reproduclblllty
of the tests, the theme 1s seldom addressed. In
the present tests, reproduclblllty 1sstudied by
performing tests under the same mltlal den-
sity and the same confmmg pressure. The
word Same should be understood m the sense
of the section on Sample preparation and
testing procedure For each test, a repeated
test 1scmed out m the present study. If a
large deviation 1s observed, a further test 1s
conducted. As an example, Figure 10 shows
five repeated tests on dense Karlsruhe
medium sand It can be seen that apart from
test No 5, the reproduclblhty 1s quite satlsfy-
mg. Upon readmg the test record, we noticed
that the supportmg vacuum was extracted too
Parameter
Scatter
AP
0 37”
AEolo3
110
Z.
2 93”
2 07”
Ae,,
0 33%
early m test No. 5. Therefore, test No. 5 1s
excluded from the evaluation. Table 5 shows
the obtamed scatter of several parameters
Lat eral expansion
Accurate measurement of the lateral stram
showed that, contrary to a widespread opm-
ion, bulging (z e , unequal expansions along
the sample height) occurs not only m the
neighborhood of the peak stram but
from the
very beganrung of t he t ruaxl al ompr esst on, see
for instance Fig. 11. It can be seen from Fig
11 that dense samples develop a stronger
nonuniform deformation than loose samples
If bulgmg occurs as a spontaneous blfurca-
tlon (cf. [25]) it should be avoidable by
proper lubncatlon - at least m the mltlal
stage of the compression However, our tests
show that although the lubrlcatlon suppresses
considerably the amount of bulgmg (to a
degree which cannot be perceived by the
naked eye), slight bulgmg 1sstfl present from
the begmnmg of ax& compression. This fact
has also been reported m [8] If bifurcation
(1 e
,
non-uruqueness of the sample deforma-
tion path and onset of mhomogeneous
deformation) has to be excluded, the reason
for bulgmg has to be sought m some mlt1a.l
mhomogenelty of the sample and (assummg
that, owing to our precautions, the mlt1a.l
density 1s constant throughout the sample)
this can only be the mhomogenelty of the
mltlal stress field due to gramty.
In the meanwhile, it has also been theoret-
ically and numencally corroborated (as will
be shown m a forthcommg pubhcatlon) that
this mltlal stress mhomogenelty, however
small, 1sresponsible for bulgmg which grows
with mcreasmg deformation. It IS only at the
final stage of the tests, when bulgmg 1svisible
to the naked eye, see Fig. 12. The falure
mode m Fig. 12 has been observed m more
than thtiy tests on dense sand samples and m
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D, = 96 5 O/o U3 =300 kPa
I I I
11
I
I I
I
L
I
I I I
5
10
radial dIsplaceme& [ mm
1
lot
Dr = 11 8
a3 = 300 kPa
0
2
4 6
radial displacement I mm
1
b)
Fig 11 Evolution of the lateral deformation durmg trlaxlal compression for (a) dense Karlsruhe medium sand
and (b) loose Karlsruhe medium sand
(4
(b)
Fig 12 A sample of dense Karlsruhe medium sand (a) before and (b) after the test The test was termmated
at e1 = 12% A vacuum of 100 kPa was applied to support the sample
most of the tests on other mater& Note
I e
, the
expansion m the middle 1s larger than
that m most of the previously used experr-
that m the lower part of the sample. Thus
mental techniques no means were provided to however, does not contradict the above
follow separately the lateral deformations of reasoning about the mfluence of gravrty. As
the upper, middle, and lower parts of the discussed m the section on Fnctlon between
sample the end plates and the sample, the boundary
In
several tests with loose sand samples, condltlons are not ideal. Fnctlon exists at
a shght
barrelling has also been observed,
the end plates, which hmders the lateral
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110
expansion of the sample. As loose samples are
much weaker than dense samples, the mflu-
ence of the friction might overwhelm the
gravity and become dominant In addition,
the mitral density mhomogeneities are more
pronounced m loose samples
This fmdmg imposes the necessity for some
deconvolution technique (z e
,
back calcula-
tion towards the results of a fictitious homo-
geneously deformed sample) of the data
obtamed Of course, this cannot be under-
taken without some assumptions concernmg
the real (but unknown) deformation field.
Takmg mto account that at the lower sample
end the mitral axial stress is, due to gravity,
somewhat higher than at the upper end and
that bulgmg is always manifested as a greater
lateral expansion at the
lower
part of the
sample, it is assumed that both the axial and
the lateral deformations proceed faster at the
lower than at the upper sample end. This also
means that the axial stram e1 is not homo-
geneously distributed over the sample and
that the quantity log,,[ (h, - u r/h,-,] is merely
a mean value 5i taken over the sample height.
