ASYMPTOTIC BEHAVIOUR OF SAND IN PLANE-STRAIN COMPRESSION TESTS

8/13/2019 ASYMPTOTIC BEHAVIOUR OF SAND IN PLANE-STRAIN COMPRESSION TESTS

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Studia Geotechnica et Mechanica, Vol. XXIX, No. 34, 2007

ASYMPTOTIC BEHAVIOUR OF SAND IN PLANE-STRAINCOMPRESSION TESTS

DARIUSZ WANATOWSKI

Lecturer, Nottingham Centre for Geomechanics, School of Civil Engineering,

The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.

E-mail: [email protected]

Visiting Fellow, School of Aerospace, Civil and Mechanical Engineering, University of New South

Wales at Australian Defence Force Academy, Canberra, ACT 2600, Australia

Streszczenie:

Abstract: Experimental data are presented, which allows the asymptotic behaviour of sand underplane-strain conditions to be studied.K0consolidated plane-strain tests were conducted using a new

plane-strain apparatus under strain path-controlled conditions. The stressstrain behaviour of sand in

very loose and medium dense states under plane-strain conditions was characterized. The results

show that the stress-strain behaviour of sand under plane-strain conditions is strain path-dependent.

When the strain increment ratio imposed on specimens (dv/d1)iis higher (i.e., more positive) than

a threshold value, strain hardening behaviour will prevail. On the other hand, when the (dv/d1)i is

lower (i.e., more negative) than a threshold value, strain softening will occur. The threshold strain in-

crement ratio is defined as (dv/d1)f, that is the strain increment ratio at failure as measured in

a drained test. It is also observed that the asymptotic behaviour of sand cannot be predicted by the

Cam clay models or the Rowes stressdilatancy equation. This is consistent with the observations

made by other researchers under axisymmetric conditions.

Rezume


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D. WANATOWSKI2

LIST OF SYMBOLS

D dilatancy factor (= 1 dv/d1),dv/d1 dilatancy ratio,

dv/d1 strain increment ratio,

e void ratio,

p mean effective stress, )(3

1321 p' ++= ,

q deviatoric stress, [ ]21

213

232

221 )()()(

2

1q ++= ,

1 axial strain,

v volumetric strain,

effective friction angle of soil, effective stress ratio, = q/p, angle of the shear band orientation.

SUBSCRIPTS

asy asymptotic,c consolidated,

cs critical state,

f failure,

i imposed.

1. INTRODUCTION

When the stressstrain behaviour of soil is studied experimentally, laboratory tests

can be controlled using either stress path or strain path method. The stress path

method was originally described by LAMBE[18] and used to predict the settlement of

foundations. Since then, this method has become very popular for both practising

engineers and researchers. A typical example of a stress path test is the conventionaldrained triaxial compression test, in which the cell pressure is held constant and the

axial load is increased. This corresponds to a controlled stress path test with a 3d =0. The strain path method was originally proposed by CHUand LO[5] to study strain

softening behaviour of sand. In this method, a strain path can be specified by the

strain increment ratio dv/d1. By using digital pressure volume controllers [24] linked


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Asymptotic behaviour of sand in plane-strain compression tests 3

to a computer, dv/d1 can be precisely controlled and all possible drainage conditionscan be simulated [5]. When dv/d1> 0, the soil specimen will compress and water

will flow out of the sample with axial deformation. On the other hand, when dv/d1 0) strain path. All the tests were conducted in

deformation-controlled mode at a rate of 0.05 mm/min. A summary of all the plane-

strain tests conducted is given in table 2.

-3

-2

-1

0

1

2

3

0 2 4 6

1(%)

v(%)

dv/d1= +1.0 +0.6

-0.4

-0.6

0.0

+0.2

+0.4

-0.2

-1.0

Fig. 3. A range of strain paths imposed in plane-strain tests


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D. WANATOWSKI8

T a b l e 2

Summary of plane-strain tests conducted

Test

No.

