LII IRAIZDIPAI *“CT”nnC cr\a I r\fiAl IVATlr\.I ALI_ Fbln-ll~-.-rl~.l I.“I.IL~\I\~~~L I”‘IL I “““3 run LUCIMLILH I IUN /+NlJ e)lrlJKLH I IUN

OF GRANULAR BODIES International Workshop

Gdarisk-Sobieszewo, September 25-30, 1989

SHEAR BANDING DEPENDENCY ON MEAN STRESS LEVEL IN SAND

J. DESRUES’, W. HAMMAD’

INTRODUCTION

In order to study the influence of mean stress level on strain localization in soil me- chanics, an experimental study was performed on saturated samples of Hostun RF sand in a plane strain biaxial apparatus in the Laboratoire 3S/IMG, Grenoble, France.

It is well known that, in triaxial test, the confining pressure has an important effect on the global volumetric strain response of the specimen. An increase of the confining pressure induces a transition of the volumetric strain of a dense sample from a dilating regime to a less dilating one and eventually to contraction. It is also clear that the transi- tion from dilatant to contractant overall volumetric response would be obtained by de- creasing the initial sample density from a test to another. We present in this paper the re- sults of biaxial tests realised on both dense and loose samples at different cell pressures.

Experimentally, the plane strain biaxial test has certain advantages in the study of shear bands: this phenomenon is generally present in a biaxial test which is not quite the case in the axisymmetrical triaxial test ([2,8]; in addition, the specimen’s deformation being restricted in one direction in the biaxial test and the specimen having a prismatic rectangular shape, a thorough visualization of the displacement field of the saqple is easily realised in the plane perpendicular to the direction of zero deformation.

This study was undertaken as a continuation of previous work realised at the Institut de MCcanique de Grenoble [2,4,6] in experimental investigation of the main features of strain localization in soils.

The results reported in this paper were presented in the 2nd International Workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies in Sobi- eszewo/Gdalisk, Poland in August 1989. We do not mention here of the results obtained after that date, although considerable development of the experimental data on strain lo- calization has been performed since 1989 in our laboratory. Among other complementary data, the results of the stereophotogrammetric method used on the tests discussed here can be found in reference [3] and in papers published subsequently in the literature.

In this paper, we make a brief presentation of the biaxial set up and sample prepa- ration before presenting the obtained results. Then, the analysis of stress and strain rela- tions, deformation characteristics, onset of the localization, shear band orientations and test repeatability follows.

’ Laboratoire 3S/lMG, GRECO Gt?omattriaux, B.P. 53 - 3804 I Grenoble cedex 9 - FRANCE

58 NUMERICAL METHODS FOR L~CYALIZA~ON AND BIFURCATWN OF GRANULAR BODIES

MATERIALS AND METHODS

Materials and specimen preparation

Two types of materials were tested, the Hostun RF sand and the Manche sand. The Hostun RF sand is a fine granular siliceous sand, uniformly graded, with

DYJ = 0.32 mm, uniformity coefftcient = 1.7, minimum and maximum volumetric weight of 13.24 and 15.99 kN/m3, respectively, and grain specific gravity of 2.65.

The Manche sand is a calcareous sand, formed mainly of pieces of sea shells, with a grain size ranging 0.1 to 10 mm, minimum and maximum volumetric weight of 10.8 and 13,74 kN/m3 respectively.

Table 1. Dense Hostun RF sand (relative density 0.95)

test init. & 0,’ El Q,IG, 4 ” 0 d4-c

dens. peak peak peak peak orient. th.orient.

