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Acta Geotechnica (2006, Vol. 1, No. 1, pp. 59-73)

Experimental evidence of a unique flow rule of non-cohesive soils under

high-cyclic loading

T. Wichtmann∗, A. Niemunis, Th. Triantafyllidis

Institute of Soil Mechanics and Foundation Engineering, Ruhr-University Bochum, Universitatsstraße 150, 44801 Bochum, Germany

Abstract

The presented results of cyclic triaxial tests on sand demonstrate that the cumulative effects due to smallcycles obey a kind of flow rule. It mainly depends on the average stress ratio about which the cycles are performed.This so-called ”cyclic flow rule” is unique and can be well approximated by flow rules for monotonic loading.Amongst others it is shown that the cyclic flow rule is only moderately influenced by the average mean pressure,by the strain loop (span, shape, polarization), the void ratio, the loading frequency, the static preloading and thegrain size distribution curve. A slight increase of the compactive portion of the flow rule with increasing residualstrain (due to the previous cycles) was observed. These experimental findings prove that the cyclic flow rule is anessential and indispensable concept in explicit (N-type) accumulation models.

Key words: Cyclic flow rule; Direction of strain accumulation; Cyclic triaxial tests; Sand

1 Introduction

Cyclic loading leads to an accumulation (of stress orstrain) in soils. This paper deals with high-cyclicloading (N ≥ 104) with relatively small amplitudes(εampl < 10−3). In FE predictions of residual settle-ments two numerical strategies can be distinguished,the implicit and the explicit one. In the implicit methodeach cycle is calculated with a σ-ε constitutive model(e.g. with an elastoplastic multi-surface model, e.g.[11], an endochronic model, e.g. [20] or the hypoplasticmodel with intergranular strain [5,8,14,21]) and manystrain increments. Due to the unintentional accumu-lation of systematic (e.g. numerical) errors [12] andthe huge calculation effort [22], the application of thismethod is restricted to a low number of cycles (say,N < 50).

For high-cyclic loading explicit models are more suit-able. Several such models were proposed in the litera-ture [1,3,4,6,7,10,15–18]. They treat the accumulationof residual strain or stress under cyclic loading similarto the problem of creep under constant loads. The basicconstitutive equation is similar to that of viscoplastic

∗Corresponding author. Tel.: + 49-234-3226080; fax: +49-

234-3214150; e-mail address: [email protected]

models with the number of cycles N replacing time t:

σ = E : (ε − εacc) (1)

with the Jaumann stress rate σ of the Cauchy stressσ, the stretching ε (in our case obtained as the ma-terial derivative of the Hencky’s logarithmic strain ε,but with positive compression), the given accumulationrate ε

acc and the pressure-dependent elastic stiffness E.In the context of cyclic loading ”rate” means a deriva-tive with respect to the number of cycles N (insteadof time t), i.e. t = ∂ t /∂N . Explicit models predictthe average accumulation curve without following thestrain path during the particular cycles. The given ac-cumulation rate ε

acc can be written as a product of ascalar intensity of strain accumulation εacc and a ten-sorial direction of strain accumulation (cyclic flow rulem = unit tensor):

εacc = εacc

m (2)

The following sections present the results of numer-ous drained cyclic triaxial tests in order to examinethe variability and to establish an empirical formulafor m(σav, ...). The influence of the average stress, ofthe stress loop (span, shape, polarization), the void ra-tio, the loading frequency, the static preloading and

1

Acta Geotechnica (2006, Vol. 1, No. 1, pp. 59-73) 2

the grain size distribution curve is studied. It is demon-strated that m depends essentially on the average stressratio ηav = qav/pav. This dependence can be well ap-proximated by the flow rules of constitutive models formonotonic loading (e.g. modified Cam clay model, hy-poplastic model).

These experimental findings facilitate the develop-ment of explicit accumulation models. The flow rule m

can be treated separately from the intensity εacc. Re-grettably, only few models [1, 15, 18] incorporate suchkind of flow rule.

2 Notation

The sign convention of soil mechanics (compressionpositive) is used. The notation is given for the tri-axial test with the axial stress σ1 and the lateral stressσ2 = σ3. The Roscoe invariants p (mean pressure) andq (deviatoric stress) of the Cauchy stress σ are

p =1

3(σ1 + 2 σ3) (3)

q = σ1 − σ3 (4)

Alternatively, the isomorphic variables P =√

3 p andQ =

√

2/3 q may be used [13]. The obliquity of stressin the p-q-plane (Fig. 1) is described by the stress ra-tio η = q/p. The inclinations of the critical state line(CSL, from monotonic tests) and the failure line in thep-q-plane (Fig. 1) are Mc = (6 sin ϕ)/(3 − sin ϕ) forcompression and Me = −(6 sin ϕ)/(3 + sin ϕ) for ex-tension with the critical friction angle ϕ = ϕc for theCSL and the peak friction angle ϕ = ϕp for the maxi-mum shear strength.

Figure 1 shows a typical stress path of a cyclic tri-axial test in the p-q-plane. An average stress σ

av is su-perposed by a cyclic portion. An oscillation of the axialstress σ1(t) and the lateral stress σ3(t) without a timeshift results in in-phase stress cycles with a certain in-clination tan α = qampl/pampl in the p-q-plane (Fig. 1).For the special case σ3 = const. (tan α = 3) the am-plitude ratio ζ = qampl/pav is used. If σ1(t) and σ3(t)are applied with a time shift more complex stress paths(e.g. ellipses in the p-q-plane, out-of-phase-cycles) canbe tested.