Preparation

methodec

Drc(%)

Density

state*cp

(kPa)

qc(kPa)

(dv/d1)i

SP02

SP04

SP05

SP08

SP26

SP09

SP12

SP13

SP14SP16

SP17

SP19

SP20

SP22

SP23

U02

U08

WS

MT

WS

MT

WS

WS

MT

WS

WSMT

WS

MT

WS

MT

WS

WS

MT

0.684

0.897

0.677

0.890

0.731

0.679

0.887

0.682

0.6940.888

0.682

0.878

0.681

0.903

0.686

0.695

0.902

61

5

62

7

48

62

8

61

587

61

10

61

3

60

58

4

m. dense

v. loose

m. dense

v. loose

m. loose

m. dense

v. loose

m. dense

m. densev. loose

m. dense

v. loose

m. dense

v. loose

m. dense

m. dense

v. loose

201.0

195.9

198.2

200.6

404.5

200.0

199.8

201.5

202.9197.8

200.5

199.6

206.2

197.5

202.8

201.9

198.4

211.1

169.8

220.3

172.4

421.7

222.4

169.0

210.7

218.1174.3

227.3

174.3

226.4

171.9

226.6

218.8

168.4

0.2

0.2

0.4

0.4

0.4

0.6

0.6

1.0

+0.2+0.2

+0.4

+0.4

+0.6

+0.6

+1.0

0.0

0.0

Note: *Density description of SKEMPTON[32] is adopted: very loose (Dr< 15%); loose (Dr= 1535%);medium loose (Dr= 3550%); medium dense (Dr= 5065%); dense (Dr> 65%).

5. RESULTS

Drained tests. The drained behaviour of Changi sand has been reported elsewhere

[40], [41]. Therefore, only a brief summary is presented in this section as a basis for

further analysis. Stressstrain curves obtained from drained plane-strain tests con-

ducted on medium dense, medium loose and very loose Changi sand are plotted in

figures 4a to 4c, respectively. All the tests wereK0-consolidated to cp = 100, 200 and

300 kPa and then sheared under drained conditions with 3 maintained constant. Itcan be seen from figures 4a and 4b that during drained shearing of medium dense and

medium loose specimens, the deviatoric stress firstly reached a peak, and then re-

duced gradually to a constant ultimate value. It should be pointed out that the peak

deviatoric stress in these tests was accompanied by shear band formation. The q1curves of drained tests conducted on very loose sand are presented in figure 4c. It is

observed that the deviatoric stress approached a constant value at the end of each test.

As reported by [40] the volumetric strain in these tests also approached a constant

value. Furthermore, shear bands have not been detected either. Therefore, it can be

concluded that the critical state is reached in these tests.


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0

250

500

750

1000

0 5 10 15

Axial Strain, 1 (%)

DeviatoricStress,q(kPa)

pc'=100 kPa, ec=0.678

pc'=299 kPa, ec=0.669

pc'=201 kPa, ec=0.665

Initial state after Koconsolidation

Failure state (shear band)

Medium Dense

0

250

500

750

1000

0 5 10 15

Axial Strain, 1 (%)


pc'=102 kPa, ec=0.784

pc'=201 kPa, ec=0.771

pc'=300 kPa,ec=0.760


Failure state (shear band)

Medium Loose

a) b)

0

250

500

750

1000

0 5 10 15Axial Strain, 1 (%)


pc'=99 kPa, ec=0.932

pc'=298 kPa, ec=0.900

pc'=202 kPa, ec=0.914

Very Loose


Failure state (NO shear band)

0.8

1.0

1.2

1.4

1.6

1.8

0.6 0.7 0.8 0.9 1.0ec

f

f=1.53 (m.dense)

f=1.35 (m.loose)

f=1.16 (v.loose) = cs

c) d)

0.00

0.25

0.50

0.75

1.00

1.25

0.60 0.70 0.80 0.90 1.00Void Ratio, ec

(-d/d1)f

pc'=100kPa

pc'=200kPa

pc'=300kPa

e)

Fig. 4. Drained behaviour of the Changi sand: a) medium dense, b) medium loose,

c) very loose, d) summary of failure states, e) relationship between (dv/d1)fand ec


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D. WANATOWSKI10

A summary of all failure states obtained from the plane-strain tests is shown infigure 4d. Under plane-strain conditions, the slopes of the failure lines (f) for very

loose, medium loose and medium dense sands are 1.16, 1.35, and 1.53, respectively.