WI&l kPa % deg. deg.

shf40 15.83 0.673 100 3.0 6.35 46.7 14.2 20.0 21.7

shf44 15.88 0.668 100 3.1 6.70 47.8 13.9 21.5 21.1

shf41 15.86 0.670 200 4.5 6.26 46.4 13.-S 24.5 21.8

shf45 15.97 0.659 200 4.6 6.44 47.0 12.6 24.5 21.5

shf42 1 15.81 1 0.675 1 400 1 6.5 1 5.87 1 45.1 1 11.5 1 26.0 1 22.5

shf46 15.87 0.669 400 5.8 5.90 45.3 10.5 25.5 22.4

shf43 15.95 0.661 800 7.5 5.20 42.6 8.4 28.5 23.7

shf47 15.90 0.665 800 7.5 5.17 42.5 8.0 31.0 23.8

Table 2. Loose and very loose Hostun RF sand (relative density 0.25 and 0.0)

shf55 14.06 0.884 200 6.1 3.39 33.0 0.0 26.5 28.5

shf49 13.97 0.896 400 8.0 3.39 33.0 0.0 28.0 28.5

shf54 14.04 0.887 400 9.0 3.43 33.3 0.0 29.0 28.4

shf53 14.06 0.884 800 11.5 3.03 30.2 -3.0 33.0 29.9

shf52 13.95 0.899 800 12.0 3.26 32.0 -2.7 32.0 29.0

shR8 13.08 1.026 400 10.3 2.88 29.0 -3.2 29.5 30.5

SHEAR BANDING DEPENDENCY ON M..ANSXESS LEVEL INSAND 59

Table 3. Loose calcareous Manche sand (relative density 0.35)

I I . I I I I I I I 1

Two series of tests were performed on Hostun RF sand, one on dense specimens and the other on loose ones with relative densities of about 0.95 and 0.25 respectively. In each series the confining pressure was varied: 100, 200, 400, 800 kPa. Each of the two ,,-:,.. . ..^.. ..-,,-&-.I :- --A-.. .^ ---C- &LA _I.&-:-- ..I ̂ . . _. . ̂ _ 501 KS w&m IqJGcwal LII “lwa L” sxJ11111111 IlIt: uu1a111t;u c;UI vci.

The Manche sand was tested at a relative density of about 0,35 with 03 =lOO, 200 and 400 kPa. Tables 1, 2 and 3 show the initial densities and voids ratios of the speci- mens as well as the applied confining pressure.

Fig. 1. Geometry of the biaxial specimen and loading conditions

The specimens were shaped as a rectangular prism having approximate dimensions of 335 x 103 x 35 mm (Fig. 1); the high slenderness ratio of 3.3 was chosen to allow free development of shear planes away from the end platens. The tested sand was enveloped by a rubber membrane. Lubricated ends were used to reduce as much as possible the end restraint effect on the specimen. For the same reason, the two plane strain deformation faces were lubricated with silicon grease. The air pluviation method was used for sand deposition. Once placed in the cell and submitted to a small cell pressure, the specimens were saturated with deaerated water, after circulation of carbon bioxid gas in the dry material in order to improve the solubility of the interstitiel gas before water tilling.

The biaxial plane strain apparatus

The apparatus used in this study is formed by a pressurising cell (Fig. 2b) in which a specimen is placed between two thick rigid parallel glass plates (Fig. 2a) imposing no deformation in the perpendicular direction. In the vertical direction a displacement con- trolled loading is applied. In the third direction, the specimen is submitted to the contin- ing cell pressure from which the fluid pore pressure is deduced to define the effective confining pressure.

60 NUMEIKAI. METHODS FOR LOCAIXATION AND BIFURCATWN 01: GRANULAR BODIES

plone stroln device rear gloss plate

electrlcol motor

screw jack -oxiol displacement

transducer

-load cell

-load beom -horizontol displacement

transducer

-porthole

Fig. 2. Biaxial apparatus: a) front view of the plane strain device mounted on the base of the apparatus, b) pressure cell and loading device

During the test, the axial force and displacement are measured, as well as the volu- metric strain and the cell pressure. Photographs are taken during the test to record the shear band locations and directions in the specimen; these photos can also be used to measure the displacement fields by the false relief stereophotogrammetric method [5].

As the axisymmetric triaxial test, the biaxial test is an element test, i.e. a test starting from an initially quasi-homogeneous specimen, subjected to boundary conditions such that ideally the stress and the strain fields are likely to remain homogeneous along the test. It is well known that shear banding occurs in these tests, and the motivation of these experimental study is to observe the onset of localization, and to describe its evolution under increasing confining pressure.

