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Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

A computational procedure for the prediction of settlement ingranular materials under cyclic loading

A. Karrech a, D. Duhamel a,*, G. Bonnet b, J.N. Roux c, F. Chevoir c, J. Canou d,J.C. Dupla d, K. Sab a

a Laboratoire Analyse des Materiaux et Identification, Ecole Nationale des Ponts et Chaussees (ENPC), Institut Navier (IN), 6 et 8 Avenue Blaise Pascal,

Cite Descartes, Champs sur Marne, 77455 Marne-La-Vallee, Cedex 2, Franceb Laboratoire de Mecanique, Universite de Marne-La-Vallee, France

c Laboratoire des Materiaux et Structures du Genie Civil, ENPC, IN, Franced Centre d’Enseignement et de Recherche en Mecanique des Sols, ENPC, IN, France

Received 17 July 2006; received in revised form 29 June 2007; accepted 5 July 2007Available online 10 August 2007

Abstract

Granular materials have been subject of considerable interest in the recent years because of the richness of their physical behavior.Coupled with experimental approaches, the discrete element methods are now recognized as powerful tools to understand several mech-anisms related to these materials such as convection, compaction and settlement. However, these methods are very costly in terms ofcalculation time.

In this paper, we introduce a computational method for the simulation of three-dimensional granular bed responses under long termcyclic loading. It sequentially uses a molecular dynamics scheme, a time averaging technique, and a relaxation method in order to predictthe long term flow. The suggested approach is then applied to specific cases in order to verify its efficiency and accuracy.� 2007 Elsevier B.V. All rights reserved.

Keywords: Settlement; Granular materials; Molecular dynamics; Time averaging; Relaxation

1. Introduction

Granular materials have been found to be very usefulfor road pavements and railway track platforms, becauseof their porosity, vibrational characteristics and rigidity.However, with the increase in vehicle speeds and high com-fort standards, a better understanding of granular behaviorunder dynamic loading is required. Therefore, accurateprediction of local long term residual displacements andinduced forces is necessary in assessing the capacity ofinfrastructures and preventing their differential settlement.These types of materials can exhibit astonishing androughly understood phenomena under vibration. Severalexperimental devices and numerical methods were devel-

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.07.011

* Corresponding author. Tel.: +33 1 64 15 37 28; fax: +33 1 64 15 37 41.E-mail address: duhamel@lami.enpc.fr (D. Duhamel).

oped in order to investigate the behavior of granular mate-rials subjected to cyclic loading. In the early seventiesShenton [1] studied granular residual deformations underrepeated loading using triaxial testing. He obtained anempirical logarithmic law which describes granular settle-ment with respect to the number of cycles. Guerin [2] devel-oped an experimental setup and studied the settlement ofdimensionally reduced ballast samples. Results showedthe existence of two settlement phases, the first one takesplace at the beginning of the process and corresponds tothe compaction regime whereas the second takes placebeyond a relatively high number of cycles; it correspondsto the stabilized regime. Testing results showed that thesecond regime can be described with a power law relatingthe elastic deflexion to the settlement rate with respect tothe number of cycles. Later on, Bodin [3] studied the samephenomenon under vertical and lateral excitations, she

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 81

ended up with a similar empirical law which includes a cou-pling effect between lateral and vertical permanent dis-placements. More recently, Al-Shaer [4] produced adimensionally reduced railway platform, on which he stud-ied the granular settlement in the vertical direction for dif-ferent train speeds. The obtained results show that thesettlement speed increases non-linearly with respect to theacceleration intensity. In addition, the power laws obtainedby Guerin [2] and Bodin [3] at relatively low accelerationintensities (accelerations lower than the 10 ms�2) are nolonger valid.

In general, experimental tests provide reliable resultsthat can describe the global behavior of granular materials.However, in many cases experiments are limited in terms ofnumber of geometrical and mechanical parameters. Inaddition, without a high degree of sophistication, the exist-ing experimental setups do not allow access to local stress,displacement and acceleration fields, thereby, they jeopar-dize the understanding of the granular behavior. Numeri-cal methods can represent trustworthy complements. Oneof the recent interesting models specially developed forgranular materials was suggested by Nguyen [5]. The par-ticularity of this phenomenological continuum model liesin its unidirectional aspect. The non-tension effectsobserved in granular materials are considered when devel-oping the constitutive law. More recently, Abdelkrim [6]suggested a computational step by step algorithm whichuses a cyclic constitutive law relating the progressive accu-mulation of irreversible displacement to the cyclic stressgenerated by the traffic loading, applied it to the granularmaterials considered as a continuum elastic medium, andpredicted the permanent displacement of the followingstep. When repeated for a large number of cycles, this pro-cedure can produce interesting results in a relatively limitednumber of iterations. However, this algorithm assumes theexistence of residual stresses under repeated loads. Thisassumption can be relaxed in case of discrete element anal-ysis as permanent deformations are not necessarily relatedto permanent stresses, but can be attributed to the granularrearrangement, and migration under repeated loading.Saussine et al. [7] used a full contact dynamic to describe2D polygonal granular materials settlement under repeatedloading and showed an interesting agreement between thediscrete element approach and the experimental results.

