Strain accumulation in sand due to cyclic loading: drained triaxial tests

Soil Dynamics & Earthquake Engineering (2005, in print)

Strain accumulation in sand due to cyclic loading: drained triaxial tests

T. Wichtmanni), A. Niemunis, Th. Triantafyllidis

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

Abstract

Our aim is the prediction of the accumulation of strain and/or stress under cyclic loading with many (thousandsto millions) cycles and relatively small amplitudes. A high-cycle constitutive model is used for this purpose. Itsformulas are based on numerous cyclic tests. This paper describes drained tests with triaxial compression anduniaxial stress cycles. The influence of the strain amplitude, the average stress, the density, the cyclic preloadinghistory and the grain size distribution on the direction and the intensity of strain accumulation is discussed.

Key words: Cyclic tests; Triaxial compression; Uniaxial cyclic stress paths; Accumulation; Residual strain;Explicit model; Sand

1 Introduction

Cyclic loading leads to an accumulation of strainand/or excess pore water pressures in soils becausethe generated stress or strain loops are not perfectlyclosed. Such cumulative phenomena are of importancein many practical cases. In the seismic areas a lim-ited number of load cycles with large amplitudes (i.e.εampl > 10−3) during an earthquake may lead to anaccumulation of high pore water pressures and con-sequently to a dramatic loss of the bearing capacityof structures (liquefaction). Also offshore foundationsmay loose their bearing capacity in a similar way dur-ing a storm. Under drained conditions if the amplitudeis small (i.e. εampl < 10−3) and the number of cycles ishigh the accumulation of settlements becomes the mainconcern (long-term serviceability, e.g. railways, water-gates, tanks). Some structures are extremely sensitiveto differential settlements, which must be kept withinan extremely small range in order to ensure the opera-tional requirements. In this case an accurate predictionis required for several decades of intensive traffic.

For the finite element (FE) prediction of residualsettlements two different numerical strategies can bedistinguished. The implicit (incremental or low-cycle)models need hundreds of load increments per cycle andthe residual settlement appears as a by-product of clas-

i)Corresponding author. Tel.: + 49-234-3226080; fax: +49-234-3214150; e-mail address: [email protected]

sically calculated stress - strain loops. The accumula-tion results from the fact that the cycles are not per-fectly closed. The implicit approach is suitable for a rel-atively low number of cycles (say N < 50) and generalpurpose constitutive models can be used (usually multi-surface plasticity models, Mroz and Norris [1], Dafalias[2]).

The present research is devoted to another so-calledexplicit (high-cycle) model. In such a model only thefirst two cycles are calculated conventionally with strainincrements (see Fig. 1). The strain amplitude is deter-mined from the second cycle. The first, so-called irregu-

lar cycle is not suitable for this purpose, because it maydiffer significantly from the following so-called regular

ones. The accumulation due to the following cycles isdetermined from a direct (explicit) formula. The mate-rial model predicts the residual strain due to a packageof e.g. ∆N = 20 cycles by the piece without tracingthe oscillating strain path during the individual cycles.The strain amplitude is assumed to be constant duringthis calculation. In order to update the strain ampli-tude after a possible densification or re-distribution ofstress, the explicit calculation can be interrupted by aso-called control cycle (semi-explicit approach). For ahigh number (e.g. several thousands) of cycles explicitmodels turn out to be more useful and accurate be-cause the accumulation of systematic errors in implicitmodels is of the order of the physical accumulation.

1

Soil Dynamics & Earthquake Engineering (2005, in print) 2

t, N

"cyclic creep"explicit accumulation

�

irregular cycle

implicit cycle (calculation of strain amplitude)

regular cycles

Fig. 1: Calculation procedure of explicit models

The explicit constitutive model is based on exper-imental observations. We have performed numerouscyclic triaxial and cyclic multiaxial direct simple sheartests. The results of the tests with triaxial compressionand uniaxial stress cycles are presented in this paper.The multiaxial tests are described in [3].

2 Notation, preliminaries

We use the effective Cauchy stress σ (compression pos-itive) and small strain components. The Roscoe vari-ables are

p = tr σ/3 = (σ1 + 2σ3)/3 (1)

q =√

3/2 ‖σ∗‖ = σ1 − σ3 (2)

and the work conjugated strain rates are

εv = tr ε = ε1 + 2ε3 (3)

εq =√

2/3 ‖ε∗‖ = 2/3 (ε1 − ε3) (4)

The star t∗ designates the deviatoric part of t. We de-note the axial component with the index t1 and the ra-dial components with t2 and t3. Using the word ”rate”we mean the derivative with respect to the number ofcycles N , that is t = ∂ t /∂N and not with respect totime.

Stress obliquity is expressed with η = q/p or us-ing the yield function Y = −I1I2/I3 of Matsuoka andNakai [4] as

Y =Y − 9

Yc − 9with Y =

27(3 + η)

(3 + 2η)(3 − η)(5)

and with Yc = (9 − sin2 ϕc)/(1 − sin2 ϕc). The criticalfriction angle is denoted by ϕc and the one at peak asϕp. The inclinations of the critical state line (CSL, seeFig. 2) are

Mc =6 sin ϕc

3 − sin ϕc

and Me = −6 sin ϕc

3 + sin ϕc

(6)

for compression and extension corresponding both toY = 1.

In all tests the average stress σav was kept constant

and the axial stress component σ1 was cyclically variedwith an amplitude σampl

1 (see Fig. 2). It is convenientto introduce the normalized size of the stress amplitude

ζ = σampl1 /pav = qampl/pav . (7)

q = � 1

pav p

qav

q

11

1Mc( � c)

Mc( � p)

Me( � c)

Me( � p)

� av = qav / pav

average stress � av

ampl ampl

p = � 1ampl ampl / 3

CSL

Fig. 2: State of stress in a cyclic triaxial test

The strains under cyclic loading can be decomposedinto a residual and a resilient portion denoted by thesuperscripts tacc and tampl, respectively. We abbrevi-ate the norm of the former as

εacc = ‖εacc‖ =√

(εacc1 )2 + 2(εacc

3 )2 (8)

with the direction of strain accumulation (cyclic flowrule)

ω = εaccv /εacc

q . (9)

εaccv and εacc

q are the cumulative volumetric and devi-atoric strain components, respectively. The strain am-plitude generated by an uniaxial stress loop is describedby

εampl = ‖εampl‖ (10)

or the volumetric (εamplv ) and deviatoric (εampl

q ) com-ponents. Alternatively also the shear strain amplitudeis used:

γampl =√

3/2 ‖(ε∗)ampl‖ = (ε1 − ε3)ampl (11)

The tests were done at various density indices ID =(emax−e)/(emax−emin) = Dr %d,max/%d (Dr = relativedensity, %d = dry density).