The axial strain is denoted by ε1 and the lateral oneby ε2 = ε3. The strain invariants (εv = volumetricstrain, εq = deviatoric strain) are

εv = ε1 + 2 ε3 (5)

εq =2

3(ε1 − ε3) (6)

q

pav p

qav

q

1

1

1Mc(

c)

Mc(

p)

Me( c)

av = qav/pav

average stress

av

ampl

pampl

CSL

1

1Me(

p)

Figure 1: Cyclic stress path in the p-q-plane

Their rates are work-conjugated to the Roscoe invari-ants p and q. The axial strain is calculated from ε1 =ln(h0/h) and the volumetric one from εv = ln(V0/V )with the initial height h0 of the sample, the actualheight h, the initial volume V0 and the actual volumeV . The isomorphic strain invariants are εP = 1/

√3 εv

and εQ =√

3/2 εq . The total strain is

ε =√

(ε1)2 + 2(ε3)2 =√

(εP )2 + (εQ)2 (7)

In the case of cyclic loading the strain ε is composed ofan accumulated (residual) portion (εacc) and an elastic(resilient) portion (εampl) as shown in Fig. 2. The ratios

Ω =εacc

v

εaccq

and ω =εacc

v

εaccq

(8)

are used as scalar measures of the flow rule m .The density is described by the index ID = (emax −e)/(emax − emin) with void ratio e. The initial valueafter consolidation prior to cyclic loading is denoted byID0.

irreg

2 ampl

acc

tirregular cycles

regular cycles

Figure 2: Development of strain in a cyclic triaxial test


εv [%]ε1 [%]

q [MPa] q [MPa]

Figure 3: Contractive or dilative soil behaviour under cyclic loading in dependence on qav after Luong [9]: a)q-ε1-loops, b) q-εv-loops

0 0.5 1.0 1.5 2.0-1.0

-0.5

0

0.5

Average stress ratio av = qav/pav [-]

Vol

umet

ric s

trai

n

v

[%]

acc

dense sand

loose sand

a)

= Mc( c)

after N = 100 cycles

0 50 100 150 200 250 300 3500

50

100

150

200

250

q [k

Pa]

p [kPa]

accv

acc

b) flow rule mod. Cam-claymeasured direction of acc.

Figure 4: Studies on the direction of strain accumulation after Chang & Whitman [2]: a) residual volumetric strainsεacc

v after 100 cycles as a function of the average stress ratio ηav, b) measured directions of strain accumulation inthe p-q-plane compared to the flow rule of the modified Cam clay model

3 Literature survey

There are few studies on the direction of strain accu-mulation under cyclic loading and those publicationsmostly deal with a low number of cycles. In a drainedcyclic triaxial test Luong [9] applied packages of 20 cy-cles each at different average deviatoric stresses qav.The right part of Fig. 3 shows the measured q-εv-loops.For low values of qav the material behaviour was con-tractive while it was dilative at larger qav. Luong de-fined a so-called ”characteristic threshold (CT) line”in the p-q-plane separating the contractive (σav below

CT line) from the dilative (σav above CT line) materialbehaviour. This (border)line was found independent ofthe soil density.

Another important observation on the direction ofstrain accumulation was made by Chang & Whitman[2]. In a series of cyclic triaxial tests on medium coarseto coarse sand the average mean pressure pav was con-stant while the stress ratio ηav was varied from test totest. Four tests were performed on loose and four otherones on dense specimens. In Fig. 4a the residual vol-umetric strain after 100 cycles is shown as a function


of ηav. Independently of the sand density, εaccv = 0

was observed for ηav ≈ Mc(ϕc). Thus, Chang & Whit-man [2] assumed the CT-line of Luong [9] to be identicalwith the critical state line (CSL). For ηav < Mc(ϕc) adensification and for ηav > Mc(ϕc) a dilative materialbehaviour was measured.

In other tests Chang & Whitman [2] observed thatthe ratio γacc/εacc

v with γ = ε1 − ε3 = 3/2εq increasedwith ηav. A good approximation of the measured di-rection of strain accumulation by the flow rule of themodified Cam clay model

ω =Mc(ϕc)

2 − (ηav)2

2ηav(9)

has been demonstrated for different sands (Fig. 4b).No influence of the average mean pressure pav and theamplitude ratio ζ = qampl/pav on ω could be detected.Also the influence of the number of cycles was reportednegligible. However, the tests of Chang & Whitman [2]were restricted to 1,050 cycles. It is therefore not clearif the test results can be extrapolated to larger numbersof cycles. Furthermore, only the case σ3 = constant wastested.

The studies of Luong [9] and Chang & Whitman [2]as well as our own experiments presented within thispaper are restricted to relatively small amplitudes. Forlarge stress amplitudes with qmin ≈ 0 and qmax ≈Mc(ϕp), Suiker et al. [19] reported a dilative behaviour(during the first approx. 1,000 cycles) although the av-erage stress (defined as qav = 1

2(qmax +qmin)) lay below

the critical state line. In that case, the residual strainand also the flow rule εacc

v /εaccq depends on the time the

stress path is resting on the failure line.