This means futhermore that, whereas the
lower part of the sample has reached, say, the
peak deformation and the hnut state, the
upper part is still m an earher stage of the
deformation
The deconvolution can be undertaken
under the assumption that the genume upper,
middle, and lower axial deformations fulfil
the conditions
El,U/EZ,U = Cl/T,
El.JEZ,I = Zllf2
el.dE2.1 = Cl/52
with P2= (e2,u + Q + e2J/3.
The subscripts u, 1, 1 denote the upper,
mtermediate and lower collars, respectively
This procedure leads to three stress-stram
curves, one for each part of the sample, which
comcide more or less, see for example Fig 13.
Limit state
The stress-stram curves of tnaxial compres-
sion are expected to obtam a maximum value
which is called peak. The correspondmg stress
state 1scalled a
Zzmzt tate.
Often, the peak is
followed by a decrease of the stress deviator
lul - us1 upon continued deformation This
stress decay is termed
softenmg.
It should be
0
0 5 10 15
AXIAL STRAIN WI
Fig 13 Deconvoluted stress strain and volumetric
stram curves for dense Karlsruhe medwm sand
noted that a too drastic softenmg should be
attributed to pronounced mhomogeneities of
the deformation rather than to the material
behavlour. Actually, a test should be termi-
nated as soon as the mhomogeneities become
pronounced, smce any contmuation of this
test is meanmgless (the measurements
obtamed cannot be evaluated m the sense of a
unique stress stram curve)
It is commonly expected that a contmued
deformation will lead eventually to the so-
called critical state, where no further volume
changes (dilatancy) occur. However, m the
course of tnaxial compression this critical
state is usually not obtamed withm the range
of feasible homogeneous deformations. As
mentioned above, the deformation of the
sample cannot be increased arbitrarily with-
out the onset of mevitable mhomogeneities.
It must be added that for loose sand sam-
ples and for samples tested at high confining
pressure as well as for other granular materials
consistmg of soft grams, e.g wheat and
rape, a limit state m the above sense is
not
obtamed and the stress-stram curves mcrease
contmuously as shown m Fig. 15(d) and (e)
Agam it could be argued that after a sufficient
stram the peak would, probably, be reached.
However, this cannot be achieved due to the
limited range of feasible deformation A
senous difficulty arises from this fact m the
determmation of the friction angle.
Collapse
A curious effect was observed dunng tests
with the synthetic granulate
polystyrol,
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111
AXIAL STRAIN [XI
Fig 14 Trlaxlal test on polystyrol
whose grams are angular and hard. This effect
mmics the collapse of loess soil upon munda-
tion. A sudden collapse (also called “stick-
slip”) takes place as the deviatonc stress I u1 -
usI attams a certam value as shown m Fig. 14.
The collapse is accompanied by an abrupt
reduction m axial stress and a hght sound
emission
Whether collapse occurs seems to depend
upon the shape and hardness of the grams. In
addition, the gram size distribution might be
also a controllmg factor. Indeed, the gram
size distribution of polystyrol has been found
to be extremly uniform, as shown m Fig. 6.
Besides, collapse has also been found to occur
m potato powder [ 261.
BAROTROPY AND PYKNOTROPY
Baro tr opy
The term barotropy 1sused to signify the
dependence of the mechamcal behaviour of
the materials on the stress level [ 271 If the
relations descnbmg barotropy are known, the
results obtamed can be extrapolated towards
low pressure levels, which are of mterest for
silo design but also extremely difficult m
experimentation.
In the present tests, barotropy is mvesti-
gated by conductmg tests with samples of the
same m1tia.ldensity under varymg confmmg
pressures. The test results with Karlsruhe
medium sand, sugar, wheat, rape and luran
are shown in Fig 15. For clarity, only the
stress stram and volumetnc strain curves
plotted usmg the mean value of the stress and
stram over the sample height are shown.
Given m Fig. 16 is the dependence of the
fnction angle cp,derived from Fig. 15, on u3
for Karlsruhe medium sand, sugar, wheat,
rape and luran. As no hmit state can be
reached except for dense Karlsruhe sand, the
friction angle 1sevaluated at the axial stram
of 10%. It can be seen from Fig 16 that the
friction angle decreases with mcreasmg con-
fmmg pressure. The fact that the fnction
angle depends on the confmmg pressure is a
common feature at least for the granular
materials covered by the present tests. For
rape and luran, we have almost a hnear depen-
dence of cpon u3.