3 Need for a systematic study

Most of the studies on the cyclic behaviour of sandconcern a low number of cycles (e.g. N < 1, 000) andit is unclear, if the observations can be extended tothe high-cycle behaviour. The few studies going be-yond N = 1, 000 are either not well documented (e.g.the strain amplitudes are not available) or their testingprogram is incomplete. Some of the published resultsare contradictory.

There is a general consent that with increasing num-ber of cycles N the residual strain εacc increases with-out a clear limit while the rate of accumulation εacc de-creases. Different approximations of the experimentalcurves εacc(N), however, were proposed. In cyclic triax-ial tests Lentz and Baladi [5,6] observed the increase ofthe residual vertical strain εacc

1 to be proportional to thelogarithm of the number of cycles N . Similarly Suiker[7] found from cyclic triaxial tests on gravel and sandthat the accumulation rate εacc decreases proportionalto 1/N , however, with two proportionality coefficientsc1 for N < 1, 000 and c2 for N > 1, 000 with c1 > c2.Helm et al. [8] observed similar proportionality coeffi-cients but with c2 > c1 (cyclic triaxial tests on mediumcoarse and fine sand). An accumulation faster than thelogarithm of the number of cycles was also observed byGotschol [9] in cyclic triaxial tests on gravel.

The effect of the strain amplitude was mainly stud-ied in direct simple shear (DSS) tests. Youd [10] re-ported a considerable increase of the rate of accumula-tion with increasing shear strain amplitude γampl. Foramplitudes γampl < 10−4, however, he observed no ac-cumulation of residual strain. Similar experimental re-sults were obtained by Silver and Seed [11]. Sawickiand Swidzinski [12,13] performed DSS tests with dif-ferent shear strain amplitudes. For a given initial den-sity they demonstrated a unique curve εacc

v = C1 ln(1+C2N) (which they called ”common compaction curve”)wherein N = N (γampl/2)2. These test results may beaffected by the disadvantages of DSS tests, namely theinhomogeneous distribution of strain over the specimenvolume and the accumulation of lateral stresses, whichare usually not measured (c.f. Budhu [14]). Further-more, the applied number of cycles was rather smallin all these tests. The common compaction curve wasnot confirmed by our triaxial tests with a large numberof cycles, see Section 6. From cyclic triaxial tests per-formed by Marr and Christian [15] one could conclude

a relationship εacc ∼ (σampl1 )b with the exponent b in

the range 1.91 ≤ b ≤ 2.32. Marr and Christian [15]provided data up to 10,000 cycles.

In the above mentioned studies only deviatoric orpre-dominant deviatoric strain amplitudes were ap-plied. Ko and Scott [16] performed complementarytests (on cubical specimens) using isotropic stress am-

plitudes. After a small initial compaction no furtheraccumulation was observed. Ko and Scott [16] sug-gested to neglect the effect of the volumetric amplitudeon the accumulation rate. Experiments of Wichtmannet al. [3] have demonstrated this false. The volumet-ric strain amplitude does affect the accumulation, al-though not as strongly as the deviatoric component.The influence of the space encompassed by the strainloop and its polarization (Pyke et al. [17], Ishihara andYamazaki [18], Yamada and Ishihara [19], Wichtmannet al. [3]) has been reported elsewhere.

Several experimental studies with cyclic simple sheartests (e.g. Silver and Seed [11], Youd [10], Sawicki andSwidzinski [12]) ended with the conclusion that the av-erage mean pressure pav does not influence the accumu-lation of residual strain. Our own tests, Section 7, showa significant influence of the average stress. Marr andChristian [15] observed a slightly higher accumulation

rate at a higher pav (keeping ζ = σampl1 /pav constant)

and an increase of the accumulation rate with the av-erage stress ratio ηav. However, the different strainamplitudes in the tests due to the stress-dependenceof the stiffness were not taken into account (the strainamplitudes may differ significantly with pav or ηav, seeFigs. 11 and 15).

From numerous cyclic simple shear tests (e.g. Silverand Seed [11], Youd [10]) and cyclic triaxial tests (e.g.Marr and Christian [15]) it is evident that the initial soildensity is an important parameter for the prediction ofaccumulation.

The influence of the frequency of cyclic loading onthe accumulation rate was found to be negligible insome studies (Youd [10] for 0.2 ≤ f ≤ 1.9 Hz, Shenton[20] for 0.1 ≤ f ≤ 30 Hz) and to be significant in others(Kempfert et al. [21]).

The direction of accumulation (cyclic flow rule) wasstudied by Luong [22] in drained cyclic triaxial tests.Small packages with 20 load cycles were applied succes-sively to the same sand specimen. The average stresswas varied from package to package and the accumu-lated strain was monitored. The so-called ”character-istic threshold” (CT) line in the p - q - plane waspostulated such that σ

av below the CT line leads todensification and σ

av above the CT line to dilativeaccumulation. Chang and Whitman [23] found thatthe direction of accumulation could be well expressedusing the flow rule of the modified Cam Clay modelω = (Mc

2 − (ηav)2)/(2ηav). They proposed to identifyLuong’s CT line with the critical state line (CSL). TheCT line was shown to be independent of pav, of thestress amplitude and the density. However, no experi-mental data for the cyclic flow rule for N > 1, 050 waspresented. We investigated whether the slight increaseof ω with N in the diagrams of Chang and Whitman


[23] continues beyond N > 1, 050.

4 Testing device, preparation of samples and

tested materials

force transd.

ball bearing

displ. transd.

pressure transds. (pore + cell press.)

differential pressure transducer

cell pressure

back pressure

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

plexiglas cylinder

water in the cell

drainage

pneumatic cylinder

ball bearing

load piston

inner rodsouter rods

membrane

Volume measuring unit

Fig. 3: Scheme of cyclic triaxial test device

In our study we used five triaxial devices of the typepresented in Fig. 3. The axial load was applied by apneumatic loading system and measured at a load cellinside the pressure cell below the lower specimen endplate. The axial deformation of the specimen was mon-itored by a displacement transducer of a high accuracywhich was fixed to the load piston. Due to the smearingof the end plates the bedding error was carefully sub-tracted. The inhomogeneity of the resilient and thecumulative deformation in a triaxial sample was in-vestigated by Niemunis [24] using the PIV-technique.These inhomogeneities remain even after a large num-ber of cycles, both in the amplitude and in the accu-mulation rate. Local axial strain measurements with ashort distance (< 5 cm) between the reference pointscan therefore lead to errors. The integral value of thewhole sample is thought to be more representative (al-though there may be some disturbance at the top ofthe sample).