According to the above mentioned studies, granularmaterials settlement under repeated loading can be due tosome or each one of the following issues: (i) material com-paction due to the cyclic loading, (ii) migration and flow ofgranular particles to less loaded zones, and (iii) granularoverwork and failure, (iv) lateral and vertical loading cou-pling effect [8,9]. In this paper, we focus only on the firsttwo issues. It is assumed that the ground/granular interfaceoverlapping is negligible as compared to the permanentdeformation. The granular particles are assumed to be rigidwith elastic contacts and the excitation is taken as purelyvertical. With the advent of discrete element methods, itis possible to simulate complex flows of granular materials

and to cover a high range of parameters. Besides, unlike thecontinuum models which rely on homogenization tech-niques [10], by definition discrete methods allow indepen-dent particles flow. Therefore, granular rearrangementscan be observed and quantified in order to evaluate theireffect in the settlement mechanism.

Since its introduction by Cundall and Strack [11] formechanical engineering applications, the molecular dynam-ics (MD) method is gaining momentum and becoming aninteresting computational tool for granular materials anal-ysis. Associated with appropriate time averaging and relax-ation technique, the molecular dynamics represents aninteresting mean for studying the settlement mechanismunder dynamic loading, in more detail than the experimen-tal tests would usually allow.

The present paper, therefore, introduces a computa-tional method which associates a molecular dynamicsscheme, the engine of the suggested method, with timeaveraging and relaxation techniques, in order to describethe granular settlement under dynamic loading. In Section2, the molecular dynamics scheme adopted for this study isbriefly introduced. In addition, a particular interest is givento the contact models which govern the granular interac-tions. In Section 3, the estimation technique is explainedand tested in terms of agreement with full calculations. InSection 4, the relaxation method is derived and explainedthrough two simple examples. Finally, the whole procedureis presented in a sequential flow chart in Section 5, more-over, results regarding the convergence of the suggestedmethod and settlement mechanisms are presented for dif-ferent field parameters.

2. Discrete element simulation model

Consider a system S of N isotropic rigid particles withelastic contacts, equal mechanical characteristics, ran-domly distributed diameters db2[j1, Nj]. These particles areplaced into a cylindrical container of diameter D. The ratioD/d is high enough to avoid any size effects. In our simula-tions, this parameter is about 25, unless differently indi-cated. The container is open from the top in order toapply the appropriate boundary conditions according tothe desired modes of vibration.

2.1. Equations of motion

The molecular dynamics, as adopted by the mechanicalengineering researchers [11–13] consist of modeling the gran-ular motion using Newton equations, in order to describe theflow. For a given particle, b of mass mb and moment of iner-tia Ib, the equation of motion can be written as follows:

F b þP

a6¼b2SF ab ¼ mb€rb;

Mb þP

a6¼b2SMab ¼ Ib

€hb;

8><>: ð1Þ

82 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

where (Fb, Mb) represent the external actions, (Fab, Mab)denote the contact actions of a particle a on b, andð€rb; €hbÞ are the particle accelerations. Several numerical fi-nite difference schemes such as Verlet, leap-frog, or predic-tor–corrector algorithms can be used to integrate the abovementioned second-order equations of motion. The predic-tor–corrector algorithm is used herein. It consists of firstpredicting the displacements and velocities using a Taylordevelopment:

rpbðt þ DtÞ ¼ rbðtÞ þ _rbðtÞDt þ €rbðtÞ

Dt2

2þ OððDtÞ3Þ;

_rpbðt þ DtÞ ¼ _rbðtÞ þ €rbðtÞDt þ OððDtÞ2Þ:

ð2Þ

The predicted configuration is used to calculate the con-tact interactions at t + Dt. The resulting accelerations aregenerally different than the predicted accelerations. The dif-ferences between them can be written as follows:

Da ¼ €rcbðt þ DtÞ � €rp

bðt þ DtÞ: ð3Þ

The difference calculated in the above mentioned equationis used to correct the predicted fields. The new configura-tion as well as its corresponding velocities and accelerationsread:

rcbðt þ DtÞ ¼ rp

bðt þ DtÞ þ c0

Dt2

2Da;

_rcbðt þ DtÞ ¼ _rp

bðt þ DtÞ þ c1DtDa;

€rcbðt þ DtÞ ¼ €rp

bðt þ DtÞ þ c2Da:

ð4Þ

According to Allen and Tildesley [14], the correctorcoefficients are evaluated in order to optimize the algorithmconvergence. In the case of three order predictor–correctorscheme, the coefficients are c0 = 0, c1 = 1 and c2 = 1.

1 A grey to dark rock composed principally of plagioclase feldspar,hornblende, and/or pyroxene. Because of its hardness, diorite is widelyused in road pavements and railway track platforms.