These values correspond to the friction angles (f) of 36.0, 43.4, and 49.7, respec-

tively. Since the failure line for very loose sand is also the CSL, cs=1.16 and cs=

36.0 [40], [41]. A further summary of the failure states is given in figure 4e, where

the relationship between maximum dilatancy ratio at failure (dv/d1)fand void ratio

(ec) is given. It should be stressed that this relationship is affected by cp .

0

250

500

750

1000

1250

0 2 4 6 8 10

Axial Strain, 1 (%)


dv/d1= +1.0

+0.2+0.6

+0.4

0.0 (undrained)

- 0.2

- 0.4

- 0.6- 1.0pc' = 200 kPa

Peak point

(shear band)

PT point

a)

-400

-200

0

200

0 2 4 6 8 10

Axial Strain, 1 (%)

ExcessPoreWaterPressure,u(kPa)

pc' = 200 kPa

dv/d1= +1.0

+0.4+0.2

+0.6

0.0 (undrained)

- 0.2- 0.4

- 0.6- 1.0

Peak point (shear band)

PT point

b)


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D. WANATOWSKI12

Strain path tests on medium dense sand. The influence of a strain path on thestressstrain behaviour of sand in plane-strain tests is clearly demonstrated in figure

5, where the results of a series of strain path tests conducted on medium dense sand

with different strain increment ratios dv/d1, ranging from +0.1 to 0.1 are presented.

The void ratios ecof specimens afterK0consolidation, were in the range from 0.677

to 0.695, corresponding to relative densities Drc= 5862%. All the specimens were

prepared by the WS method. Although all the specimens were in the narrow density

range, different types of behaviour, mainly strain hardening and limited strain soften-

ing, were observed. The response produced for more compressive (positive) strain

paths was strong strain hardening compared to that for more dilative (negative) strain

paths. Therefore, for a given axial strain, the higher the strain increment ratio im-

posed, the stiffer the stressstrain curve obtained, as shown in figure 5a. It can also beseen from figure 5a that stressstrain curves for the strain paths with dv/d1> 0 are

different from those with dv/d10. For the strain path tests with dv/d1> 0, strainhardening behaviour was only observed. As shown in figure 5a, peak state is not

reached by any of the dv/d1> 0 paths before the termination of the test. There is no

visible shear band either. For the strain paths within the range of 0.6 dv/d10the strain hardening behaviour is also observed. However, it is followed by the peak

state and the strain softening behaviour in the post-peak region. It should be pointed

out that the peak deviatoric stress in these tests coincides with shear band formation.

Consequently, banding softening behaviour [9] is observed in 0.6 dv/d10 paths.It should also be stressed that the banding softening could have been possibly ob-

served in the dv/d1> 0 strain paths if the shearing was further continued. However,

the lateral pressures (2) in these tests reached the capacity of the lateral pressure

transducers. Thus, the dv/d1 > 0 strain path tests had to be terminated. It is also

shown in figure 5a that the limited strain softening was observed in the dv/d1= 1

path test. In this test, a material strain softening initially occurred and then the phase

transformation state (PT) characterised by the minimum value of deviatoric stress was

reached. After that a strain hardening behaviour was observed and eventually the

banding strain softening occurred.

The excess pore water pressures developed in the tests are shown in figure 5b. It

can be seen that the higher the strain increment ratio imposed, the more negative the

pore water pressure change. It is noted that the excess pore water pressure developed

in an undrained test is not the highest, as often assumed. The pore water generation

for the undrained path is in between that for a negative (dilative) strain path anda positive (compressive) strain path.