4 Test device, testing procedureand tested materials

A scheme of the used triaxial cell is given in Fig. 5.A detailed description of this test device (four of themwere available in the present study) can be found in [23].Specimens (diameter d = 10 cm, height h = 20 cm)were prepared by dry pluviation out of a funnel throughair into split moulds. Afterwards the specimens weresaturated with de-aired water. A back pressure of 200kPa was used in all tests. The procedure of specimenpreparation is explained more detailed in [23]. In orderto minimize friction the end plates were lubricated anda thin latex membrane was applied. In the analysis ofthe tests the deformation of this membrane (beddingerror) was carefully subtracted from the deformationof the soil skeleton.

load cell

displ. transducer

press. transd. (cell-, back pressure)

differential pressure transducer

back pressure

soil specimen (d = 10 cm, h = 20 cm)

drainage

Axial load Fav Fampl+ -cell pressureσav σampl+ -3 3

(pneumatic loading system)

ball bearing

Volume measuring unit:

reference pipe

plexiglas cylinder

water in the cell

ball bearing

load piston

membranemeasuring pipe

Figure 5: Scheme of triaxial cell

0.1 0.2 0.6 1 2 60

20

40

60

80

100

Fin

er b

y w

eigh

t [%

]

Diameter [mm]

Sand Gravel

coarsefine finemedium

d50 = 0.35 U = 1.9 emin = 0.630 emax = 0.930

d50 = 1.45 U = 1.4 emin = 0.623 emax = 0.886

d50 = 0.52 U = 4.5 emin = 0.422 emax = 0.691

d50 = 0.55 U = 1.8 emin = 0.577 emax = 0.874

12 34

Figure 6: Tested grain size distributions

The axial load was applied by a pneumatic load-ing system and was measured inside the pressure cell(Fig. 5). The axial deformation of the specimen wasmeasured by means of a displacement transducer whichwas attached to the load piston. The volume changeswere determined via the pore water. For this purposea differential pressure transducer was employed. Alsothe cell pressure and the back pressure were measured.During cyclic loading the signals of all transducers wererecorded by means of a data acquisition system. Inorder to reduce the amount of data five complete cy-cles were sampled in certain intervals. The distancebetween these recordings was increased logarithmicallywith the number of cycles N .

Specimens were consolidated for one hour under theaverage effective stress σ

av. After that resting periodσ1 and σ3 were cyclically varied. Since usually large


deformations occur during the first, so-called irregular

cycle (see Fig. 2) in order to prevent an unintentionalpore-pressure build-up, a low frequency of fB = 0.01Hz was used during this cycle. Afterwards usually 105

cycles were applied with a loading frequency fB = 1Hz.

The tests were performed on a quartz sand with sub-angular grain shape. The four tested grain size distri-bution curves are presented in Fig. 6. Most of the testswere performed on the medium to coarse sand No. 1.

5 Test results

5.1 Influence of the average stress

Numerous tests were performed with in-phase stresscycles (σ3 = constant) at different average stresses σ

av.The average mean pressure (50 kPa ≤ pav ≤ 300 kPa)and the average stress ratio −0.88 ≤ ηav ≤ 1.375 werevaried from test to test. The tested average stressescover the cases of triaxial compression (ηav > 0) aswell as triaxial extension (ηav < 0). In Fig. 7 theyare illustrated in the p-q-plane. The specimens wereprepared with similar initial densities (0.57 ≤ ID0 ≤0.69). In the tests with ηav > 0 amplitude ratios ζ =qampl/pav = 0.3 were tested. In the tests with triaxialextension a smaller ratio ζ = 0.2 was chosen due tothe smaller distance of σ

av to the yield surface. Forthis reason, in the tests with ηav = -0,75 and ηav =-0,88 even amplitude ratios of ζ = 0.1 and ζ = 0.05,respectively, were applied.

First the tests with pav = 200 kPa are considered. InFig. 8 for selected average stress ratios ηav, the resid-ual deviatoric strain εacc

q in the regular cycles is plot-ted versus the accumulated volumetric strain εacc

v . Thedata points correspond to the numbers of cycles N =2, 5, 10, 20, 50, 100, ......, 105. From Fig. 8 it is ap-parent that the direction of strain accumulation sig-nificantly depends on the average stress ratio ηav. Atan isotropic average stress (ηav = 0) a pure volumet-ric accumulation (densification) takes place while therate of deviatoric strain vanishes (εacc

q = 0). Withincreasing absolute value of the average stress ratio|ηav| = |qav/pav| the deviatoric component of the di-rection of strain accumulation increases in comparisonto the volumetric portion. At an average stress on the(monotonic) critical state line (ηav = Mc(ϕc) = 1.25or ηav = Me(ϕc) = −0.88, respectively) only the devi-atoric strain accumulates and a vanishing rate of vol-umetric strain (εacc

v = 0) could be observed (at leastduring the first cycles). The latter remains valid de-spite the material is not at the critical density. ForMe(ϕc) ≤ ηav ≤ Mc(ϕc) a densification of the sand andwithin the overcritical regime a dilative behaviour was

q [k

Pa]

p [kPa]

= 0.125

= -0.125 = -0.25 = -0.375

= -0.5 = -0.625

= -0.75

= M

e (c ) = -0.88

= 0.25

= 0.375 = 0.5

= 0.625 = 0.75 = 0.875 = 1

.0 = M

c(

c) =

1.2

5

= 1

.125 = 1

.313

= 1

.375

300100 200

peak shear strength for ID = 0.65

peak shear strength for ID = 0.65

tria

xial

com

pres

sion

tria

xial

ext

ensi

on

Figure 7: Tested average stresses

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1.5

-1.0

-0.5

0.5

0

1.0

1.5

2.0

2.5

3.0

Accumulated volumetric strain acc [%]v

Acc

umul

ated

dev

iato

ric s

trai

n

acc

[%]

q

all tests: Nmax = 105, pav = 200 kPa, ID0 = 0.57 - 0.69, fB = 1 Hz

av = 0.5

av = 0.25

av = 0

av = -0.25

av = -0.5

av = -0.625

av = -0.75

av = -0.815

av = -0.88

av = 0.