2.2. Contact modeling

Granular materials represent the assemblies of solid dis-crete particles. At the microstructural level, the granularbehavior is mainly governed by the contact nature betweenthe material constituents. Consequently, the description ofthe granular flow highly depends on the understandingof the granular interactions. The interactions consideredherein are dissipative, non-linear, and loading path depen-dant. In the normal direction, Hertz–Mindlin law [15,16], isused to describe the elastic normal interactions:

F en ¼

4

3E�

ffiffiffiffiffiR�p

d3=2n : ð5Þ

The equivalent Young modulus between two particles aand b is defined by E� ¼ E

2ð1�m2Þ, and the equivalent particle

radius is defined by 1R� ¼ 1

Raþ 1

Rb. The Hertz law induces a

circular contact surface of radius, a, which depends onthe normal penetration dn. Assuming a uniform tangentialdisplacement, uncoupled normal and tangential actions,and no slip at the contact interface, Johnson [17] showed

that the tangential elastic contact force can be written asfollows:

F et ¼ 8G�

ffiffiffiffiffiR�p

d1=2n dt: ð6Þ

The equivalent shear modulus between two particles a andb is defined by G� ¼ G

2ð2�mÞ. Note that the advantage of theabove mentioned tangential contact law is its easy imple-mentation as compared to more realistic but complex tan-gential contact models which are available in the literature[18]. The tangential displacement is evaluated using the rel-ative tangential velocity of two overlapping particles sincet0, dt ¼

R t

t0vtab

dt. The normal and tangential forces actingon the particles also enclose viscous terms that depend onthe mass and normal velocity. In addition, a coulomb fric-tion is included through the threshold Ft = lFn, where l isthe friction coefficient. Finally, the forces acting on a par-ticle a can be written as follows:

F n ¼ F en �

m2

cnvn;

F t ¼F e

t � m2ctvt if kF tk 6 lF n;

signðF et ÞlF n � m

2ctvt if kF tk > lF n:

� ð7Þ

The phenomenological constant coefficients cn,t describethe normal and tangential viscous damping. In case of lin-ear contact, these coefficients are related to the restitutioncoefficients en,t and can be written as cn;t ¼ 2an;t

ffiffiffiffiffiffiffiffiffiffimkn;t

p,

where an,t represent the dissipation coefficients defined byan,t = �ln(en,t)/(p

2 + ln2(en,t))1/2 as derived by Da Cruz

[13]. In our case, the viscous dissipative damping is as-sumed to remain proportional to

ffiffiffiffiffiffiffiffiffiffimkn;t

p, and the propor-

tionality constants an,t are introduced as input parameters.Roux and Chevoir [19] and Elata and Berryman [20]

showed that in case of Hertz–Mindlin contacts, some load-ing paths can generate fictive energy in the system. In orderto address this issue, when the normal effort Fn decreasesby DFn, the value of the tangential force when the normalforce was Fn � DFn is assigned to Ft. These contact actionsare embedded in each iteration, in order to update the cor-rection terms of the predictor–corrector algorithm.

2.3. Granular sample preparation

The system parameters used for simulations are summa-rized in Table 1. It is worthwhile noticing that the surfaceinteraction between the granular particles and the wallare modeled using the same contact equations presentedin Section 2.1, where the wall is assumed to be made of steelwith a Young modulus, E = 210 GPa and a Poisson’s ratio,m = 0.3. The granular particles are made of diorite1. UnlikeAlanso-Marroquin and Hermann [21] and Sitharam [22]who used reduced rigidities which produce time steps from1�3 s to 1�5 s, the simulations conducted in this study usethe material properties, contact parameters and dimensions

Table 1System parameters and material properties used for simulation

Dimensions Properties

Initial relative density /i 0.5615 Particle density qp 2710 kg/m3

Mean particle radius �r 3.2 mm Particle Young modulus E 46.9 GPaRadius standard deviation rr 5.6510�4 Particle Poisson’s ratio m 0.25Radius of the cylindrical container R 75 mm Friction coefficient l 0.5Depth of the sample H 60 mm viscous coefficients an, at 0.44

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 83

presented in Table 1. This means that the step time is ofabout 10�7 s. This step time is of the same order as theone used by Lobo-Gerrero et al. [23], it ensures realisticdeformation and granular flow with respect to time(Fig. 1).

Although the particles are randomly distributed in termsof positions and shapes, the cylindrical sample is assumedto be symmetric with respect to x and y axes, as can be seenin Fig. 2. Therefore, only one quarter of the sample is mod-eled. The above mentioned figure also shows the boundaryconditions concerning the planes of symmetry (ux = 0 atx = 0 and uy = 0 at y = 0) and the wall frontier (wall-par-ticles interactions will be described later). The cyclic loadis described with a sinusoidal law as can be seen in theabove mentioned Fig. 2.

We first start with the description of the sample prepa-ration. The preparation process consists in starting withan initial configuration where the particles are randomlyspread in the container with an initial density /i of about0.05. The particles are then subjected to the gravity fieldin order to settle them down until the full equilibrium. Inorder to avoid any instability due to the preparation pro-cess, an equilibrium criterion in terms of the potential

(Ep(t)) and kinetic (Ec(t)) energies is adopted, kEcðtÞkkEpðtÞk 6 �,

where � is a small constant of about 10�6. At the end ofthe preparation process, the ratio defined beforehand, thecoordination number, the apparent weight and the densityof the sample are stable (Fig. 3).