For technical reasons it was easier to perform testswith saturated specimens than with dry ones. The re-sults do not differ significantly. The volume changesof fully saturated specimens were measured by thesqueezed out pore water using a differential pres-sure transducer. The lateral deformations of the dryspecimens were determined with six local non-contact

displacement transducers (LDT) distributed over thespecimen surface. Cell pressure and back pressure weremonitored by means of pressure transducers.

Specimens were prepared by pluviating dry sand outof a funnel through air into half-cylinder moulds. Thefunnel was being lifted so that the drop height was keptconstant. Different initial densities were achieved byvarying the outlet diameter of the funnel. If the spec-imen was supposed to be water-saturated it was firstflushed with carbon dioxide and then saturated withde-aired water. Back pressures of 200 or 300 kPa wereused. The quality of saturation was approved if B >0.95 (B = parameter of Skempton).

Starting from a small isotropic effective stress, thecell pressure was isotropically increased to σav

3 and thenthe axial stress was raised to σav

1 . During the applica-tion of σ

av the deformations were continuously mea-sured. The cyclic loading with σampl

1 was commencedafter a one-hour consolidation period at σ

av. In all tests(except those on the influence of the loading frequency,Section 9) the specimens were subject to 105 cycles at afrequency of 1 Hz. The signals of all transducers wererecorded over the period of five complete cycles afteran idle (non-recording) phase of ∆N cycles. The gaps∆N between the readings were inversely proportionalto the accumulation rate (increase with N).

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 d50 = 1.45,

U = 1.4

d50 = 0.52, U = 4.5

d50 = 0.55, U = 1.8

12 34

Fig. 4: Tested grain size distribution curves (d50: meangrain diameter, U = d60/d10: uniformity index)

This experimental study was performed with aquartz sand with subrounded grains. In all tests ex-cept those on the influence of the grain size distribution(see Section 11) the sieve curve No. 1 in Fig. 4 with theminimum and maximum void ratios emin = 0.577 andemax = 0.874 (German standard code DIN 18126) anda critical friction angle ϕc = 31.2◦ was used.


5 The high-cycle accumulation model

In this section the explicit accumulation model (seeNiemunis et al. [25]) is briefly presented. The generalstress-strain relation has the form

σ = E : (ε − εacc) (12)

wherein E denotes an elastic stiffness. The rate of strainaccumulation ε

acc is proposed to be

εacc = εacc m (13)

= fp fY fe fπ (fampl f AN

︸︷︷︸

gA

+ fampl f BN

︸︷︷︸

gB

) m

with the direction expressed by the unit tensor m (fortriaxial tests ω =

√

3/2 tr (m)/‖m∗‖ holds) and withthe intensity εacc = ‖εacc‖. While the cyclic flow rulem is a function of the average stress ratio ηav = qav/pav

only, the intensity of strain accumulation εacc dependson the amplitude, the shape and the polarization of thestrain loop, the average stress σ

av, the void ratio e andthe applied number of cycles N , i.e. the cyclic preload-ing history. In the proposed expression (13) the respec-tive functions describe the following contributions:

fampl: amplitude, shape and polarization of thestrain loop, summarized in the scalar εampl

fp: average mean stress pav

fY : average stress ratio Y av

fe: average void ratio e

fAN , fB

N : cyclic preloading= number of cycles N , if εampl = constant

fπ: polarization changes

These functions (except fπ) and the cyclic flow rulem were derived on the basis of the performed cyclictriaxial tests and are discussed in the following sections,see also Table 1.

The number of cycles N is not a suitable state vari-able for cyclic preloading since it contains no informa-tion about the amplitude of the previous cycles. Thus,a new state variable gA =

∫gA dN =

∫fampl fA

N dNwas introduced which considers both N and εampl inthe past.

In this paper only the special case of a one-dimensional cyclic strain path is studied. In order tocapture more complex strain loops the accumulationmodel uses a tensorial amplitude definition (see Niemu-nis [24] or Niemunis et al. [25]). For a one-dimensionalstrain loop, however, this new amplitude definition isidentical to the classical one.

The function fπ was determined from cyclic multiax-ial direct simple shear (CMDSS) tests and is presentedelsewhere (Wichtmann et al. [3]). For the case thatthe polarization of the strain loop does not change dur-ing cyclic loading (e.g. in the performed cyclic triaxialtests) fπ = 1 holds.

Function Mat. constants

fampl =(

εampl/εamplref

)2

εamplref 10−4

f AN = CN1CN2 exp

(

−gA

CN1 fampl

)

CN1 3.4 · 10−4

f BN = CN1CN3 CN2 0.55

CN3 6.0 · 10−5

fp = exp

[

−Cp

(pav

pref

− 1

)]

Cp 0.43

pref 100 kPa

fY = exp(CY Y av

)CY 2.0

fe =(Ce − e)2

1 + e

1 + eref

(Ce − eref)2Ce 0.54

eref 0.874

fπ = 1 if polarization = constant,

else see Wichtmann et al. [3]

Table 1: Summary of the partial functions fi and a list ofthe material constants Ci for the tested sand

6 Influence of the strain amplitude

The effect of the strain amplitude εampl on the accumu-lation rate was studied in a series of tests with stressamplitudes varying between 0.06 ≤ ζ ≤ 0.47 (12 kPa≤ qampl ≤ 94 kPa). The average stress was pav = 200kPa, ηav = 0.75 in all tests and the initial relative den-sity lay within 0.55 ≤ ID0 ≤ 0.64, wherein ”initial”means after the irregular cycle. Specimens were testedin the dry condition.

Keeping ζ(N) constant a decrease of the strain am-plitude εampl with N was noticed (Fig. 5), especiallyfor larger amplitudes and during the first 100 cycles(so-called conditioning phase).