The details of the plane strain biaxial set up can be found in reference [2].

RESULTS AND ANALYSIS

Dense Hostun RF sand

The curves of stress ratio 0, /CT 3 versus axial strain E, and the corresponding volumetric strain for dense Hoshm sand are shown on fig. 3. It must be stressed here that the interpretation of the test in terms of homogeneous stress and strain is valid only be- fore the peak; the stresses and strains computed after the peak from overall measurement are purely indicative, but they allow us to make comparisons between the post peak re- sponse in the different tests.

The stress ratio versus strain curves are characterized by a marked peak. It has been shown in previous studies [2,4,12] using the stereophotogrammetric method [5] on the same biaxial apparatus that the peak indicates the onset of the localization process; more precisely, the drop of the axial load recorded in a displacement-driven loading program was found to take place when the incipient shear band reaches a complete extension throughout the specimen. According to the observations by Lade [7], we can notice in

SHEAR BANDING DEPENDENCY ON MEAN STREST LEVEL IN SAND 61

Fig. 3 that the sudden drop in the axial load occurs while the stress strain curve has still a positive slope.

The variation of the cell pressure results in an important evolution of the curves: a noticeable retardation of the localization and a decrease in the peak value are observed as CS~ increases. On the other hand, the drop is steeper with increasing cell pressure. After the peak, a slight positive slope is observed on the stress-strain curve, decreasing with increasing confining pressure. This can be due to the reaction of the rubber membrane at the shear band level; this reaction becomes relatively less important at high confining pressure levels.

Fig. 3. Specimen response under plane strain tests on dense (DR = 0.95) Hostun RF sand for 4 different cell pressures: 100,200,400, 800 kPa. The stress and strain after peak are only indicative, cf text: a) stress ratio

Q , I (r 3 versus axial deformation E, , b) volumetric strain E” versus axial deformation E,

Global volumetric strain measurements, during the tests on dense sand specimens, show an initial contraction followed by dilatancy, and then an abrupt stop. The same kind of response is commonly observed in axisyrnmetric triaxial tests on dense sand at low cell pressure, although the transition to the final plateau is more clear here than in triaxial tests. The transition occurs simultaneously with the drop in the stress-strain curve; as discussed above, this event corresponds to the complete development of the shear band. When considering in detail the volumetric curves, it can be noted that during the drop of the axial load, a significant increase of the dilatancy slope is observed. The increased

62 N~JME~KAL METHODS FOR LOCALMTI~N AND BIFURCMION 01: GRANUUR BODIES

sloie is related to the dilatancy in the shear band during its propagation, so it does not represent an elementary characteristic value; in fact it depends on the ratio of the vol- umes of the material inside and outside the shear band. The slope having a significant value and used in the calculation of the dilatancy angle is that one calculated just before the peak of the stress ratio -strain curve.

Loose Hostun RF sand

For the loose Hostun RF specimens, the stress ratio vs axial strain curves (Fig. 4a) are well graded again with respect to the cell pressure o 3 . Each curve shows a progres- sive evolution up to a transition point, after which the measurements show slight irregu-

0.00

E, Fig. 4. Specimen response under plane strain tests on loose (DR = 0.25) Hostun RF sand for 4 different cell pressures: 100,200,400,800 kPa. The stress and strain after peak are only indicative, cf text: a) stress ratio

cs , Id 3 versus axial deformation E , , b) volumetric strain E ,, versus axial deformation E ,

larities, with a tendency to a mere decrease of the axial stress. This transition point of transition corresponds to the onset of the localization. Thus, for discussion of the effect of mean stress on localization, this point can be considered as the equivalent of the ob- served peak in dense tests. The influence of the increase of the cell pressure is revealed by a retard of the localization, even more evident than in the case of dense specimens. As

SHEAR BANDING DEPENDENCY ON MEAN S~‘RUSS LEVEL IN &ND 63

far as volumetric response is concerned, all the tests - excepted the test shf48 realised at 100 kPa - show a continuously contracting behaviour in terms of volumetric strain (Fig. 4b) giving a negative value of the dilatancy angle all along the homogeneous phase of deformation. So, at the moment of the onset of localization, the rate of volumetric strain was negative or zero, which means that we had a non dilating behaviour when the local- ization process started. (see Table 2, exception test shf48). CareM examination of the curves however suggests that a very small amount of dilatancy could have taken place just after the localization.