As preliminary verifications, it can be noticed in Fig. 3athat the apparent weight is close to 13.6 N, the cumulativeweight of the granular sample. The small difference is dueto the reaction forces at the wall–sample interface [24].On the other hand, Fig. 3b shows the variation of the coor-dination number versus time, and it obviously increases

Fig. 1. Contact model for deformable bodies.

with respect to time. In addition, it can be seen that thecoordination number approaches 5, when the sample stabi-lizes. Actually, in order to obtain a stable sample, each par-ticle should have a sufficient number of constraints.According to Alexander [25], in a frictionless assembly,the coordination number in a D dimensional space shouldbe at least equal to z ¼ 2D. However, in a frictional assem-bly, Edwards [26] asserted that this number is at least equalto z ¼ 1þD. The samples prepared herein are subjected toa friction coefficient of 0.5, therefore the obtained resultsare in agreement with the above mentioned propositions.According to Suiker and Fleck [27], the coordination num-ber of 3D granular systems depends on the friction coeffi-cient. In addition, it falls between 4 and 6 when theparticles are relatively rigid, monodisperse, rotating, andin neutral equilibrium. This behavior is mainly due to thefrictional energy dissipation which increases with l andinduces a slowing down of the rearrangement process. InFig. 3c the granular relative density is evaluated withrespect to time. It can be noticed that the relative densitystabilizes at the end of the preparation process and asymp-totically tends to /f = 0.532. The former value justifies theuse of the material properties presented in Table 1. Actu-ally, the authors noticed that reducing the material proper-ties results in soft contact stiffness and induces high particlepenetrations. This leads to densities which are too highwith no physical meaning. As a matter of fact, a randomlydistributed granular package can be stable under loadingstarting from a compaction relative density /lp of0.555 ± 0.005 (loose packing [28]) until /dp = 0.64 (densepacking). Crystalline solids with consistent three-dimen-sional orders in their internal structure can reach a maxi-mum relative density of /cl = 0.74. The former valueseems to be the limit that cannot be overstepped for realmaterials. In practice, densities from /dp and /cl are diffi-cult to reach unless artificially arranged in order to get anordered structure.

3. Residual displacement estimation

When used during loading cycles, the molecular dynam-ics method briefly described in the last section providesenough information to estimate the granular configuration

2 The free surface of the sample is non uniform after preparation, thisinduces an overestimated sample height and consequently gives anunderestimated relative density. The more accurate relative density isabout / = 0.5615.

a

YX

Zoo

oo

o oo o

o oo o

o oo o

o oo o

F = F0 (1 + sin ω t )

b

Fig. 2. Loads and boundary conditions (a) applied to the granular material sample (b).

0 0.5 1 1.50

10

20

30

40

50

60

t (s)

App

aren

t wei

ght,

(N)

0 0.5 1 1.50

1

2

3

4

5

t (s)

Coo

rdin

atio

n N

umbe

r, <

z>

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t (s)

Rel

ativ

e de

nsity

, φ

0 0.5 1 1.50

0.5

1

1.5

2

2.5

t (s)

Ene

rgie

s, (

J)

Kinetic Energy, Ec

Potential Energy, Ep

a b

c d

Fig. 3. Sample preparation under gravity field variation of: (a) apparent weight, (b) coordinations number, (c) granular relative density, and (d) kineticenergy (resp. potential energy) versus time.

84 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

after a finite number of cycles. In this section, we explainthe time averaging technique and we compare its resultsto a full time integration for several numbers of cycles.

3.1. Estimation method

Let us subject the sample S to a cyclic load of period T,then the particles flow according to the initial chain of con-tacts, internal interactions, material properties, and bound-ary conditions. The displacements of the particles withrespect to time can be additively decomposed as follows:

8t 2 ½0; T �; 8b 2S;

ubðtÞ ¼ urbðtÞ þ uv

bðtÞ;

(ð8Þ

where the superscripts (r) and (v) denote, respectively, theresidual and reversible displacements. It should be empha-sized that the loading cycle results in dynamic and visco-elastic effects that continue to act even after removing theexciting load. This means that they may continue to moveand to explore new positions until a final equilibrium isreached. Once the dynamic and viscoelastic effects vanish,the residual displacement of the particle b can be simply

t

ur(t)

u

t

ur(t)

u

uv(t)

ur r(pT)

Fig. 4. Overall displacement evolution with respect to time, illustration of the residual and reversible displacement.

3 The term ‘‘training data’’ is widely encountered in neural networkapplications. It will be used herein to denote the displacements of theparticles with respect to the number of cycles. These displacements arecalled training data since they will be used first for estimation and then forextrapolation.

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 85

described as the difference between the initial and final (ob-tained after cyclic loading and relaxation) displacements:

urbðT Þ ¼ lim

s!þ1ðubðT þ sÞ � ubð0ÞÞ; ð9Þ

where s represents the relaxation time. It can be easily no-ticed that anticipating the residual displacement at the endof the loading cycle is not an obvious task. On the otherhand, the above mentioned definition is not suitable forthe calculation of instantaneous residual displacements,especially when the sample is subjected to several loadingcycles with no relaxing time in between.

In order to overcome this problem and suggest anappropriate residual displacement description, let us inves-tigate a typical overall displacement of granular samplessubjected to repeated loading. According to observationsmade on granular materials, the overall displacement canbe illustrated by a typical description presented in Fig. 4.In this representation, the first term of Eq. (8) representsthe residual displacement which increases slowly with timeand the second denotes the fluctuating reversible displace-ment with respect to time.