Fig. 6 presents the resulting accumulation curvesεacc(N) in a semi-logarithmical diagram. Only the ac-cumulation during the regular cycles is of interest andthe diagrams in this paper show only this portion. Theaccumulated strain was found to increase almost pro-portionally with the logarithm of the number of cyclesup to N ≈ 10, 000 and then over-proportionally. Theapproximation of these curves by the functions fA

N and

fBN is discussed in Section 10. From Fig. 6 it is obvious

that higher amplitudes cause larger accumulation ratesεacc.

In Fig. 7 the accumulated strain after different num-


100 101 102 103 104 1050

1

2

3

4

5

6 54

12 64 24 74 34 85 44 94

Str

ain

ampl

itude

� ampl

[10-

4 ]

Number of cycles N [-]

all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.55 - 0.64

qampl [kPa] =

Fig. 5: Strain amplitude εampl versus the number of cyclesN

100 101 102 103 104 1050

0.4

0.8

1.2

1.6

2.0

Acc

umul

ated

str

ain

� acc

[%]



qampl [kPa] = 948574645545352412

Fig. 6: Accumulation curves εacc(N) in tests with differentstress amplitudes qampl

bers of cycles is plotted versus the square of the strainamplitude. Since εampl varies slightly with N (Fig. 5)

the mean value εampl = (∫ N

1εampl dN)/N is used in

Fig. 7. The accumulated strain εacc is normalized withthe void ratio function fe (Section 8) in order to removethe effect of the slightly varying initial densities and thedifferent compaction rates in the nine tests. The barover fe means that fe is calculated with a mean value of

the void ratio e = (∫ N

1e dN)/N . The curves in Fig. 7

reveal that the accumulation rate εacc increases propor-tionally to (εampl)2, independently of N . The followingamplitude function

fampl =

(

εampl

εamplref

)2

(14)

is therefore proposed wherein the reference amplitudeis εampl

ref = 10−4. Function (14) is in agreement with the

exponents that have been determined from the tests ofMarr and Christian [15] in terms of stress amplitudes.

0 0.5 1.0 1.5 2.00

2

4

6

8

N = 100,000 N = 50,000 N = 10,000 N = 1,000 N = 100 N = 20


� acc

/ fe

[%]

( � ampl)2 [10-7]

Fig. 7: Accumulated strain εacc/fe as a function of thesquare of the strain amplitude (εampl)2

10-8 10-6 10-4 10-20

0.4

0.8

1.2

1.6

2.0

Acc

umul

ated

str

ain

� acc

[%]


qampl [kPa] = 948574645545352412

N = N (� ampl)2~

Fig. 8: Accumulated strain εacc in dependence on N =N (γampl)2 as proposed by Sawicki and Swidzinski [12,13]

In Fig. 8 the accumulated strain is shown as a func-tion of N = N (γampl)2 and according to the ”commoncompaction curve” proposed by Sawicki and Swidzinski[12,13] the curves εacc(N) should coincide. Evidently,they do not.

Fig. 9 presents the strain ratio ω as a function of thestrain amplitude εampl. At higher values of εampl thestrain ratio is almost independent of εampl. At smallerstrain amplitudes the accumulated volumetric and de-viatoric strains were very small and therefore the valuesof ω could be inaccurate. Thus the direction of accu-mulation is thought to be independent of the strainamplitude. This is in accordance with the experimen-tal results of Chang and Whitman [23] for lower cyclenumbers. Plots of ω over N demonstrate an increase of


the volumetric component of the direction of accumu-lation with the number of cycles.

0

0.4

0.8

1.2

1.6

2.0

0 1 2 3 4

Accumulation up to: N = 20 N = 10,000 N = 100 N = 100,000 N = 1,000


Strain amplitude � ampl [10-4]

Str

ain

ratio

ω =

� acc

/

� acc

[-]

q

v

Fig. 9: Strain ratio ω as a function of the strain amplitudeεampl

7 Influence of the average stress

The average stress was varied between 50 kPa ≤ pav ≤300 kPa and 0.25 ≤ ηav = qav/pav ≤ 1.375 keepingζ = 0.3 constant. Specimens were prepared with theinitial densities 0.57 ≤ ID0 ≤ 0.69 and tested in thesaturated condition. The obtained results contradictthe assumption made by several researchers (Silver andSeed [11], Youd [10], Sawicki and Swidzinski [12]), thatthe average stress does not influence the accumulationprocess.

200 300100

q [k

Pa]

η = 0.75

p [kPa]

200

q [k

Pa]

p [kPa]

CSL

Peak

stre

ngtha) b)

Fig. 10: Stress paths in tests on the influence of the averagestress, a) variation of pav, b) variation of ηav

The cyclic stress paths of a series of six tests withan identical average stress ratio (ηav = 0.75) but differ-ent average mean pressures are presented in Fig. 10a.Figs. 11 - 14 contain the results of these tests. Thestrain amplitudes slightly increased with increasing av-erage mean pressure (Fig. 11) due to the well knowndependence of stiffness G ∼ pn (here: n = 0.75), i.e.slightly underproportional to p.

Fig. 12 presents the accumulated strain as a functionof the average mean pressure pav for different numbers

0 100 200 3000

1

2

3

4

Str

ain

ampl

itude

s [1

0-4]

Average mean pressure pav [kPa]

all tests: ηav = 0.75,

� = 0.3,

ID0 = 0.61 - 0.69

� ampl

� ampl� amplv

� amplq

Fig. 11: Strain amplitudes in tests with different averagemean pressures pav but ζ = const (mean values during 105

cycles)

0 50 100 150 200 250 300 3500

0.5

1.0

1.5

2.0

N = 100,000 N = 50,000 N = 10,000 N = 1,000 N = 100 N = 20


� acc

/ (f a

mpl

f e)

[%] 0.58

0.43

0.320.300.25

0.66

Cp = all tests: ηav = 0.75, � = 0.3,

ID0 = 0.61 - 0.69

Fig. 12: Accumulated strain εacc/(fampl fe) in dependenceon the average mean pressure pav

of cycles. For comparison the residual strain has beenpurified from side effects, i.e. divided by the amplitudefunction fampl and by the void ratio function fe. Itwas interesting to observe that the accumulated strainsignificantly increases with decreasing (!) average meanpressure. This pressure dependence is stronger for largeN . The function fp is proposed in the simple form

fp = exp

[

−Cp

(pav

pref

− 1

)]

. (15)

The atmospheric pressure pref = patm = 100 kPa is usedas a reference for which fp = 1 holds. Equation (15)has been confirmed by the data in Fig. 12 for differ-ent numbers of cycles. The corresponding curves areshown as solid lines in Fig. 12 and the parameter Cp isgiven at each curve. Cp turns out to increase with N .However, in order to keep the number of material con-stants manageable, a constant mean value of Cp = 0.43


is proposed. The diagram in Fig. 13 is similar to that inFig. 12 with the only difference, that the dependencieson Y av (see remarks later in this Section) and N (seeSection 10) have been removed. The curve resultingfrom equation (15) with Cp = 0.43 in Fig. 13 demon-strates, that the loss of accuracy due to the assumptionCp = const is acceptable.