75! av = 0

.875

" av =

1.0

# av =

1.1

25$ av

= 1

.25

% av =

1.3

13&

av = 1.375

Figure 8: εaccq -εacc

v -strain paths for pav = 200 kPa anddifferent average stress ratios −0.88 ≤ ηav ≤ 1.375


0 0.2 0.4 0.6 0.8 1.00

0.4

0.8

1.2

all tests: Nmax = 105, = 0.3, ID0 = 0.61 - 0.69, fB = 1 Hz

pav [kPa] =

50 100 150 200 250 300

acc [%]v

acc

[%]

q

1

1/ = 1.43

a)triaxial compression av = 0.75

0 0.1 0.2 0.3 0.4 0.5 0.6-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

all tests: Nmax = 104, = 0.2, ID0 = 0.62 - 0.66, fB = 0.1 Hz

pav [kPa] =

50 100 150 200 250 300

acc [%]v

acc

[%]

q

1

1/ = -0.90

b) triaxial extension av = -0.5


v -strain paths for different average mean pressures: a) tests with ηav = 0.75 (triaxial compression),b) tests with ηav = −0.5 (triaxial extension)

0

0

100

100

200

200

300 400

400

500

300

η = -0.375

η = -0.50

η = -0.88 = Me (

c )

η = -0.25

η = -0.125

η = Me (

p )

η = -0.75

η = -0.625

-100

-200

-300

M c(

p)

η =

1.37

5η

= 1.

313

N = 10 - 20N = 50 - 100N = 500 - 1,000N = 5,000 - 10,000N = 50,000 - 100,000

accumulation between the cycles

η = 0.75

η = 1.

00

η = 0.50

η = 0.25

η =

1.25

= M

c(

c)

dila

tanc

y

dilatancy

0

0

100

100

200

200

300 400

400

300

Average mean pressure pav [kPa] Average mean pressure pav [kPa]

Dev

iato

ric s

tres

s qa

v [k

Pa]

Dev

iato

ric s

tres

s qa

v [k

Pa]

η = -0.375

η = -0.50

η = -0.88 = Me (

c )

η = -0.25

η = -0.125

η = Me (

p )

η = -0.75

η = -0.625

-100

-200

-300

M c(

p)

η =

1.37

5η

= 1.

313

N = 20N = 100N = 1,000N = 10,000N = 100,000

accumulation up to cycle:

η = 0.75

η = 1.

00

η = 0.50

η = 0.25

η =

1.25

= M

c(

c)

all tests: ID0 = 0.57-0.69

all tests: ID0 = 0.57-0.69di

lata

ncy

dilatancy

dens

ifica

tion

dens

ifica

tion

b)a)

vacc

qacc

vacc

qacc

Figure 10: Direction of strain accumulation m for different average stresses σav, shown as a unit vector in the

p-q-plane, a) determined from the total strain accumulated up to a certain number of cycles N , b) determined fromthe strain increments between two data recordings


ω =

acc

/

acc

[-]

vq

1/ω

=

acc

/

acc

[-]

qv

-10

-8

-6

-4

-2

0

2

4

6

8

10

contractancydilatancy

accumulation up to:

N = 100,000 N = 10,000

N = 1,000 N = 100 N = 20

hypoplasticity mod. Cam Clay

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6


Average stress ratio av = qav/pav [-]Average stress ratio av = qav/pav [-]


0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-0.4

0

0.4

0.8

1.2

1.6

2.0

2.4

all tests: pav = 200 kPa, = 0.3,

ID0 = 0.57 - 0.67


ID0 = 0.57 - 0.67contractancydilatancy

av = Mc( c)

accumulation up to: N = 100,000 N = 10,000 N = 1,000 N = 100 N = 20 hypoplasticity mod. Cam Clay

a)b)

d)c)

Ω =

acc

/

acc

[-]

v

q

1/Ω

=

acc

/ ac

c [-

]

qv

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0

0.8

1.6

2.4

3.2


av = Mc( c)


ID0 = 0.57 - 0.67

accumulation between: N = 50,000 - 100,000 N = 5,000 - 10,000 N = 500 - 1,000 N = 50 - 100 N = 10 - 20 hypoplasticity mod. Cam Clay

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6-12

-8

-4

0

4

8

12

16


av = Mc( c)


ID0 = 0.57 - 0.67

accumulation between: N = 50,000 - 100,000 N = 5,000 - 10,000 N = 500 - 1,000 N = 50 - 100 N = 10 - 20 hypoplasticity mod. Cam Clay

av = Mc( c)

Figure 11: Direction of strain accumulation in dependence on the stress ratio ηav for pav = 200 kPa and triaxialcompression: a) strain ratio ω = εacc

v /εaccq , b) reciprocal value 1/ω, c) strain ratio Ω = εacc

v /εaccq , d) reciprocal value

1/Ω

observed (see e.g. the test with ηav = 1.375). The mea-sured dependence m(ηav) agrees well with the results ofthe experiments of Luong [9] and Chang & Whitman [2](Section 3).