It can be seen that the differential residual displacementover a cycle p is considerably smaller than the reversibledisplacement as argued by Abdelkrim [6]:

kurbððp þ 1ÞT Þ � ur

bðpT Þk � maxðkuebðtÞkÞ;

t 2 ½pT ; ðp þ 1ÞT �: ð10Þ

Therefore, the overall displacement can be seen as a rapidlyfluctuating periodic function representing the reversibleterm, a slowly increasing function representing the residualterm, and a correcting term representing the error that canbe encountered. It is worthwhile to notice that the errorterm represents the unpredicted or not explained variationin the dependent variable. It is conventionally called ‘‘er-ror’’ whether it is really measurable or not. Since thereversible term is periodic, the displacement can be de-duced from Eq. (8) as follows:

ubðt þ T Þ ¼ ubðtÞ þ urbðt þ T Þ þ nb: ð11Þ

Over several loading cycles, p = 1, . . ., k, the displace-ment of the particle b under consideration can be writtenas follows:

ubðt þ kT Þ ¼ ubðtÞ þXk

p¼1

urbðt þ pT Þ � ur

bðt þ ðp � 1ÞT Þh

þnðpÞb � nðp�1Þb

i: ð12Þ

Now, the objective is to estimate the unknown functionur

bðtÞ, where nb is a zero mean random error. The estima-tion is made based on a finite number of training data3

u(t + T), . . ., u(t + kT). It provides the best functions ~urbðtÞ

expressed in terms of a set of parameters a which can beobtained by the minimum square root method.

Several functions can be chosen to describe and estimatethe residual displacement ur

bðtÞ. For instance, this functioncan be assumed to be linear with respect to time, in orderto simplify the presentation without loss of generality.Then, the accuracy of the estimation can be tested bystudying the variation of the following functions withrespect to the number of cycles:

guðkÞ ¼1

k

Xb;p;d

½urb;dðt þ pT Þ � ur

b;dðt þ ðp � 1ÞT Þ�2 !1

2

ð13Þ

and

fnðkÞ ¼1

k

Xb;p;d

½npb;d �

2

!12

; ð14Þ

where b denotes the particles of the systems, p representsthe index of a given training datum, and d one of the threedegree of translation. The two above mentioned functionsare simply the expressions of the second norm used tostudy the accuracy of the estimations. In Fig. 5, it can be

0 100 200 300 4000

1

2

x 10

Number of calculated cycles, k

Est

imat

ion

accu

racy

ki = 100

gu(k)

fξ(k)

0 100 200 300 4000

1

x 10

Number of calculated cycles, k

Est

imat

ion

accu

racy

ki = 500

gu(k)

fξ(k)

a b

Fig. 5. Accuracy of the estimation procedure for a linear estimator with two different initial numbers of calculated cycles, ki.

86 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

seen that gu becomes stable when k increases, moreover, theerror terms decrease in norm average fn when k increases.

The number of training data required for the estimationof the trend function can be expressed in such a way thatthe accuracy is controlled. In general, applying the law oflarge numbers provides an upper bound which convergesin 1ffiffi

np . In our case, a higher bound can be obtained since

the errors are always bounded (the errors are within themaximum and minimum displacements obtained along k

loading cycles). The sequence of error terms ðnðpÞb Þp¼1;...;k

can be seen as a set of random independent variables,where nðpÞb 2 ½a; b�. The probability of obtaining a functionwhich estimates the residual displacement with an error �can be written as

Pr1

k

Xp

nðpÞb

���������� P �

" #6 2 e

� 2k�2

ðb�aÞ2 : ð15Þ

The above mentioned inequality is a theorem which isknown as the ‘‘Chernoff Bound’’. Its proof can be foundin [29]. According to Eq. (15), it is possible to control thenecessary number of cycles, k, by choosing an error levelof � and an associated probability. Indeed, fork P ðb�aÞ2

2�2lnð2pÞ, Pr½k 1

k

PpnðpÞb kP �� 6 p. The trend function

can be estimated using the available training data withinthe preselected error.

So far, the accuracy of the estimations is establishedthrough a parametric approach. The obtained model esti-mating the residual displacements over k loading cycles willbe used in the following section in order to extrapolate theresults and obtain an estimated configuration after addi-tional h loading cycles.

3.2. Extrapolation

Using the estimation model obtained beforehand, thefinal configuration as well as the residual displacementsevaluated during the above mentioned k cycles, it is possi-

ble to estimate the (k + h)th configuration, where h 2 N. Inparticular, if the residual displacement is assumed to be lin-ear with respect to time, this configuration can be describedas follows:

ubðt þ ðk þ hÞT Þ ¼ ubðt þ kT Þ þ hk

Xk

p¼1

½urbðt þ pT Þ

� urbðt þ ðp � 1ÞT Þ�: ð16Þ

The first right-hand term of the above mentioned equationis known from the last step, whereas, the second representsthe estimation of the residual displacement at the (k + h)thcycle. The estimation suggested beforehand assumes thatthe residual displacement is linear with respect to the num-ber of cycles. A better estimation can be made using a moreappropriate settlement law that has been obtained empiri-cally by Shenton [1]. This law reads ~ur

bðt þ pT Þ ¼ ab lnðt þ pT Þ þ bbðt þ pT Þ. The parameters ab and bb can be de-duced from the molecular dynamics calculations using theleast square method (minimizing the quadratic error be-tween the empirical low and the calculated residual dis-placements). Therefore, the displacement can be writtenas follows:

ubðt þ ðk þ hÞT Þ ¼ ubðt þ kT Þ þ ab lnðt þ ðk þ hÞT Þþ bbðt þ ðk þ hÞT Þ; ð17Þ

where the parameters ab and bb are calculated using theleast squares method which consists in minimizing thesquares of errors expressed by:

R ¼Xk

p¼1

ðnpbÞ

2 ¼Xk

p¼1

ðurbðt þ pT Þ � ~ur

bðt þ pT ÞÞ2: ð18Þ

The above mentioned expression is a second-order poly-nom, its minimum can be obtained by deriving it withrespect to its variables ab and bb:

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 87

oRoab¼ �2

Xk

p¼1

½urbðt þ pT Þ � ab lnðt þ pT Þ

� bbðt þ pT Þ�Lnðt þ pT Þ ¼ 0;

oRobb¼ �2

Xk

p¼1

½urbðt þ pT Þ � ab lnðt þ pT Þ

� bbðt þ pT Þ�ðt þ pT Þ ¼ 0:

ð19Þ

This system of two unknowns and two equations can besolved to obtain the following expression of the estimationparameters, using the information collected during themolecular dynamics calculations:

ab ¼Pp¼k

p¼1t2p

Pp¼kp¼1ur

bðtpÞ lnðtpÞ �Pp¼k

p¼1tp lnðtpÞPp¼k

p¼1urbðtpÞtpPp¼k

p¼1ln2ðtpÞPp¼k

p¼1t2p �

Pp¼kp¼1tp lnðtpÞ

� �2

ð20Þand

bb ¼Pp¼k

p¼1 lnðtpÞ2Pp¼k

p¼1urbðtpÞtp �

Pp¼kp¼1tp lnðtpÞ

Pp¼kp¼1ur

bðtpÞ lnðtpÞPp¼kp¼1ln2ðtpÞ

Pp¼kp¼1t2

p �Pp¼k

p¼1tp lnðtpÞ� �2

;

ð21Þwhere tp = t + pT. To distinguish between the two ap-proaches, Eq. (16) will be termed as linear estimation andEq. (17) will be called logarithmic estimation. It is obviousthat for both of the above mentioned approaches, the clo-ser h/k is to zero, the better the estimated configuration is.However, the objective of the procedure is to simulate thegranular flow when h is close to or higher than k, therefore,the parameters will be selected in such a way that the esti-mation procedure gives sufficiently accurate results.

3.3. Numerical application

In order to examine the reliability of the time averagingtechnique, estimated configurations are compared to fulltime calculated solutions. The prepared stable sample isnow subjected successively to ki (initial) and k loadingcycles, as described beforehand. The information collectedfrom the molecular dynamics calculation is used in order toestimate the configuration after h cycles as described inthe last section. In order to systematically evaluate theagreement between the calculated and estimated displace-ments, the following estimation of the relative error Er isdefined:

ErðuÞ ¼ku� ukkuk : ð22Þ

The second norm, as defined in Eqs. (13) and (14), isused in the above mentioned equation in order to calculatethe relative error. Note that u is an N · 3 matrix containingthe coordinates of the particles. It denotes the calculatedresidual displacements. Similarly, u represents the esti-mated residual displacements, b is a particle of the system

S, and N is the number of particles. Figs. 6 and 7 showthat an interesting agreement between the calculated andestimated configurations is obtained, especially when thecalculated number of cycles k is higher than 20, either forthe linear or logarithmic estimation approaches. It can beseen in the above mentioned figures that the error in thesecases does not exceed 2% (in Figs. 6 and 7, the errors arepresented in a logarithmic scale for clarity).

It can also be noticed that the initial number of cycles ki

highly affects the accuracy of the estimation techniques.Fig. 6 shows that the maximum relative error decreasesfrom 14% to 2.5% when ki increases from 10 to 500 initialloading cycles, in the case of linear estimation. Similarly,Fig. 7 shows that the maximum relative error decreasesfrom 14% to 1% when ki increases from 10 to 500 initialloading cycles, in the case of a logarithmic estimation. Inaddition, Figs. 6 and 7 show that the relative estimationerrors decrease with the number of calculated cycles k. Thisis obviously due to the fact that the more calculated pointsare available, the better are the estimations. Moreover, itcan be seen that the relative error increases with the lengthof the estimation interval h. More interestingly, it can benoticed that the logarithmic estimation is more accuratethan the linear estimation. Actually, with its two parame-ters, this approach is more appropriate for time averaging.Therefore, it will be used in the last section in the full set-tlement calculation procedure.

In spite of the accurate residual displacements that can beobtained through the averaging technique, the obtained con-figuration can exhibit local excessive penetrations betweenthe particles, especially for large numbers of cycles h. Thiskind of configuration may be unsuitable for further molecu-lar dynamics calculations. In order to overcome this prob-lem, a relaxation technique which consists of correcting theestimated configuration is suggested in the following section.