0

0.5

1.0

1.5

2.0

0 50 100 150 200 250 300 350


all tests: ηav = 0.75, � = 0.3, ID0 = 0.61 - 0.69

� acc

/ (f a

mpl

f e f Y

f N)

[-]

N = 100,000 N = 50,000 N = 10,000 N = 1,000 N = 100 N = 20

fp with Cp = 0.43, Eq. (15)

Fig. 13: Accumulated strain εacc/(fampl fe fY fN ) in de-pendence on the average mean pressure pav

The direction of accumulation was found to be rela-tively insensitive to changes in the average mean pres-sure (Fig. 14). A slight increase of the strain ratioω with a decreasing average mean pressure and withN could be observed. The increase with N was ob-served for dry (Fig. 9) and for fully saturated specimens(Fig. 14).

0 50 100 150 200 250 300 3500

0.4

0.8

1.2

1.6Accumulation up to:

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

Str

ain

ratio

ω =

� acc

/

� acc

[-]

q

v


all tests: ηav = 0.75, � = 0.3, ID0 = 0.61 - 0.69

Fig. 14: Strain ratio ω as a function of the average meanpressure pav

The results of a series of eleven tests with 0.375 ≤ηav ≤ 1.375 (0.088 ≤ Y av ≤ 1.243) and pav = 200 kPa= const and ζ = const (see the scheme of the stresspaths in Fig. 10b) are presented in Figs. 15 - 17.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60

1

2

3

4

5

Str

ain

ampl

itude

s [1

0-4]

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

all tests: pav = 200 kPa, � = 0.3, ID0 = 0.57 - 0.67 � ampl

� ampl� amplv

� amplq

Fig. 15: Strain amplitudes in tests with different averagestress ratios ηav (mean values during 105 cycles)

0 0.2 0.4 0.6 0.8 1.0 1.20

0.5

1.0

1.5

2.0

2.5

3.0

N = 100,000 N = 50,000 N = 10,000 N = 1,000 N = 100 N = 20

all tests: pav = 200 kPa, � = 0.3,

ID0 = 0.57 - 0.67 1.9

1.8

1.4

2.1

2.1

1.8

CY =

acc

/ (f a

mpl

f e)

[%]

Average stress ratio Yav [-]

Fig. 16: Accumulated strain εacc/(fampl fe) in dependenceon the average stress ratio Y av

The strain amplitudes slightly decrease with increas-ing ηav as shown in Fig. 15. The accumulation increaseswith Y av, Fig. 16. In the test with Y av = 1.243 (η =1.375) the stress cycles touched the failure line Mc(ϕp)and the rate of accumulation exploded reaching nearly20 % residual strain after 105 cycles. If the stress cyclesdo not surpass this limit the function fY can be quitewell approximated by

fY = exp(CY Y av

)(16)

The material constant CY = 2.0 is approximately inde-pendent of the number of cycles (Fig. 16). For isotropicstresses (Y av = 0) fY = 1 holds.

Fig. 17 shows the strain ratio ω plotted versus ηav.A decrease of ω with ηav is evident. The critical stressratio Mc(ϕc) = 1.25 was determined from static tests.It coincides with a purely deviatoric accumulation. Forηav < Mc(ϕc) densification and above the critical state


0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8-0.4

0

0.4

0.8

1.2

1.6

2.0

2.4Accumulation up to:

N = 100,000 N = 10,000 N = 1,000 N = 100 N = 20 hypoplasticity mod. Cam Clay

all tests: pav = 200 kPa, � = 0.3,

ID0 = 0.57 - 0.67

� av = Mc( � c)

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

Str

ain

ratio

ω =

� acc

/

� acc

[-]

q

v

densification

dilatancy

Fig. 17: Strain ratio ω as a function of average stress ratioηav

line a dilative behaviour was observed. This is in ac-cordance with the experimental results of Luong [22]and Chang and Whitman [23]. The dependence ω(ηav)conforms with the flow rule of the modified Cam Claymodel, as already observed by Chang and Whitman[23], or with the hypoplastic theory, Niemunis [24],Fig. 17. Both flow rules (for monotonous loading) de-scribe well the direction of accumulation observed inthe cyclic triaxial tests, especially for the higher num-bers of cycles, e.g. N = 105.

Next, Fig. 18, average mean pressures pav = 100 kPaand pav = 300 kPa and different stress ratios ηav > 0(ζ = 0.3 = const) were tested in order to study if amore complex description than fp fY is needed. Fig. 18confirms, that the accumulation increases with decreas-

ing average mean pressure pav and increasing deviatoricstress ratio Y av. For pav = 100, 200 and 300 kPa thesame approximation (16) could be used with CY = 2.0.Analogously to Fig. 12, the parameter Cp of Eq. (15)increases with N for all tested average stress ratios ηav

and took values 0.25 ≤ Cp(N = 105) ≤ 0.66. There-fore, strictly speaking, Cp is not a constant. It has lo-cal extremes for ηav = 0.75 (maximum) and ηav = 0.25(minimum). For simplicity we disregard the variabilityof Cp with N and ηav in the accumulation model.

Fig. 19 presents curves of identical residual strainsafter N = 105 cycles with εampl = 3 · 10−4 and ID0 =0.60. In a wide p-q-range the curves εacc = const arealmost parallel to the maximum shear strength inclina-tion η = Mc(ϕp).

The direction of accumulation at different averagestresses σ

av is shown in Fig. 20. It is depicted as a unitvector starting at (pav, qav) and having the inclinationεacc

q /εaccv towards the p-axis.