The effective stress paths obtained from the strain paths tests are plotted in figure

5c. Except the test with dv/d1 = 1.0, all the other paths show a tendency to ap-

proach the asymptotic line. It can be observed that the higher (i.e., the more positive)

the strain increment ratio, the lower the asymptotic stress ratio asy. This can also be


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0

200

400

600

800

0 5 10 15

Axial Strain, 1 (%)


pc' = 200 kPa

+0.2

dv/d1= 0.0 (undrained)

- 0.2- 0.4- 0.6

+0.4

+0.6

a)

-200

-150

-100

-50

0

50

100

150

0 5 10 15

Axial Strain, 1 (%)

ExcessPoreWaterPressure,u

(kPa)

pc' = 200 kPa


+0.6

+0.4

+0.2

- 0.6 - 0.4- 0.2

b)


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D. WANATOWSKI14

0

200

400

600

800

0 200 400 600 800

Mean E ffective S tress, p' (kPa)

Devia

toricStress,q(kPa)

pc ' = 200 kPa

+0.2

+0.4

0.0

- 0.2, -0.4, - 0.6

dv/d1= +0.6

CSL

M = 1.16

50

100

150

200

50 100 150 200 250

CSL

0.0

- 0.2, -0.4, - 0.6

c)

0.8

1.0

1.2

1.4

0 5 10 15

Axial Strain, 1 (%)

EffectiveStressRatio,=

q/p' pc' = 200 kPa


+0.6

+0.4+0.2

- 0.2- 0.4

- 0.6

M = cs= 1.16

d)

Fig. 6. Strain path-dependent behaviour of very loose sand:

a) stressstrain curves, b) excess pore water pressure versus axial strain curves,

c) effective stress paths, d) effective stress ratio versus axial strain curves


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seen from the q/pversus 1curves shown in figure 5d. However, due to shear bandformation, strain softening and thus a reduction in q/poccurred in some tests (figures5a and 5d). It is also observed that although asyincreases with an increase in the di-

latancy ratio (dv/d1), the asyapproached in the tests with dv/d1= 0.4, 0.6, and

1.0 is about the same value of max = 1.45. Therefore, it appears to be a limiting

value, max. When (dv/d1) exceeds a certain value, asywill only reach this limitingvalue.

Strain path tests on very loose sand. The results of a series of strain path tests con-

ducted on very loose sand with strain increment ratios dv/d1, ranging from +0.6 to 0.6

are presented in figure 6. The void ratios ecof specimens afterK0consolidation, were in

between 0.887 and 0.903, corresponding to relative densities Drc = 310%. All the

specimens were prepared by the MT method. The effect of strain paths on the stress

strain behaviour of very loose sand can be seen in figure 6a, where the stressstrain

curves are plotted. By imposing a more compressive (positive) strain increment ratio,

a sand specimen shows a stiffer behaviour. Similarly, when the strain increment ratio

becomes more negative (dilative) a specimen exhibits more pronounced strain softening.

It is also observed that under an undrained condition (dv/d1= 0) strain softening occurs

for very loose sand. The curves representing excess pore water pressure versus axial

strain for very loose sand are presented in figure 6b. The positive pore water pressure

development was observed when dv/d1< 0. Negative pore water pressure response was

obtained when dv/d1> 0. The more negative the dv/d1imposed, the more positive theexcess pore water pressure change.

The effective stress paths obtained from the tests on very loose sand are presentedin figure 6c. The CSL line obtained from drained plane-strain tests on very loose sand

is also plotted in figure 6c for comparison. The stress paths obtained from compres-

sive strain paths exhibit strain-hardening behaviour, while the stress paths obtained

from dilative strain paths exhibit strain-softening behaviour. Nevertheless, it can be

seen from figure 6c that the stress paths resulting from tests with dv/d1< 0.2 ap-

proached the CSL. The stress paths obtained from tests with dv/d1= +0.4 and +0.6

approached the corresponding asymptotic lines. The q/p1curves are presented infigure 6d. It is seen that all the curves tend to approach asymptotically a constant

valueof q/p. The limiting stress ratio maxfor loose sand appears to be cs= 1.16.

6. SHEAR BAND FORMATION

Shear band formation in drained and undrained plane-strain compression tests

have been observed by a number of researchers [2], [11], [12], [13], [25]. However,

the shear band formation in other than dv/d1 = 0 (undrained) strain paths has not

been reported before.


