Powder Technology, 60 (1990) 99 - 119

Recent Results of Triaxial Tests with Granular Materials

D KOLYMBAS and W WU

Znstztute for Sozl Mechanzcs and Rock Mechanzcs, Unzverszty of Karlsruhe, Kazserstr 12, D-7500 Karlsruhe 1

VRG)

(Recewed December 13,1988, m rewed form June 22,1989)

99

SUMMARY

In this paper are presented some recent traaxurl test results obtamed w&h dry sand, sugar, rape, wheat and synthetic granulates The device used was a trlaxwl apparatus specaally desrgned to test dry sdo mater&s The results are reported wrth a mew to faclll- ta tmg development and checkmg of appropn- ate constltu twe equations. This IS only pos- sible if special precautions have been taken to suppress error sources and guarantee a homogeneous deformation The results pre- sented here reveal some characterlstlcs of the sample behavlour, namely (I) even durang the ml teal lsotroplc consohda tlon the samples behave anasotropwally, (11) the mhomogene- ous sample deformation sets m from the begmnmg of the traaxlal compression and, therefore, the test results cannot be evaluated without a deconvolutlon technique, and (ur) with loose sands and granulates constltu ted from soft grams, as well as at high stress levels, a peak state 1s not obtained and, there- fore, any reference to a friction angle 1s questionable A simple deconvolutlon tech- nique 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 1s the so-called trlaxzul 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

1 Loadnlg frame

2 Lorbdmg piston

3 Pressure cell

1 Top cap

5 Bottom cap I 6 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 ~110s has 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 pressure Air 1s used 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,

101

Fig 4 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 1s referred 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

Frlctlon 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 1s m accor- dance with the observations of Lmton et al [lo] and Ueng et al [4]

Correctzon for the beddmg error 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; (T IS 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

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 the effects of the lateral 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

006 02 06 2 6 20 gram size [mm 1

Fig 6 Gram size dlstrlbutlon curves of the materials tested

symbol

A

.

Cl

n

v

.

0

materlal

k%Fuhe Sugar

Rape

Polystyrol

Luran

Lupolen

Wheat

Accuracy of the measurement of the m&al densz ty 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 A W 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 1s measured 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 1s proportional 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 procedure 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 1s mounted on the piston) and the side walls of the auxiliary mould 1s estabhshed Subsequently, the rubber membrane 1s fixed 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

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 1s more 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 measurement 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 Cauchys 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 tz ta tzve descrzp tzon of the tests The followmg parameters are used to

describe the stress stram and volumetnc strain curves quantitatively.

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%

n 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 n corrected ;/D I uncorrected

0 0 5 1 0 1 5

E,C%l

D, ~00% /

n corrected

0 uncorrected /

/., q

/. n , 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

107

(a) Fig

5 10

AXIAL STRAIN [Xl

-21

--l!

-4

t 15

0

D, = IL 2 % r

U3 -2OOkPa

5 10 15 20

AXIAL 5TRAIN [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 termma tlon 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 result 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 1s shown m Fig. 9. It can be seen that the mfluence of the beddmg error on the result 1s very 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.

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%

ReproducMlty of the tests 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 1s studied 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 1s cmed 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

Lateral 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 the truaxlal compresston, 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 1s stfl 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, 1s responsible for bulgmg which grows with mcreasmg deformation. It IS only at the final stage of the tests, when bulgmg 1s visible 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

D, = 96 5 O/o U3 =300 kPa

I I I 11 I I I I L I I I I

5 10 1 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

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 1s called a Zzmzt state. 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,

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 tropy The term barotropy 1s used 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.l density 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 1s evaluated 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 cp on 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 found 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

112

(4

3

AXIAL STRAIN tX1

4 1'0 115 $0 AXIAL STRAIN WI

&=I6 2 %

AXIAL STRAIN [Xl

d:l:t:llt:i~l:l:l:l:l~l:~l 1'0 (e) AXIAL STFAIN [Xl

Fig 15 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

113

ENSE KARLSRUHE SAND

2 46 a 10

d3 1100kPa 1

Fig 16 Dependence of p on u3

2 46 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 10

a3 [lOOkPol

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 if on the relative density 0, defined by

D, = %mx(r - YInin 1

~(YlllOX - %li*) (14)

can be derived from Fig. 21 and IS given m Figs. 22 to 26, respectively.

0 LOOSE KARLSRUHER SAND

0 SUGAR

n WHEAT a RAPE

d 2 4'6 8 1'0

u3 I lOOkPa 1

Fig 19 Dependence of tie on u3

DENSE KARLSRUHE SAND

d 2 4 6 B 10 a, I100 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 cp and $ 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 0 D, I%1

Fig 22 Dependence of cp on Q for Karlsruhe sand

20 40 60 80 lb0

D, I % 1

Fig 23 Dependence of 9 on D, for Karlsruhe sand

115

I II I I ' 60

I I

20 40 60 160 D, LohI

Fig 24 Dependence of Eolas on Dr for Karlsruhe sand

Q. I1

__

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 Dr 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 cp on 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

t 30 ,

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

116

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

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

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 -

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 cp of 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 SUbJeCt to 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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119

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