A slight increase of the compactive portion of thedirection of strain accumulation with the number ofcycles could be detected. This is depicted e.g. inFig. 8 for the test with ηav = Mc(ϕc) = 1.25, whichshows an increasing deflection of the strain path fromthe vertical. Similar results were obtained by Suikeret al. (2005), who measured an initial dilative andadjacent compactive behaviour for stress cycles withηmin ≈ 0 and ηmax ≈ Mc(ϕp). In our tests, the sameresults were obtained on dry specimens (determinationof lateral strains from local measurements with non-contact displacement transducers) and on the saturatedones (measurement of volume changes via the pore wa-ter). Thus, it is unlikely that this effect is causedby measurement technique errors. Moreover, this ten-dency seems to be independent of the loading frequency

(Section 5.5), i.e. the duration of the test. An increaseof the water level in the measurement pipe, i.e. an in-crease of the squeezed-out water volume (see Fig. 5),due to a densification of the specimen leads to a slightincrease of the back pressure in the specimen and thusto a reduction of the axial and lateral effective stresses,i.e. a reduction of pav at qav = constant. However, thisincrease of the stress ratio ηav should cause an oppo-site rotation of m (decrease of ω with N). Anyway, inthe test with ηav = Mc(ϕc) = 1.25 the accumulation ofvolumetric deformation is small and the water level inthe measurement pipe rarely changes its height.

The performed tests foster the observations byChang & Whitman [2] that for a given ηav, the averagemean pressure pav does (at least in the tested range 50kPa ≤ pav ≤ 300 kPa) hardly influence the direction ofstrain accumulation. This is demonstrated in Fig. 9afor the tests with ηav = 0.75 (triaxial compression) andin Fig. 9b for ηav = -0.5 (triaxial extension). The strainpaths in the εacc

q -εaccv -diagram coincide approximately


-1.0 -0.8 -0.6 -0.4 -0.2 0-5

-4

-3

-2

-1

0

1

accumulation up to: N = 10,000 N = 1,000 N = 100 N = 20 hypoplasticity mod. Cam Clay


all tests: pav = 200 kPa, = 0.05 - 0.2,

ID0 = 0.60 - 0.66

av = Me( c)


Average stress ratio av = qav/pav [-] Average stress ratio av = qav/pav [-]


ω =

acc

/

acc

[-]

vq

1/ω

=

acc

/

acc

[-]

qv

-1.0 -0.8 -0.6 -0.4 -0.2 0-5

-4

-3

-2

-1

0

1


av = Me( c)

a) b)Ω

=

acc

/

acc

[-]

v

q

-1.0 -0.8 -0.6 -0.4 -0.2 0-5

-4

-3

-2

-1

0

1 av = Me( c)

all tests: pav = 200 kPa, = 0.05 - 0.2,

ID0 = 0.60 - 0.66


-1.0 -0.8 -0.6 -0.4 -0.2 0-5

-4

-3

-2

-1

0

1


d)c)

accumulation between: N = 5,000 - 10,000 N = 500 - 1,000 N = 50 - 100 N = 10 - 20 hypoplasticity mod. Cam Clay

accumulation between: N = 5,000 - 10,000 N = 500 - 1,000 N = 50 - 100 N = 10 - 20 hypoplasticity mod. Cam Clay

1/Ω

=

acc

/ ac

c [-

]

qv av = Me( c)

all tests: pav = 200 kPa, = 0.05 - 0.2, ID0 = 0.60 - 0.66

all tests: pav = 200 kPa, = 0.05 - 0.2, ID0 = 0.60 - 0.66

accumulation up to: N = 10,000 N = 1,000 N = 100 N = 20 hypoplasticity mod. Cam Clay

Figure 12: Direction of strain accumulation in dependence on the stress ratio ηav for pav = 200 kPa and triaxialextension: a) strain ratio ω = εacc

v /εaccq , b) reciprocal value 1/ω, c) strain ratio Ω = εacc

v /εaccq , d) reciprocal value

1/Ω

for different average mean pressures pav. From Fig. 9it can be concluded that the increase of εacc

v /εaccq dur-

ing cyclic loading is correlated with the residual strainεacc (due to previous cycles) rather than with N . Al-though all tests have been carried out up to N = 105,the smaller pressures pav lead to a larger strain accu-mulation and hence to a larger compactive portion ofm at equal N .

Figure 10 summarizes the measured directions ofstrain accumulation for all tested average stresses σ

av

and depicts them as unit vectors in the p-q-plane. Theorigin of each vector lies in (pav,qav) of the respectivetest. Figure 10a was generated using the strains ac-cumulated up to a certain number of cycles N . Thus,the inclination of the vectors to the p-axis is 1/ω =εacc

q /εaccv . In Fig. 10b the directions of strain accumu-

lation were calculated from the rates, i.e. the vectorsare inclined by 1/Ω = εacc

q /εaccv . The vectors in Fig. 10

are shown for different numbers of cycles. This is indi-cated by the different grayscales. The rotation of thevectors towards the positive p- or εacc

v -axis with increas-

ing number of cycles is of course more pronounced ifshown with rates (Fig. 10b) than with the total strains(Fig. 10a).

Chang & Whitman [2] reported, that the flow rule ofthe modified Cam clay model, i.e. a model for mono-tonic loading, approximates well also the direction ofstrain accumulation under cyclic loading (Fig. 4b). Fig-ure 11a shows the ratio ω = εacc

v /εaccq as a function of

ηav for the tests with pav = 200 kPa and ηav > 0 (triax-ial compression). For small stress ratios ηav one cannotprecisely estimate ω from this diagram and thereforeFig. 11b contains a corresponding illustration of thereciprocal value 1/ω. Figure 11c and 11d present di-agrams for the ratio of the rates Ω = εacc

v /εaccq or its

reciprocal value 1/Ω, respectively. The scatter is largerin the rate diagrams. Beside the measured directions ofstrain accumulation, also the flow rules of the modifiedCam clay model (Eq. (9)) and the hypoplastic model(see e.g. Eq. (4.73) in [13]) are drawn into Fig. 11. Bothflow rules (for monotonic loading) approximate well thedirection of strain accumulation under cyclic loading.