0100

200

300

0.2

0.4

0.6

0.8

0

0.4

0.8

1.2

� acc /

f am

pl [%

]

Yav [-]p av

[kPa]

0.330

0.107

0.073

0.052

0.138

0.238

0.297

0.628

0.710

0.601

0.450

0.348

0.218

0.129

0.0590.055

0.097

0.117

0.209

0.096

0.156

all tests: � = 0.3,

ID0 = 0.57 - 0.69, N = 100,000

Fig. 18: Accumulated strain εacc after 100,000 load cyclesfor different average stresses σ

av

0 100 200 300 400

100

200

300

400q

= σ

1 -

σ 3 [k

Pa]

p = (σ1 + 2σ3) / 3 [kPa]

1 %

0.7

%

5 %

εampl = 3 10-4,

ID0 = 0.60, N = 100,000

1

1

2 %

0.5

%� ac

c = 0

.4 %

Mc( � c)

Mc( � p)

Fig. 19: Curves of identical accumulation in the p-q-plane

8 Influence of the density

In another series of tests the initial void ratio was variedbetween 0.58 ≤ e0 ≤ 0.80 (0.24 ≤ ID0 ≤ 0.99). Theaverage stress (pav = 200 kPa, ηav = 0.75) and theamplitude ratio (ζ = 0.3) were kept constant this time.The specimens were water-saturated.

The condition ζ = qampl/pav = const implies slightlyhigher strain amplitudes for lower initial densities(Fig. 21), because the stiffness decreases with the voidratio e. As it could be expected, the strain accumu-lation is larger in loose soils (Fig. 22). The following


0

0.1

0.2

0.3

0.4

0.56 0.60 0.64 0.68 0.72 0.76 0.80 void ratio e [-]1.06 0.92 0.79 0.65 0.52 0.38 0.25 relative density ID [-]

N = 100,000 N = 50,000 N = 10,000 N = 1,000 N = 100 N = 20

all tests: pav = 200 kPa, ηav = 0.75, �

= 0.3

Ce = 0.43

Ce = 0.48Ce = 0.51

Ce = 0.53

C e = 0.54C e

= 0.54

εacc

/ f a

mpl

[%]

Fig. 22: Accumulated strain εacc/fampl in dependence on the void ratio e or the relative density index ID, respectively

00

100

100

200

200

300 400

400

500

300

pav [kPa]

qav [k

Pa]

η = 0.75

η = 1.00

η = 0.50

η = 0.25

M c (

� p)

η =

1.37

5

η =

1.31

3

all tests: � = 0.3,

ID0 = 0.57 - 0.69

η =

1.25

= M

c (

� c)

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

Accumulation up to:

Fig. 20: Direction of accumulation in the p-q-plane for dif-ferent average stresses

function was found to describe this dependence:

fe =(Ce − e)2

1 + e

1 + eref

(Ce − eref)2. (17)

The material constant Ce = 0.54 (= void ratio forwhich εacc = 0 holds) is approximately independent ofthe number of cycles (Fig. 22). The reference void ratiocorresponding to fe = 1 is chosen to be eref = emax =0.874.

In Fig. 23 curves of identical accumulation εacc =const are plotted in the e-ln(p)-diagram for N = 105,

Initial void ratio e0 [-]

0.56 0.60 0.64 0.68 0.720

1

2

3

4

5

Str

ain

ampl

itude

s [1

0-4]

all tests: pav = 200 kPa, ηav = 0.75, � = 0.3� ampl

� ampl� amplv

� amplq

Fig. 21: Strain amplitudes in tests with different initial voidratios e0 (mean values during 105 cycles)

10050 200 500

Mean pressure p [kPa]

0.58

0.62

0.66

0.70

Voi

d ra

tio e

[-]

εampl = 3 10-4, ηav = 0.75, N = 100,000

εacc = 2 %

1.5 %

1 %

0.7 %

0.5 %

0.3 %

0.2 %

0.1 %

0.05 %

Fig. 23: Curves of identical accumulation in the e-ln(p)-diagram


εampl = 3 · 10−4 and ID0 = 0.60. From Fig. 23 it isobvious that the rising inclination of the curves withεacc = const is opposite to the inclination of the criti-cal state line. Therefore, it does not seem possible todescribe the accumulation rate by the distance e − ec

of the void ratio from the CSL.

The direction of accumulation was found to be in-dependent of the void ratio for the range of densi-ties studied (Fig. 24). However, for dense specimens(ID0 > 0.90) the scatter in the strain ratios ω was con-siderable (Fig. 24), probably because ω was calculatedas a quotient of very small values (the accumulationwas extremely slow).

0.56 0.60 0.64 0.68 0.720

0.4

0.8

1.2

1.6

Void ratio e [-]


Str

ain

ratio

ω =

� acc

/

� acc

[-]

q

v

all tests: pav = 200 kPa, ηav = 0.75,

� = 0.3

Fig. 24: Strain ratio ω as a function of the void ratio e

9 Influence of the loading frequency

In order to study the effect of the loading frequencysix tests with pav = 200 kPa, ηav = 0.75, ζ = 0.3,0.50 ≤ ID0 ≤ 0.60 and varying loading frequencies 0.05Hz ≤ f ≤ 2 Hz were carried out.

The loading frequency has almost no effect on thestrain amplitudes, Fig. 25. The residual strains pre-sented in Fig. 26 exhibit no dependence on the loadingfrequency. The direction of accumulation is also inde-pendent of f (Fig. 27). These observations coincidewith the work of Youd [10] and Shenton [20] but con-tradict the results of Kempfert et al. [21].

10 Influence of the number of cycles

The accumulation curves εacc(N) normalized by fampl,fp, fY , fe and fπ with fπ = 1 are shown in Fig. 28. Thecurves lie within a band, which can be approximatedby the function fN :

fN = CN1 [ln (1 + CN2N) + CN3N ] (18)

0

1

2

3

4

Str

ain

ampl

itude

s [1

0-4 ]

0.10.05 0.2 0.5 1 2

Loading frequency f [Hz]


� ampl

� ampl� amplv

� amplq

Fig. 25: Strain amplitudes as a function of the loading fre-quency f

0.10.05 0.2 0.5 1 20

0.1

0.2

0.3

0.4

0.5


� acc

/ (f

ampl

f e)

[%]

N = 104

N = 103

N = 100

N = 20


Fig. 26: Residual strain εacc/(fampl fe) for different loadingfrequencies f

0

0.4

0.8

1.2

1.6

0.1 0.2 10.50.05 2




Str

ain

ratio

ω =

� acc

/

� acc

[-]

q

v

Fig. 27: Strain ratio ω versus loading frequency f


fN =CN1CN2

1 + CN2N︸︷︷︸

fA

N

+ CN1CN3︸︷︷︸

fB

N

(19)

with the material constants CN1 = 3.4 · 10−4, CN2 =0.55 and CN3 = 6.0 · 10−5. The function fN consists ofa logarithmic and a linear part. The logarithmic partis pre-dominant for N < 10, 000 (see also Sawicki &Swidzinski [12,13]), however, the linear part is neces-sary for the description of the accumulation at highercycle numbers which is faster than logarithmic. Thefunction fN could be also fitted to the results of testsup to Nmax = 2 · 106.