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D. WANATOWSKI16

400

500

600

700

0 2 4 6 8 10

Axial Strain, 1 (%)

LateralStress,2

(kPa)

0

4(O) 5(P,B)

1

3

2

7

6

98

SP05

a)

b)


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0

250

500

750

1000

0 2 4 6 8 10

Axial Strain, 1 (%)


0.0

0.5

1.0

1.5

2.0

EffectiveStressRatio,q/p'

76

4(O) 5(P,B)

3

2

1

0 9

8

q

SP05

q/p'

c)

-100

-50

0

50

100

0 2 4 6 8 10

Axial Strain, 1 (%)

ExcessPWP,u(k

Pa)

0 5(P,B)

12

3

4(O)

67

8

9SP05

d)

Fig. 7. Shear band development in Test SP05:

a) 2versus 1curves, b) selected photographs, c) qversus 1and q/pversus 1curves,d) uversus 1curves

During all the plane-strain tests presented in this study, the process of shear band

formation has been monitored by taking photographs at a close time interval using

a high-resolution digital camera. Grid lines were marked on the membrane surface

facing the camera (3plane) to facilitate the observation of the shear band. The occur-


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D. WANATOWSKI18

rence of the shear bands was also detected by comparing the 2 1curves obtainedfrom the four individual pressure transducers. In this investigation, shear bands only

occurred for medium loose to medium dense specimen with dv/d10 imposed. Forall the other strain path tests, including those on very loose sand, shear bands were not

detected.

The results of a strain path test SP05 conducted on a medium dense specimen (ec=

0.677) using dv/d1= 0.4 are presented in figure 7. The 2 1curves obtained fromthe four load cells during Test SP05 are presented in figure 7a. The numbers marked

on the stress-strain curves shown in figure 7a refer to the points where photographs of

the specimens were taken. Selected photographs taken around the shear band being

formed are shown in figure 7b. For more details, see [39], [40]. It can be seen from

figure 7a that at the point 4, the 2values measured by each load cell start to diverge.However, from the photographs, the shear band became visible at the point 5, which

was after the point 4. Therefore, the point 4, where 2starts to diverge, can be used as

an indicator for the onset of the shear band. As shown in figure 7c, the point 5 where

the shear band became visible, is the peak point on the q1curve. It can also beseen from figure 7d that the pore water pressure started to increase just after the point

5 being is another indication that shear band had occurred. The above results indicate

that when the shear band occurs, the lateral stress distributions become highly non-

uniform. Therefore, the point, where the 2curves start to diverge, indicates the point

where shear band has been initiated. The results obtained from plane-strain tests un-

der strain path controlled conditions also show that the shear band develops in the

specimen earlier than its physical appearance on the membrane surface.

Fig. 8. A shear band observed at the end of Test SP26

The typical mode of the shear bands developed during strain path tests is identical

to that obtained under drained and undrained conditions [39], [40]. An example of the

= 55


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shear band observed at the end of Test SP26 is shown in figure 8. The photographs(figure 8) were taken after the cell was dismounted and a vacuum was applied to hold

the sample. It can be seen that the specimen slides towards the 3 direction.

Therefore, the shear band on the 3plane is nearly horizontal and on the 2plane isinclined at an angle = 55. The inclination for all the other shear bands measuredin plane-strain tests was in the range of 49.057.5for medium loose sand and in therange of 49.658.6for medium dense sand. The average values of 55.6and 56.2were obtained for medium loose sand and medium dense sand, respectively. It has

been observed that shear band inclinations were best predicted by the Roscoe solution

[39], [40].

7. DISCUSSION

The experimental results presented in this paper clearly demonstrate that stress

strain behaviour of sand under plane-strain condition is strain path-dependent. The re-

sults show that the more positive strain increment ratio leads to the stiffer response of

soil. Depending on the density, when a dv/d1more negative than a certain value is

imposed, strain softening will occur. This is consistent with the observations made for

axisymmetric conditions based on triaxial test results [3], [7], [8], [19], [35], [36].