Analogously to Fig. 11, Fig. 12 contains diagramsfor triaxial extension (ηav < 0) at pav = 200 kPa. Alsofor triaxial extension the modified Cam clay model andthe hypoplastic model deliver a good approximation ofthe flow rule under cyclic loading.

5.2 Influence of the stress/strain loop(span, shape, polarization)

First in-phase stress cycles (σ3 = constant) with pav

= 200 kPa and ηav = 0.75 (triaxial compression) areanalyzed. In the tests the stress amplitude 12 kPa≤ qampl ≤ 80 kPa was varied. The illustration of theεacc

q -εaccv -strain paths in Fig. 13a demonstrates, that

the amplitude of the uniaxial cycles does hardly in-fluence the cyclic flow rule. Similar tests at an averagestress pav = 200 kPa and ηav = -0.5 (triaxial extension,Fig. 13b) confirmed, that the ratio of the volumetricand the deviatoric accumulation rate does not signifi-cantly vary with the stress or strain amplitude.

Also the influence of the polarization of the cycles,i.e. their inclination in the stress or strain space, onm was studied. At pav = 200 kPa and ηav = 0.5in-phase stress cycles with six different inclinations0 ≤ αPQ ≤ 90 in the P -Q-plane were tested. Foreach polarization four or five stress amplitudes 20 kPa≤

√

(P ampl)2 + (Qampl)2 ≤ 100 kPa were studied. Fig-ure 14 presents the εacc

q -εaccv -strain paths. Each dia-

gram contains the paths of the tests with a certain po-larization αPQ. Although the data shows some scatterfor αPQ = 10 and 30, independently of the polariza-tion αPQ, the dependence of the direction of strain ac-

cumulation on the amplitude√

(P ampl)2 + (Qampl)2 ofthe cycles is small. The six diagrams in Fig. 14 providean information about the mean value of the inclination1/ω of the strain paths in the εacc

q -εaccv -plane. The val-

ues lie within 0.97 ≤ 1/ω ≤ 1.27 (i.e. 31% scatter, butthe εacc-dependence of m has to be kept in mind). Nosimple correlation of 1/ω with αPQ is noticeable.

Also the influence of the shape of the stress or straincycles was tested. At pav = 200 kPa and ηav = 0.5different elliptic stress paths in the P -Q-plane weretested. Circular paths with different radii (Fig. 15a),elliptic paths with Qampl = 80 kPa and different spansin the direction parallel to the P -axis (Fig. 15b) as wellas elliptic paths with P ampl = 80 kPa and differentamplitudes along the direction parallel to the Q-axis(Fig. 15c) were tested. The mean inclinations of thestrain path in the εacc

q -εaccv -plane (mean values of each

testing series) lay within 1.15 ≤ 1/ω ≤ 1.31, whichcorresponds to a scatter of 14% (considering individ-ual pairs of tests, this scatter might be even larger).However, there seems to be no simple dependence of

a)

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

0.5

1.0

1.5

2.0 qampl [kPa] = 8070605142312212

1

1/ = 1.44

acc [%]v

acc

[%]

q

all tests: Nmax = 105, pav = 200 kPa, ID0 = 0.58 - 0.61, fB = 1 Hz

triaxial compression av = 0.75

0 0.2 0.4 0.6 0.8 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0

20 30 40

50 45 42.5

qampl [kPa] =

all tests: Nmax = 104, pav = 200 kPa, ID0 = 0.59 - 0.66, fB = 0.1 Hz

acc [%]v

acc

[%]

q 1

1/ = -1.00

b) triaxial extension av = -0.5


v -strain paths for in-phase stress cy-cles with different amplitudes, a) tests with ηav = 0.75(triaxial compression), b) tests with ηav = −0.5 (triax-ial extension)

the inclination 1/ω of the εaccq -εacc

v -strain paths on theshape of the cycles.

We may thus conclude, that the influence of the size,the polarization and the shape of the stress loop (or theresulting strain loop) on m is not strong and may beneglected for the first in an explicit model.

5.3 Changes in the amplitude and theirsequence

Figure 16 presents results of tests, in which four pack-ages of cycles were applied in succession. Each pack-age contained 2.5 · 104 cycles. At an average stresswith pav = 200 kPa and ηav = 0.75 the amplitudesqampl = 20, 40, 60 and 80 kPa were tested. The se-quence of the amplitudes was varied from test to test.The strain paths in the εacc

q -εaccv -plane are depicted in


0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.30

0.1

0.2

0.3

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

20 40 60 80 100

P2+ Q2 = 20 40 60 80 100

P2+ Q2 =

20 40 60 80 100

P2+ Q2 = 20 40 60 80 100

P2+ Q2 =

20 40 60 80

P2+ Q2 = 20 40 60 80

P2+ Q2 =

PQ = 0˚

P

Q

PQ = 10˚

P

Q

PQ = 30˚

P

Q

PQ = 54,7˚

P

Q

PQ = 75˚

P

Q

PQ = 90˚

P

Q

1

1/ = 1.27

1

1/ = 1.23

11/ = 1.04

11/ = 1.12

11/ = 1.05

1

1/ = 0.97

a)

c)

b)

d)

e) f)


v -strain paths for in-phase stress cycles with different polarizations αPQ in the P -Q-plane (alltests: Nmax = 104, pav = 200 kPa, ηav = 0.5, 0.56 ≤ ID0 ≤ 0.64, fB = 0.05 Hz)