The dependence of the dilatancy angle
on the confmmg pressure is shown m Fig. 17.
Agam, the dllatancy angle 1~ calculated with
respect to the axial stram of e1 = 10% for
materials for which no limit state was
obtamed. G 1s ound to decrease with mcreas-
mg confining pressure In other words,
dilatancy 1s suppressed by mcreasmg con-
fmmg pressure The fact that both cp and
decrease with elevatmg confmmg pressure can
be explamed by the stress dllatancy theory
developed by Rowe [2&S].
A statement pertinent to the above discus-
sions should be made at this stage As shown
m the section on Quantitative description of
the tests, it is a difficult task to evaluate rate
quantities from expenmental data. Fre-
quently, the test results are fitted into a
theory, e.g the stress dllatancy theory. The
fnction angle can be evaluated with great
confidence. The dllatancy angle, however, can
only be evaluated with a poor confidence
The results depend largely on the evaluation
method, which has been rarely mentioned m
the literature
The dependence of the mitral slope of the
stress stram curve on the confmmg pressure
(see Fig 18) can be described by the empm-
cal relation proposed by Janbu [29]
n
(13)
where
K
and n are material constants,
pa 1s
the atmospheric pressure.
The dependence of the mitral dllatancy
angle on the confmmg pressure is grven m
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2
(4
3
AXIAL STRAIN tX1
4
1'0 115
0
AXIAL STRAIN WI
&=I 6 2
AXIAL STRAIN [Xl
d:l:t:llt:i~l:l:l:l:l~l:‘~‘l
'0
(e)
AXIAL STFAIN [Xl
Fig 5 Trlaxlal tests on (a) dense Karlsruhe medium sand, (b) loose Karlsruhe medmm sand, (c) sugar, (d) wheat,
(e) rape and (f) luran
Fig. 19. It can be seen from Fig. 19 that for stress level and the mltlal den&y. In fact,
Karlsruhe sand the mltml dllatancy angle
apart from matenals compnsmg compressible
remams nearly constant vrespectlve of the
or crushable particles, e.g. wheat, rape and
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3
ENSE KARL SRUHE SAND
2 4’6
a 1’0
d3
1100kPa
1
Fig 16 Dependence of ‘p on u3
2 4’6
8 lb
a3 [ lOOkhI
Fig 17 Dependence of on u3
sugar, the mitral dllatancy angle has roughly
the same value for a given material. Therefore,
we can conclude that the mitial dilatancy
EcJa3
800
700
600
R
DENSE KARLSRUHE SAND
ARLSRUHE SAND
2 4
’ 6 8
1’0
a3
[ l OOkPol
Fig 18 Dependence of E0 u3 on a3
angle 1s constant irrespective of the stress level
and mitral density.
The dependence of the axial stram eu at
the hmit state on the confmmg pressure is
given m Fig. 20 for dense Karlsruhe medium
sand. elf is proportional to the confmmg
pressure Tlus fact has also been observed by
Colhat-Dangus et al [ 301. In other words, the
material becomes more ductile with mcreasmg
confmmg pressure.
Pyknotropy
The dependence of the mechamcal behav-
iour on the mitral density is called pykno-
tropy. In the present study, pyknotropy is
investigated by conducting tests with the
same confmmg pressure while varying the
m&al density from test to test. The test
results for Karlsruhe medium sand are shown
m Fig. 21.
The dependence of the friction angle cp,
dilatancy angle ,
E,/a3,
tie and E f on the
relative density 0, defined by
D, =
mx(r - ‘YInin
~(YlllOX %li*) (14)
can be derived from Fig. 21 and IS given m
Figs. 22 to 26, respectively.
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0 LOOSE KARLSRUHER SAND
0 SUGAR
WHEAT
a RAPE
d 2 4' 6
8 1' 0
u3 I l OOkPa
1
Fig 19 Dependence of tie on u3
DENSE KARLSRUHE SAND
d 2 4 ’ 6 B 1’0
a,
I 100 kPa 1
Fig
20.
Dependence of Elf on u3 for dense Karlsruhe
medmm sand
It can be seen from Figs. 22 and 23 that
both cpand mcrease with mcreasmg relative
density 0,. This can be also explamed by the
stress dllatancy theory.
An almost hnear relation between
E J,
and D, can be seen from Fig. 24. A simple
explanation I that dense sand is stiffer
than loose sand.