0

0.2

0.4

0.6

0.8 tests for fampl tests for fp tests for fY tests for fe fN, Eq. (18)

over-logarithmic accumulation

logarithmic accumulation

100 101 102 103 104 105


� acc

/ (f a

mpl

f p f Y

f e f

� ) [%

]

Fig. 28: Normalized accumulation curves

As already mentioned the number of cycles N aloneis not a suitable state variable for the description ofcyclic preloading. The amplitude of the previous cy-cles is also of importance. To simplify the matter theaccumulation rate was splitted into a rate of struc-tural accumulation (∼ fA

N ) and a basic rate (∼ fBN ).

gA =∫

gA dN =∫

fampl fAN dN is used as a state vari-

able for cyclic preloading depending on εampl and N .Replacing N in Eq. (19) by gA one obtains

fAN = CN1CN2 exp

(

−gA

CN1 fampl

)

(20)

The severe problem of the determination of cyclicpreloading in situ is discussed by Triantafyllidis andNiemunis [26], Wichtmann and Triantafyllidis [27,28]and Triantafyllidis et al. [29].

11 Influence of the grain size distribution

The presented tests on the uniformly graded sand No. 1(Fig. 4) are being supplemented by tests on the grada-tions 2 and 3 (Fig. 4), which have similar uniformityindices 1.4 ≤ U = d60/d10 ≤ 1.9 but different meangrain diameters 0.35 mm ≤ d50 ≤ 1.45 mm. Also tests

on the well-graded sand No. 4 (U = 4.5, Fig. 4) arebeing conducted, which has a similar d50 compared tosand No. 1. Figure 29 presents the accumulation curvesfor an identical average stress, identical initial densitiesand similar strain amplitudes. The grain size distribu-tion curve strongly affects the accumulation rate. Anincrease of εacc with a decreasing mean diameter d50 atU = const was found. At d50 = const the intensity ofaccumulation significantly increases with U .

Up till now the presented accumulation model candescribe the behaviour of all sands by simply modify-ing the material constants. The tests on different gra-dations will be carried on in future. We expect thatbased on those tests we will be able to simplify themodel (some constants might become redundant) or atleast to correlate most constants to the granular prop-erties (grain size distribution, grain shape, content offines).

100 101 102 103 104 1050

1

2

3

4

5

6

7

Acc

umul

ated

str

ain

� acc

[%]


all tests: pav = 200 kPa, � av = 0.75, ID0 = 0.62 - 0.71,

� ampl = 2.7 - 3.0 10-4

No. 2: d50 = 0.35 mm, U = 1.9 No. 4: d50 = 0.52 mm, U = 4.5

No. 1: d50 = 0.55 mm, U = 1.8 No. 3: d50 = 1.45 mm, U = 1.4

Gradation:

Fig. 29: Accumulation curves εacc(N) for different grada-tions

12 Summary, conclusions and outlook

In order to develop a high-cycle model for the accumu-lation of strain and/or stress in sand under cyclic load-ing numerous cyclic triaxial tests and cyclic multiax-ial direct simple shear (CMDSS) tests were performed.This paper describes the model in brief and concen-trates on the experimental results from the tests withtriaxial compression and uniaxial stress cycles. Themain conclusions from these tests are:

• The direction of accumulation (cyclic flow rule)does exclusively depend on the average stress ratioηav = qav/pav. A slight increase of the volumetriccomponent of the direction of accumulation withthe number of cycles N was observed.

• The direction of accumulation is in agreement withthe flow rule of material models for monotonous


loading (modified Cam clay model, hypoplasticmodel).

• The accumulation rate is proportional to thesquare of the strain amplitude, εacc ∼ (εampl)2,at least within the range of the tested amplitudes0.5 · 10−4 ≤ εampl ≤ 5 · 10−4.

• Our triaxial tests contradict the so-called ”com-mon compaction curve” (Sawicki and Swidzinski[12,13]).

• The accumulation rate εacc increases with decreas-ing average mean pressure pav and with increasingaverage stress ratio ηav.

• εacc grows with increasing void ratio.

• εacc does not depend on the loading frequency(0.05 Hz ≤ f ≤ 2 Hz).

• For N > 104 the accumulated strain εacc increasesfaster than the logarithm of the number of cyclesN .

• The intensity of accumulation εacc is stronglyaffected by cyclic preloading.

• The accumulation rate depends significantly on thegrain size distribution curve. An increase of εacc

with a decreasing mean grain diameter d50 wasfound. A well-graded soil is compacted much fasterthan a poorly-graded one.

At present we are extending the model for triaxialextension. Furthermore, the influence of the polariza-tion and the shape of the strain loop on the accumula-tion rate is being tested. Results of preliminary cyclictriaxial tests with varying stress loops in the p - q -plane (uniaxial paths with different inclinations as wellas elliptic ones) and some CMDSS test data can befound in [3]. Also the boundaries of the validity of theexplicit formulas have to be checked and extended infurther experiments (e.g. it was found that the relation-ship εacc ∼ (εampl)2 overestimates the accumulation forεampl > 10−3). A correlation of the material constantsof the accumulation model with the grain character-istics (grain size distribution, grain shape, content offines) will be tried out.

Acknowledgements

The authors are grateful to DFG (German ResearchCouncil) for the financial support. This study is a partof the project A8 ”Influence of the fabric change insoil on the lifetime of structures” of SFB 398 ”Lifetimeoriented design concepts”.

References

[1] Mroz Z, Norris VA. Elastoplastic and Viscoplastic Consti-tutive Models for Soil with Application to Cyclic Loading. SoilMechanics - Transient and Cyclic Loads. Constitutive Relationsand Numerical Treatment, Pande GN, Zienkiewicz OC (eds.),John Wiley and Sons, 1982:173-218.