4. Relaxation method

The objective of this section is to prove the existence ofan optimality property that can be satisfied by the bound-ary conditions of the system S as well as the equations ofmotions (1). This property is used in order to equilibrateinstable configurations with excessive local penetrations.

4.1. Minimum energy functional

Applying an arbitrary vector ub 2 R3 to the equations ofmotion (1), it is possible to establish its dual, the virtualwork principle (v.w.p.) [30], which can be written asfollows:

8ðubÞb2S 2 ðR3ÞN ;Pb2SF bub þ

Pb2SP

a 6¼b2SF abub ¼P

b2Smbabub:

(ð23Þ

The first left-hand term of the above mentionedequation is the virtual work of the external forces, it canbe expressed by PðeÞðu1; . . . ; uN Þ ¼

Pb2SF bub. The second

20 40 60 80 10010

10

10

100

h = 20

h = 40h = 60

h = 80

h = 100

Number of calcultated cycles, k

Rel

ativ

e er

ror

Initial number of cycles ki= 10

20 40 60 80 10010

10

10

10

h = 20

h = 40

h = 60h = 80

h = 100

Number of calcultated cycles, k

Rel

ativ

e er

ror

Initial number of cycles ki= 500

Fig. 6. Linear estimation: relative error distributions for different initial number of cycles (ki), calculated number of cycles (k), and extrapolated number ofcycles (h).

20 40 60 80 10010

10

10

100

h = 20

h = 40h = 60

h = 80h = 100

Number of calcultated cycles, k

Rel

ativ

e er

ror

Initial number of cycles ki= 10

20 40 60 80 100

10

10

h = 20

h = 40h = 60

h = 80h = 100

Number of calcultated cycles, k

Rel

ativ

e er

ror

Initial number of cycles ki= 500

Fig. 7. Logarithmic estimation: relative error distributions for different initial number of cycles (ki), calculated number of cycles (k), and extrapolatednumber of cycles (h).

88 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

left-hand term expresses the virtual work of the inter-nal forces PðiÞðu1; . . . ; uN Þ ¼

Pb2SP

a6¼b2SF abub. Theright-hand term denotes the work of the inertia forcesAðu1; . . . ; uN Þ ¼

Pb2Smbabub. Note that the set of vectors

(ub)b2[j1, Nj] is arbitrary in ðR3ÞN . Therefore, it can be seenas virtual displacements or velocity applied to the systemS. In particular, this virtual displacement can be takencontinuous and in coherence with the boundary conditions(ub ¼ ud

b 8b 2Sd), where Sd denotes the frontier of Swhere displacements are applied. Similarly, the externalforces of Eq. (1) should be in coherence with the boundaryconditions ðF b ¼ F f

b 8b 2SfÞ, where Sf denotes the fron-tier of S where forces are applied.

Applying a set of displacements (ub)b2[j1, Nj] to the sys-tem of particles S, the existence of a solution for the equa-tions of motion implies that there is a set of internal forcesðF abÞb;a2½j1;N j�2 and a set of external forces (Fb)b2[j1, Nj] gov-erning the evolution problem. Taking into considerationthe above mentioned solution ðub; F b; F abÞb;a2½j1;N j�2 , it ispossible to write the v.w.p. as follows:P

b2Sf F fbub þ

Pb2Sd F bud

b þP

b2Smbðg � aÞub;

þP

b2SP

a 6¼b2SF abub ¼ 0:

(ð24Þ

where SdSSf ¼ S and g represents the gravity accelera-

tion. As a second application of the v.w.p., let us considerthe same set of forces, the solution of Eq. (1) associatedwith a different virtual displacement, which is in coherencewith the boundary conditions, ðu0b; F b; F abÞb;a2½j1;N j�2 . Thisleads to a second equation which is similar to Eq. (24).The difference between them leads to the followingequation:P

b2Sf ðF fbu0b � F f

bubÞ þP

b2Smbðg � aÞðu0b � ubÞ;þP

b2SP

a6¼b2SðF abu0b � F abubÞ ¼ 0:

(ð25Þ

The last left-hand term of Eq. (25) can be expressed asfollows:

8a; b 2S; in contact;

F abðua; ubÞ ¼ rwabðua; ubÞ;

(ð26Þ

where wab(ua, ub) represents the potential of the contactforces. The proof of the optimum existence can be estab-lished through the variation rate of this potential:

dwabðua; ubÞ ¼ wabðua þ dua; ub þ dubÞ � wabðua; ubÞ: ð27Þ

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 89

Considering a quasi-static relaxation, the conditionkFtk 6 lFn remains valid and it is possible to write the po-tential as wabðua; ubÞ ¼ kn

cþ1dcþ1

n ðua; ubÞ, where c depends onthe normal contact model, it equals 3/2 in case of Hertzlaw, dn(ua, ub) = [0.5(da + da) � kxa + ua � xb + ubk], andxb2[j1, Nj] are the initial positions of the particles. Makinguse of Cauchy–Schwartz inequality, it is possible to showthat dwab P 0 while contact exists. Therefore, it can be de-duced that:

8 Solution ðub; F b; F abÞb;a2½j1;N j�2 ;8u0 6¼ u in coherence with ud ;P

b2Sf F fbu0b þ

Pb2S

mbgu0b þP

b2SP

a6¼b2Swabðu0bÞ;

PP

b2Sf F fbub þ

Pb2Smbgub þ

Pb2SP

a6¼b2SwabðubÞ:

8>>>>><>>>>>:

ð28Þ

The obtained inequality proves the existence of a func-tional FðuÞ which reads

FðuÞ ¼Xb2Sf

F fbub þ

Xb2S

mbgub þXb2S

Xa6¼b2S

wabðubÞ: ð29Þ

This functional can be minimized in order to equilibrateinstable configurations where excessive penetrations insome zones can induce divergence of the flow.