However, under plane-strain conditions shear bands would occur in medium loose and

medium dense specimens. The occurrence of shear bands can lead to strain softening

behaviour in post-peak region. Under axisymmetric conditions, shear bands would not

occur, if free-ends were used to reduce end frictions. Therefore, the strain softeningbehaviour under plane-strain conditions as presented above is classified into two

types. The first type is the material softening occurring for very loose sand as shown

in figure 6 for tests with dv/d1of 0.0 (undrained), 0.2, 0.4, and 0.6. This type of

softening is the true element behaviour and it reflects the response of loose granular

soils to the strain path imposed. The second type is the strain softening observed in

hardening regime for medium loose and dense sand after the occurrence of shear

band, for example, test dv/d1 = 0.0 (undrained), 0.2, 0.4, or 0.6 for medium

dense sand in figure 5. This type of strain softening is caused by the formation of

shear band in the hardening regime. This type of strain softening has been termed as

banding softeningby CHUet al. [9].

The asymptotic state of sand has been investigated by CHUand LO[8] under axi-symmetric and three-dimensional stress conditions. It was found that the asyis inde-

pendent of the soil stress history and the initial effective confining stress. Chu and Lo

[8] also concluded that there is a relationship between the controlled strain increment

ratio dv/d1and the asymptotic stress ratio asy. This relationship is independent of

the initial effective confining stress, the loading mode and shearing rate and is not

significantly affected by the void ratio.


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D. WANATOWSKI20

0.0

0.5

1.0

1.5

2.0

-1.0 -0.5 0.0 0.5 1.0 1.5-dv/d1

asy

f= 1.53

asy= M0- (dv/d1)M0= 1.37 = 0.23

Failure points from drained

(d3' = 0) tests

0.760

Test data

pc' = 200 kPa

ec= 0.67-0.71

Fig. 9. Relationship between asyand (dv/d1)

The experimental data obtained from the strain path tests conducted on medium

dense sand (ec= 0.670.71) and commenced with cp = 200 kPa under plane-strainconditions is presented in figure 9. Figure 9 shows that there is a relationship between

asyand dv/d1at the asymptotic state. By fitting the line through all the data points,

the empirical equation between the asymptotic stress ratio and the strain incrementratio in strain path tests of 0.6 dv/d11 can be established:

=

1

0asy

d

dM v , (1)

where: M0= 1.37 and = 0.23 are the experimental constants. Similar relationship

can be established for any density state [39].

Equation (1) shows that the asymptotic stress ratio increases with a decrease in the

controlled strain increment ratio. However, as discussed earlier, the achievable stress

ratio is bounded by a limiting value of asy. Furthermore, it has been established by

CHU and LO [8] under axisymmetric conditions that the maximum achievable as-

ymptotic stress ratio, maxis bounded by the failure line determined by drained tests.

In other words, when the controlled strain increment ratio (dv/d1)iis more negative

than the strain increment ratio at failure (dv/d1)f, the asymptotic stress ratio can only

reach its limit value, the failure stress ratio f.


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0

100

200

300

0 100 200 300

Mean Effective Stress, p' (kPa)


dv/d1= -1.0

SP13

(ec= 0.682)

pc' = 200 kPa

FL (f= 1.53)

Fig. 10. Effective stress path of a dv/d1= 1.0 test

The effective stress path resulted from a dv/d1= 1.0 test is presented in figure

10. Fairly detailed results of this test have been given earlier in figure 5. Although the

strain increment ratio imposed in the test is more negative than (dv/d1)f= 0.76 de-

termined for the same cp = 200 kPa (see figure 4e), the effective stress path ap-proaches the failure line. Therefore, just as under the axisymmetric conditions,

(dv/d1)fseems to be a threshold strain increment ratio under plane-strain conditions.

When the controlled strain increment ratio is more negative than the strain increment

ratio at failure, which in this case is 0.76, the asymptotic stress ratio can only reach

its limit value, the failure stress ratio f. Thus, for the strain path tests with dv/d1

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ASYMPTOTIC BEHAVIOUR OF SAND IN PLANE-STRAIN COMPRESSION TESTS