-3.04 1 0.00 0.02 0.04 0.06 0.08 0.10 012 014

Fig. 5. Repeatability of the specimen response under plane strain tests on dense (RD = 0.95) Hostun RF sand for 4 different cell pressures : 100, 200, 400, 800 kPa. The stress and strain after peak are only indicative, cf text: a) stress ratio Q , I Q 3 versus axial deformation E I , b) volumetric strain E ,, versus axial deformation E,

During one of the two tests performed at (2, = 800 kPa on the loose Hostun RF sand, an antisymmetric mode of deformation in a S shaped form was observed before shear banding. To investigate about this phenomenon, it was necessary to go further ei- ther tnr.r~*rlo h;nh c.nII ~,.PEE,,,.~ 6,. tm.rsr& In.., in;tial ,.nmns~i~, Phnncinn the cm-nnrl L”..cm”J IllEjajll tic11 prJJ”lr “L &“vTUL”J l”.. 1111&1ca1 ~“.U~U”~LJ. U,,“““L’,b SIIW .xI.#“‘.U solution, a specimen preparation leading to very loose density was used (1.34 gm/cm3, relative density about 0). The method was inspired by [ 1 I]: after addition of a small moisture content, the material is deposited in layers and gently damped. A single test was realised at CJ 3 = 400 kPa. In this test, again, an antisymmetric mode of deformation was produced at 5% of axial deformation, however not precluding shear banding as the final

SHEAR BANDING DEPENDENCY ON MEAN STRESS LEVEL IN SAND 65

The curves show that the Manche sand has an extremely contracting behaviour, to- gether with a high friction angle at low pressure level (50” in biaxial at 100 kPa, while triaxial tests under the same cell pressure give 45’). As can be seen in table 3, the friction angle in biaxial tests drops rapidly with increasing cell pressure.

wnbalr a,(LPo) l ml r 200 0 400

El 0, I I 0.00 0.05 0.10

1 1 0.15 0.20 0.25 0.30

T 0.35

0.00 _( 1

Cl 0. IS I r I

a. 00 0. OS 0.10 0.15 0.20 0.25 0.30 0.35

Fig. 7. Specimen response under plane strain tests on loose (RD = 0.35) Manche sand for 3 different cell pres- sures: 100,200, 400 kPa. The stress and strain after peak are only indicative, cf text: a) stress ratio al/a, ver-

sus axial deformation EI, b) volumetric strain E” versus axial deformation &I

Generally speaking, it is remarked in the tests on this sand that the specimens have a typical tendency to loose homogeneity by producing a mode of diffise heterogeneity before the shear bands inception. This phenomenon, and also the antisymmetric defor- mation reported in tests on loose and very loose Hostun RF sand at high confining pres- sure are related to diffise bifurcation modes. Such modes have been studied theoretically by Vardoulakis [lo] for biaxial tests. The sample geometry plays an important role in the development of diffise modes. The experiments reported here, with both the Manche calcareous sand and the loose Hostun RF sand show that when the compressibility of the material is high, diffise bifurcation modes tend to become more critical than localized one (at given sample geometry), but the localized mode still remains the final one.

Shear band orientations

The orientations observed on the photos taken during the tests are presented on Fig. 8a and 8b for dense and loose Hostun sand respectively. The measures are taken at the onset of localization. The error margins are also indicated.