The relation between tiO and D,, see
Wg. 24, conforms agam the observation that
C3 = 100 kPa
AXIAL STRAIN [Xl
Fig 21 Tests on Karlsruhe medmm sand with (13 =
100 kPa and varymg mltlal densltles
20
40 ’ 60
80
I
D, I 1
Fig
22
Dependence of cpon Q for Karlsruhe sand
20 40
’ 60 80
l b0
D, I 1
Fig 23 Dependence of 9 on
D,
for Karlsruhe sand
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115
I II I I
' 60
I I
20
40 60 160
D, LohI
Fig 24 Dependence of Eo las on D r for Karlsruhe
sand
Q. I”1
__
40 -
_.
’ .
.
.
.
30 -
.
20 -
10 -
0 :‘;‘;1/11’/‘1’/
20 40 ' 60
80 lb0
Dr %I
Fig 25 Dependence of 0 on D for Karlsruhe sand
20 40 ' 60 60 lb0
Dr [ 1
Fig 26 Dependence of Elf on D r for Karlsruhe sand
the mltlal dllatancy angle 1s approxnnately
independent of the n&al density
The relation between elf and D, given m
Figure 26 shows that with mcreasmg mltlal
density the sand becomes more bnttle.
Taking barotropy mto account, the func-
tional dependence of the fnctlon angle cpon
the confmmg pressure u3 and the relative
density D, 1s shown m a three-dnnenslonal
space of 9, o3 and D, m Fig. 27, which pro-
vides an overall picture of barotropy and
pyknotropy.
As to the unportance of barotropy and
pyknotropy m silo problems, we refer to a
recent paper by Ravenet [31], where the
slgnlficance of the vanatlon of the stress level
and of the density along the silo height 1s
appreciated
L2 -
LO -
36 _
36 _
3L -
32
t0 ,
6
/
a,[ lOOkPa
Fig 27 Dependence of q on ~3 and D for Karlsruhe
sand
COMPARISON WITH OTHER STUDIES
Systematic mvestlgatlons of barotropy and
pyknotropy are rather rare Only recently
have some types of soils been mvestlgated m
this sense. The results (see also Tables 6 and
7) may be summarized as follows-
Tests by Fukushlma and Tatsuoka
Fukushnna and Tatsuoka [18] have
focused then attention on very low lateral
stresses in the range from 0.02 to 4 bar. They
mvestigated Toyoura sand with void ratios
e, = 0.85 and e, = 0.70 (in order to mvestl-
gate the effect of lateral stress, samples with
identical mltlal void ratio e, should be
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6
TABLE 6
Comprehensive representation of test series and results of other authors Frlctlon angles m parentheses mdxate
that no peak was obtamed
Authors Material Number do
ho
of tests
(cm) (cm)
e0
Fukushlma and
Tatsuoka [ 181
Hettler and
Vardoulakls [ 81
Hettler and
Gudehus [ 321
Goto and
Tatsuoka [ 91
Kltamura and
Haruyama [ 161
Colhat-Dangus
eta1 [30]
Toyoura sand
78
7 15 ca 085 05 35 5
Karlsruhe sand
4
78 28 0 565
Oostershelde sand 3 Medium
Darmstadt sand
4 Dense
Toyoura sand 38
Toyoura sand
9
15 20 7 75 07
09
0 68 0 80
Shmasu tuff
6 134 164
Hostun sand
24
20 20140 Dense
3
3
26
ca 070
0 582
0 546
Loose
40
34
05 41 6
40 38 6
05 43
30
05
40
40
05
60
10 0
05
20
40
05
43
41
41
41
44 2
39 1
37 4
(38 7)
36 6
(34 4)
43 9
50 39 2
1 42
1 34
2 38
100
2
(24)
38
100
12
(24)
48 1
20
37 2
1 36 8
25
314
compared. However, e, can only be obtamed
(z e , i311//i303) decreases with decreasing u3.