0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

0.4

0.8

1.2

1.6

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60

0.4

0.8

1.2

1.6

2.0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

0.4

0.8

1.2

1.6

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

1

1/ = 1.31

1

1/ = 1.15

1

1/ = 1.18

1 2 3 4

Path

4 5 6 7 8

Path

4 9 10 11 12

Path

20406080

20 40 60 804

20406080

20 40 60 806 7 8 45

20406080

20 40 60 80

4

12

10

11

9

1 2 3

av

a)

b)

c)


v -strain paths for different out-of-phase (elliptic) stress paths in the P -Q-plane (all tests: Nmax =104, pav = 200 kPa, ηav = 0.5, 0.58 ≤ ID0 ≤ 0.68, fB = 0.05 Hz)


0 0.2 0.4 0.6 0.80

0.4

0.8

1.2

εacc [%]v

0 0.2 0.4 0.6 0.8

εacc [%]v

εacc

[%]

q

0

0.4

0.8

1.2

εacc

[%]

q

0 0.2 0.4 0.6 0.8

εacc [%]v

0

0.4

0.8

1.2

εacc

[%]

q

0 0.2 0.4 0.6 0.8

εacc [%]v

0

0.4

0.8

1.2

εacc

[%]

q

ID0 = 0.63 ID0 = 0.61

ID0 = 0.60ID0 = 0.58

qampl [kPa] =

80604020

qampl [kPa] =

80604020

qampl [kPa] =

80604020

qampl [kPa] =

80604020

t

q20 40 60 80

t

q20 4060 80

t

q20 4060 80

t

q 20406080

a) b)

c) d)


v -strain paths in tests with four packages of cycles (all tests: Nmax = 105, pav = 200 kPa, ηav =0.75, fB = 0.25 Hz)

Fig. 16. Considering a residual volumetric strain ofεacc

v = 0.7%, the residual deviatoric strain lies withinthe range 1.0% ≤ εacc

q ≤ 1.2%, i.e. the scatter is ap-prox. 20%. A small increase of the ratio εacc

q /εaccv was

measured at the beginning of a package if preceded bypackages with smaller amplitudes.

5.4 Influence of the void ratio / relativedensity

Identical tests at different initial void ratios e0 showsimilar flow directions. This is demonstrated by thestrain paths in Fig. 17. The smaller inclination 1/ω inFig. 17 in comparison to Figs. 9a and 13a can be ex-plained by the larger residual strains in the test withID0 = 0.24. For high initial densities (ID0 > 0.9) thevalues of ω scatter significantly since the ratio is cal-culated as the quotient of two relatively small strains.However, no clear tendency could be detected.

5.5 Influence of the loading frequency

Tests with different loading frequencies 0.05 Hz ≤ fB ≤2 Hz at identical stresses (pav = 200 kPa, ηav = 0.75,qampl = 60 kPa) and with similar initial densities arepresented in Fig. 18. No significant influence of fB onthe cyclic flow rule could be found.

5.6 Influence of the number of cycles

The increase of the compactive portion of the directionof strain accumulation with N was discussed already inSection 5.1 for N = 105 load cycles. Figure 19 showsthe results of two extremely long tests each with 2 · 106

cycles. The compactive portion of m increases also forN > 105.


0 0.5 1.0 1.5 2.0 2.5 3.00

0.5

1.0

1.5

2.0

2.5

3.0

3.5

acc [%]v

acc

[%]

q 1

1/ = 1.23 0.580 / 0.99 0.650 / 0.76 0.674 / 0.67 0.715 / 0.54 0.803 / 0.24e0 /ID0 =

all tests: Nmax = 105, pav = 200 kPa, ηav = 0.75,

= 0.3, fB = 1 Hz


v -strain paths in tests with differentinitial void ratios e0

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

acc [%]v

acc

[%]

q

1

1/ = 1.76

all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.50 - 0.60

0.05 0.1 0.2 0.5 1 2

fB [Hz] =


v -strain paths in tests with differentloading frequencies fB

0 0.5 1.0 1.5 2.00

0.5

1.0

1.5

2.0

2.5

both long-time tests: pav = 200 kPa, av = 0.75, qampl = 60 kPa, fB = 1 Hz

acc [%]v

acc

[%]

q

ID0 = 0.65ID0 = 0.60


v -strain paths in two tests withNmax = 2 · 106 cycles

5.7 Influence of a static (monotonic)preloading

Also the influence of a monotonic preloading on m wastested. The specimens were loaded radially along thep-axis or along a K0-line (with an inclination η = 0.75)up to a certain preloading pressure 100, 200 or 300 kPa,Fig. 20a. After five minutes the samples were unloadedto pav = 100 kPa. Subsequently, 104 cycles with anamplitude qampl = 50 kPa were applied at pav = 100kPa and ηav = 0 or ηav = 0.75, respectively. The testswith ppreload = 100 kPa correspond to a non-preloadedspecimen. Figure 20b makes clear, that the εacc

q -εaccv -

strain paths of the three tests with the K0-preloadingnearly coincide. A similar conclusion can be drawn forthe isotropic preloading. Thus, a monotonic preloadingdoes hardly affect the direction of strain accumulation.

