Triaxial Tests on Granular Materials

8/9/2019 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


2/21

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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101

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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102

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


9/21

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.


10/21

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


11/21

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


12/21

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,


13/21

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


14/21

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


15/21

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.


16/21

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


17/21

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


18/21

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


19/21

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


20/21

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

REFERENCES

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