[2] Dafalias, YF. Bounding surface elastoplasticity-viscoplasticity for particulate cohesive media. Deformation andFailure of Granular Materials, Proceedings IUTAM Conf., Delft,1982:97-107.

[3] Wichtmann T, Niemunis A, Triantafyllidis T. The effectof volumetric and out-of-phase cyclic loading on strain accumu-lation. Cyclic Behaviour of Soils and Liquefaction Phenomena,Proc. of CBS04, Bochum, March/April 2004, Balkema, 2004:247-56.

[4] Matsuoka H, Nakai T. A new failure for soils in three-dimensional stresses. Deformation and Failure of Granular Ma-terials, Proc. IUTAM Symp. Delft, 1982:253-63.

[5] Lentz RW, Baladi GY. Simplified procedure to character-ize permanent strain in sand subjected to cyclic loading. Inter-national Symposium on soils under cyclic and transient loading,1980:89-95.

[6] Lentz RW, Baladi GY. Constitutive equation for perma-nent strain of sand subjected to cyclic loading. TransportationResearch Record, 1981;810:50-54.

[7] Suiker ASJ. Static and cyclic loading experiments on non-cohesive granular materials. Report No. 1-99-DUT-1, TU Delft,1999.

[8] Helm J, Laue J, Triantafyllidis T. Untersuchungen an derRUB zur Verformungsentwicklung von Boden unter zyklischenBelastungen. Beitrage zum Workshop: Boden unter fast zyk-lischer Belastung: Erfahrungen und Forschungsergebnisse, In-stitute of Soil Mechanics and Foundation Engineering, Ruhr-University Bochum, Issue No. 32, 2000:201-22.

[9] Gotschol A. Veranderlich elastisches und plastischesVerhalten nichtbindiger Boden und Schotter unter zyklisch-dynamischer Beanspruchung. PhD thesis, University Kassel,2000.

[10] Youd TL. Compaction of sands by repeated shear strain-ing. Journal of the Soil Mechanics and Foundations Division,ASCE, 1972;98(SM7):709-25.

[11] Silver ML, Seed HB. Volume changes in sands duringcyclic loading. Journal of the Soil Mechanics and FoundationsDivision, ASCE, 1971;97(SM9):1171-82.

[12] Sawicki A, Swidzinski W. Compaction curve as one ofbasic characteristics of granular soils. Proc. of 4th ColloqueFranco-Polonais de Mechanique des Sols Appliquee, Grenoble,1987;1:103-15.

[13] Sawicki A, Swidzinski W. Mechanics of a sandy subsoilsubjected to cyclic loadings. International Journal For NumericalAnd Analytical Methods in Geomechanics, 1989;13:511-29.

[14] Budhu M. Nonuniformities imposed by simple shear ap-paratus. Canadian Geotechnical Journal, 1984;20:125-37.

[15] Marr WA, Christian JT. Permanent displacements dueto cyclic wave loading. Journal of the Geotechnical EngineeringDivision, ASCE, 1981;107(GT8):1129-49.

[16] Ko HY, Scott, RF. Deformation of sand in hydrostaticcompression. Journal of the Soil Mechanics and FoundationsDivision, ASCE, 1967;93(SM3):137-56.


[17] Pyke R, Seed HB, Chan CK. Settlement of sands undermultidirectional shaking. Journal of the Geotechnical Engineer-ing Division, 1975;101(GT4):379-98.

[18] Ishihara K, Yamazaki F. Cyclic simple shear tests on sat-urated sand in multi-directional loading. Soils and Foundations,1980;20(1):45-59.

[19] Yamada Y, Ishihara K. Yielding of loose sand inthree-dimensional stress conditions. Soils and Foundations,1982;22(3):15-31.

[20] Shenton MJ. Deformation of railway ballast under re-peated loading conditions. Railroad track mechanics and tech-nology, Pergamon Press, 1978:405-25.

[21] Kempfert HG, Gotschol A, Stocker T. Kombiniert zyk-lische und dynamische Elementversuche zur Beschreibung desKurz- und Langzeitverhaltens von Schotter und granularenBoden. Beitrage zum Workshop: Boden unter fast zyklischerBelastung: Erfahrungen und Forschungsergebnisse, Institute ofSoil Mechanics and Foundation Engineering, Ruhr-UniversityBochum, Issue No. 32, 2000:241-54.

[22] Luong MP. Mechanical aspects and thermal effects of co-hesionless soils under cyclic and transient loading. Proc. IUTAMConf. on Deformation and Failure of Granular Materials, Delft,1982:239-46.

[23] Chang CS, Whitman RV. Drained permanent deformationof sand due to cyclic loading. Journal of Geotechnical Engineer-ing, ASCE, 1988;114(10):1164-80.

[24] Niemunis A. Extended hypoplastic models for soils. Ha-bilitation. Institute of Soil Mechanics and Foundation Engineer-ing, Ruhr-University Bochum, Issue No. 34, 2003.

[25] Niemunis A, Wichtmann T, Triantafyllidis T. Explicitaccumulation model for cyclic loading. Cyclic Behaviour ofSoils and Liquefaction Phenomena, Proc. of CBS04, Bochum,March/April 2004, Balkema, 2004:65-76.

[26] Triantafyllidis T, Niemunis A. Offene Fragen zur Mod-ellierung des zyklischen Verhaltens von nicht-bindigen Boden.Beitrage zum Workshop: Boden unter fast zyklischer Belastung:Erfahrungen und Forschungsergebnisse, Institute of Soil Mechan-ics and Foundation Engineering, Ruhr-University Bochum, IssueNo. 32, 2000:109-34.

[27] Wichtmann T, Triantafyllidis T. Influence of a cyclic anddynamic loading history on dynamic properties of dry sand, partI: cyclic and dynamic torsional prestraining. Soil Dynamics andEarthquake Engineering, 2004;24:127-47.

[28] Wichtmann T, Triantafyllidis T. Influence of a cyclic anddynamic loading history on dynamic properties of dry sand, partII: cyclic axial preloading. Soil Dynamics and Earthquake Engi-neering, accepted for publication.

[29] Triantafyllidis T, Wichtmann T, Niemunis A. On thedetermination of cyclic strain history. Cyclic Behaviour ofSoils and Liquefaction Phenomena, Proc. of CBS04, Bochum,March/April 2004, Balkema, 2004:321-32.

Strain accumulation in sand due to cyclic loading: drained triaxial tests

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