The figures indicate that the influence of crs is significant for both dense and loose specimens; however, the effect is more marked in the dense ones. The orientations are

66 NUMERICAL METHODS FOR L~~ALIZATION AND BIFURCATION OF GRANULAR BODIES

compared with two limits proposed by different authors: x14 - +/2 (so-called Coulomb orientation) and x/4 - v/2 (so-called Roscoe orientation), and a third intermediate orien- tation proposed by Arthur [I]. In the data presented here, we define Q as the internal Eric- tion angle at peak and v as the dilatancy angle just before the peak (see discussion above in paragraph 3 on the definition of dilatancy slope from the tests results). In both dense and loose specimens the results indicate that the orientation increases with increasing mean stress, from an orientation close to “Coulomb” orientation, toward the intermediate orientation. The “Roscoe” orientation is out of the range of these experimental results.

7 6 0 0) 1

=.,j Sable Hostun RF dense

____x________----------- x *___*----- 35

IS~....,.~...~~~.,.....~...,......~.., 5c 230 450 650 650

(!@a)

Sable Hostun RF l&he

.5?......'...,....-....,..-......,...... .? 50 250 450 65C 3fC

(kPa)

Fig. 8. Shear band orientation observed at different confining pressure. The orientation is measured with re- spect to the major principal stress direction (axial stress), a) dense Hostun RF sand, b) loose Hoshm RF sand

CONCLUDING REMARKS

Throughout the analysis of the plane strain tests reported in this paper we can make the following statements:

- The onset of the localization in form of shear bands is significantly affected by the confining pressure level.

- The localization is retarded by increasing mean stress level and by decreasing sample density.

SHEAR BANDING DEPENDENCY ON MEAN STRESS LEVEL IN SAND 67

- The phenomenon of shear banding is not ruled out for specimens exhibiting overall contracting behaviour in terms of volume change

- A better appreciation of the local volumetric strains in particularly in the shear band could be made using the false relief stereophotogrammetric method to analyse the photographs taken during the test.

- In plane strain conditions, diffise bifurcation modes can be observed before shear band initiation, when the mean stress level is high and the compacity low but shear banding remains the final rupture mode.

REFERENCES

1. Arthur J.R.F., Dunstan T., Al-Am Q.A.J.L., Assadi A. (1977): Plastic deformation and failure in granular media. Geotechnique 27, pp 53-74.

2. Desrues J. (1984): La localisation de la deformation dans les materiaux granulaires. These de Doc- torat es Sciences, USMG et INPG, Grenoble.

3. Hammad W. (1991): Modelisation non lineaire et etude experimentale de la localisation dam les sables. These de doctorat, UJF - MPG.

4. Desrues J. (1990): Shear band initiation in granular materials: experimentation and theory. Geoma- terials constitutive equations and modelling, Ed Felix DARVE, Elsevier, pp 283 - 310.

5. Desrues J., Duthilleul B. (1984): Mesure du champ de deformation dun objet plan par la methode stereophotogrammetrique de faux relief. Journal de Mecanique Thtorique et Appliqut?e, Vol 3, No I, pp 79 - 103.

6. Desrues J., Hammad W. (1989): Etude experimentale de la localisation de la deformation sur sable: influence de la contrainte moyenne. Proc. ICSMFE Rio de Janeiro.

7. Lade P.V. (1982): Localization effects in triaxial tests on sand. IUTAM conference on deformation and failure in granular media, Delhi.

8. Tatsuoka F., Sakamato M., Kawamura T., Fukushima S. (1986): Strength and deformation charac- teristics of sand in plane strain compression at extremely low pressures. Soils and Foundations, Vol. 26, No. I, pp 65 - 84.

9. Peters J.F., Lade P.V., Bro A. (1988): Shear band formation in triaxial and plane strain tests. Ad- vanced triaxial testing of soil and rock. ASTM STP 977, Robert T. Donaghe, Ronald C. Chaney and Marchall

i L. Silver, Eds., American society for testing and materials, Philadelphia, pp. 604-627. i IO. Vardoulakis 1. (1981): Biturcation analysis of the plane rectilinear deformation on dry sand sam-

ples. Int. J. Solid Structures Vol. 17, No. I I, pp 1085-I 101. 1 I. Canou J. (1989): These, CERMES, Paris.

12. Desrues J., Lamer J., Stutz P. (1985): Localization of the deformation in tests on sand sample. En- gineering fracture mechanics,Vol. 21, pp 909 - 921.