CSL50 100 200 300

225

150

75

37.5

1

1.25

1

= 0.75 (K0 = 0.5)

p [kPa]

q [kPa]

0 0.5 1.0 1.5 2.0-1.0

-0.5

0

0.5

1.0

1.5

2.0

acc [%]v

acc

[%]

q

b)

a)

all tests: Nmax = 104, qampl = 50 kPa, ID0 = 0.56 - 0.66, fB = 0.1 Hz

100 200 300

100 200 300

isotropic: K0:

pprestrain [kPa] =


v -strain paths for a cyclic loadingafter a monotonic (isotropic or K0) preloading


0 0.05 0.10 0.15 0.20 0.25 0.300

0.1

0.2

0.3

0.4

0.5

0 0.4 0.8 1.2 1.6 2.00

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 80

2

4

6

8

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

0.5

1.0

1.5

2.0

a) b)

c) d)

all tests: Nmax = 105, ID0 = 0.58 - 0.61, fB = 1 Hz

qampl [kPa] = 8070605142312212

qampl [kPa] = 7769605143332313


qampl [kPa] = 8777675746362815


qampl [kPa] = 7868595041312113

Grain size distr. No. 3: d50 = 1.45 mm, U = 1.4 Mc( c) = 1.37, av/Mc( c) = 0.55




1

1/ = 1.57

1

1/ = 1.41

1

1/ = 1.44

1

1/ = 1.03

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q

acc [%]v

acc

[%]

q



v -strain paths for different grain size distributions (all tests: pav = 200 kPa, ηav = 0.75)

5.8 Influence of the grain size distribu-tion curve

All preceding remarks on the cyclic flow rule referred tosand with the grain size distribution curve No. 1 (d50 =0.55 mm, U = 1.8, ϕc = 31.2) with respect to Fig. 6.Additionally, the sands with the grain size distributioncurves No. 2 (d50 = 0.35 mm, U = 1.9, ϕc = 33.3),No. 3 (d50 = 1.45 mm, U = 1.4, ϕc = 33.9) and No. 4(d50 = 0.52 mm, U = 4.5, ϕc = 33.3) were tested.In the tests the average stress (pav = 200 kPa, ηav =0.75) was kept constant and the stress amplitude 12kPa ≤ qampl ≤ 87 kPa was varied.

Figure 21 presents the εaccq -εacc

v -strain paths for thesands Nos. 1 to 4. The diagram 21b for sand No. 1repeats Fig. 13a in order to facilitate comparison. De-spite some scatter in the tests on sand No. 3, also for thethree sands tested additionally the cyclic flow rule doeshardly depend on the stress or strain amplitude. Themean inclination of the strain paths in the εacc

q -εaccv -

diagram of the medium coarse sand No. 2 (1/ω = 1.41)

is slightly smaller than the one for the medium coarseto coarse sand No. 1 (1/ω = 1.44). For the coarse sandNo. 3, m is more deviatoric (1/ω = 1.57), for the well-graded soil No. 4 it is more compactive (1/ω = 1.03).However, these differences may be contributed to thedifferent accumulation rates and the resulting differentresidual strains. The larger the residual strains are,the smaller is the mean inclination 1/ω. This becomesevident, if the εacc

q -εaccv -strain paths of the four tested

sands are presented in a common diagram for εaccv ≤ 2%

(Fig. 22). The difference in the strain paths in Fig. 22is relatively small. Thus, it can be concluded that thedirection of strain accumulation does not significantlydepend on the grain size distribution curve.

6 Summary and conclusions

Numerous drained cyclic triaxial tests with many(104 ≤ N ≤ 106) small (εampl ≤ 10−3) cycles demon-strate a unique flow rule m for non-cohesive soils under


0 0.4 0.8 1.2 1.6 2.00

0.5

1.0

1.5

2.0

2.5

acc [%]v

acc

[%]

q

all tests: Nmax = 105, pav = 200 kPa, av = 0.75, ID0 = 0.58 - 0.72, fB = 1 Hz

No. 4

No. 2No. 1

No. 3

Grain size distribution

Figure 22: Summary of the εaccq -εacc

v -strain paths of thefour tested grain size distributions for εacc

v ≤ 2% in acommon diagram

cyclic loading. The observed flow rule depends mainlyon the average stress ratio ηav = qav/pav. A slight in-crease of the compactive portion with the accumulatedstrain εacc (due to the previous cycles) was measured.For ηav = constant, the ratio εacc

v /εaccq does hardly de-

pend on the average mean pressure pav. In the case ofin-phase cycles the influence of the amplitude is small.The direction of the cycles in the stress or strain spacehas some influence on εacc

v /εaccq , but no simple depen-

dence could be found. For out-of-phase cycles (e.g. el-liptic stress paths in the p-q-plane), the dependenceof εacc

v /εaccq on the shape of the stress or strain loop

may be neglected in explicit models. Furthermore, thecyclic flow rule is only slightly affected by the void ra-tio, the loading frequency, the static preloading andthe grain size distribution curve. The flow rule undercyclic loading can be well approximated by flow rulesfor monotonic loading (e.g. by the flow rules of themodified Cam clay model or the hypoplastic model).The flow rule m should become a well established con-cept in explicit modelling of accumulation due to smallcycles.

Acknowledgements

This study was conducted as a part of the project A8”Influence of the fabric change in soil on the lifetime ofstructures”, supported by the German Research Coun-cil (DFG) within the Collaborate Research Centre SFB398 ”Lifetime oriented design concepts”. The authorsare indepted to DFG for this financial support.

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