with a scatter and, therefore, it varied wlthm They attributed this “apparently contradic-
the ranges 0.660.. .0.687 and 0.824.. .0.898). tory phenomenon” (we cannot detect any
It was found that the barotropy of cp (I.e., contradiction herem) to the lack of any
a9/ao3)
(compressive stress is taken positive)
membrane correction, which they consider
mcreases with decreasmg u3 and that the
necessary for lateral stresses below 0 1 bar
barotropy of the deformation characteristics After membrane correction, they detected
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117
TABLE 7
Comprehensive representation of test series and results of the present mvestlgatlon Frlctlon angles m parentheses
mdlcate that no peak was obtamed In this case, the frlctlon angle IS calculated with reference to the axial stram
of El = 10%
Authors
Material Number
of tests
do
(cm)
ho
(cm)
D, cp
TlZar) (“)
Kolymbas and Wu
Karlsruhe sand 51 10
10 co 980 05
Sugar 10 10
10
10 0
38 8
ca 162 05
(33 3)
10 0
ca 254 05
80
Wheat 8 10 10 ca 683 05
40
Rape 8 10 10 co 12.0 10
40
Luran 6 10 10 ca 741 05
20
(15 8)
45 1
(29 0)
(36 0)
(28 4)
(310)
(25 4)
(28 0)
(215)
(21 3)
that barotropy becomes considerably smaller
for lateral stresses below 0.5 bar. It seems that
the experiments were carried out with the
utmost precision and accuracy. Nevertheless
Fukushlma and Tatsuoka remark the follow-
mg pomts:
-At extremely low pressures, the stress
becomes very non-umform, smce the self-
weight of the sample becomes mcreasmgly
important (the mevitable mhomogeneous
deformation of the sample has not been
mentioned).
Tests
by
Hettler et
al
- Bulgmg occurs as is clearly visible m their
Photo 1. This phenomenon has not been
taken mto account m evaluating the test
results.
- The lateral membrane buckles at large
stram and low pressure.
- No correction for beddmg error was pro-
vided for.
Hettler
et al [8
321 investigated very large
and extremely squat samples (mitial diameter
d, = 78 cm, mitml height h, = 28 cm) of vari-
ous types of sand. Owmg to the large dlmen-
sions of the samples, the number of tests is
hnuted. In some of their tests, a correctron of
the beddmg error has been undertaken by the
use of a bouton mounted at the lateral mem-
brane of the sample. However, it cannot be
assured that the motion of this bouton is
identical with the one of the adjacent sand
particle. It appears strange that with Karlsruhe
sand no barotropy was detected m the u3-
ranges 0.5.. .3 bar and 0.5.. .4 bar, whereas a
pronounced barotropy was detected m the
range 0.5.. .lO bar. Barotropy was clearly
observed with sands from Oostershelde and
Darmstadt. With loose samples from Degebo-
sand, a peak was not obtained.
- In many loose samples, a peak of the stress
Another important and controversial
stram curve was not obtamed.
fmdmg of Hettler
et al is
that the mcipient
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118
ralal strams (z e
,
the radial strams occunng
at the begmnmg of the tnaxial compression)
are null We could not confirm this statement
As shown m Fig. 28, the radial expansion
sets on as soon as the devlatonc loadmg
is applied This observation is not mfluenced
by bedding error.
- 0
s
2
0
-
Dr : 96
5
0 01
02 03 OL 05
E,
I 1
Fig 28 Imtlal radial stram us axlal stram
New (1988) ASTM state of the art
In a senes of papers presented m 1986 m
[ 331, barotropy and pyknotropy of soils were
systematically mvestigated [9,16, 301. The
fmdmgs are m close agreement with those
presented here (see also Tables 6 and 7). In
particular, the lack of peak of the stress-
stram curve at high stress levels is stated m
[301 to be the true elementary response of
the material
ACKNOWLEDGEMENTS (added m proof )
The authors are mdebted to Prof. F
Tatsuoka, Umversity of Tokyo, who read the
manuscript and pointed to discrepancies be-
tween the friction angles cpof dense Karlsruhe
sand at u3
= 100 kPa as they have been stated
(1) m our Figs. 9,15a, 16 and m Table 4, (u)
m Fig 22. The remark of Prof. Tatsuoka gave
nse to a retrospective mvestigation m the
course of which we found that the several
charges of our Karlsruhe sand are SUbJeCto
a considerable scatter. Of course, this finding
refers also to previous pubhcations on Karls-
ruhe sand. However, we maintam that withm
each test series reported m this paper (see
Figs. 15(a) and 21) the same sand type has
been used. Thus, our partial results referrmg
to barotropy and pyknotropy retam their
vahdity .
LIST OF SYMBOLS
A
4
d
d,5’
D,
EO
-%I
J-1
h0
PP
r
t0
At
Ul
u3
V
W
Y
El
Elf
E3
(T
0
cp
mstantaneous area of sample
mitral diameter of sample
mean gram diameter
thickness of collar
relative density
mitial slope of stress-&ram curve
elastic modulus of rubber membrane
axial force
mitral height of sample
atmospheric pressure
curvature radius
mitral thickness of rubber membrane
compression of rubber membrane
axial displacement
radial displacement
volume of sample
weight of sample
specific weight
axial stram
axial stram at peak (failure)
radial stram
normal stress
dilatancy angle
u&al dilatancy angle
friction angle
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