4.2. Numerical application

In this section, we consider particular instable configura-tions characterized by local excessive penetrations and sub-ject them to the relaxation technique which consists ofminimizing the functional FðuÞ while keeping fixed allthe particles of the frontier, in order to find out the optimalparticle positions guaranteeing a stable system S. As a firstexample, let us consider a granular cell of nine particles incoherence with the material properties defined in Table 1.This cell is presented in Fig. 8. The results show that thisapproach equilibrates the penetrations and provides a sta-ble configuration.

Now, let us consider a second example corresponding toan estimated configuration from simulations carried out as

Fig. 8. Relaxation of a granular cell of nine particles: (a) e

described in the last section. The configuration under con-sideration is obtained after ki = 20 initial cycles, k = 20additional calculated cycles, and a linear extrapolationover h = 20 cycles. In terms of settlement calculations, suchan estimation is in agreement with the calculated results asdiscussed in the last section. However, it can be seen inFig. 9a that it exhibits excessive particle penetrations.Although limited in number, these anomalous penetrationsmay be important enough to induce flow divergence.

The relaxation method for this case consists of fixing allthe particles of the frontier and finding the optimal posi-tions by minimizing the functional (29) using the gradientmethod. It leads to an equilibrated configuration with thepenetration distribution presented in Fig. 9b. It can benoticed that the distribution is improved and the maximumpenetration is reduced from 1.8 · 10�3 m to 6 · 10�5 m.

The example presented herein proves that the relaxationtechnique improves the estimated configuration in terms ofpenetration magnitudes and distributions. In the followingsection, the estimation and relaxation techniques will beused in a full procedure to describe the settlement of gran-ular samples.

5. Settlement procedure

In this section the molecular dynamics algorithm, theaveraging technique and the relaxation method, describedbeforehand are used in a computational procedure in orderto simulate granular materials settlement under repeatedloading. The flow chart of the procedure is explained, thena convergence study is conducted. Finally, a particular fieldcase is studied in order to show the advantages and limitsof the suggested procedure.

5.1. Flow chart of the computational procedure

The flow chart displayed in Fig. 10, transfers the basicidea of the concept into a programmable algorithm. Theanalysis starts with an initial configuration where the par-ticles are randomly distributed in the cylinder with no pen-etration in between. This initial configuration is then

stimated configuration and (b) corrected configuration.

Granular flow calculation during ki loading cycles

Granular flow calculation during the cycle p

Evaluation and storage of the residual displacements (equations (9)-(11))

p=p+1

Estimation of the residual displacements trend (equation 11)

Estimation of the configuration number (h+k) (see equations (15) and (16))

Relaxation method (functional presented in equation (27) is minimised using Newton’s method )

Sample preparation under gravity, using the molecular dynamics, (equations (1)-(7))

Generation of a randomly distributed intial configuration of a small density ( < 0.1)

Equilibrium

p < kcr

Fig. 10. Flow chart of the settlement computational procedure.

0 0.2 0.4 0.6 0.8 1 1.2

x 10

0

200

400

600

800

1000

1200

Penetration, δn(m)

Num

ber

of c

onta

cts

0 1 2 3 4

x 10

0

20

40

60

80

100

120

Penetration, δn(m)

Num

ber

of c

onta

cts

a b

Fig. 9. Relaxation of a full sample: (a) estimated configuration and (b) corrected configuration.

90 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 91

subjected to the gravity field until equilibrium (Fig. 3). Theobtained sample is subjected to ki initial loading cycles,before starting the sequential procedure, since the firstcycles are always characterized by high settlement slopesas can be seen in Fig. 4. The sequential procedure consistsfirst in subjecting the sample to k loading cycles whichupper bound can be determined knowing the probabilityand error levels defined in Eq. (15). During these k loadingcycles, training data are evaluated using the moleculardynamics and stored at the end of each cycle. The collectedinformation is then used to establish trend functionsdescribing the behavior of the granular sample in termsof residual displacement. The obtained functions are thenused for logarithmic extrapolation up to (h + k) cycles, asdescribed by Eqs. (17), (20), and (21).

Although the estimations and extrapolations are testedin terms of residual displacements accuracy, possible localanomalies in terms of excessive penetrations may show up,as can be seen in Fig. 9. Addressing these possible issuesconsists of applying the relaxation technique to theobtained configuration. The former step produces stableconfigurations which are suitable for further moleculardynamics calculations.

In Section 2, it has been shown that the accuracy of theestimation technique depends on the values of ki, k and h.The relative error decreases with the increase of k and/orthe decrease of h. In addition, the estimation accuracyincreases with the total number of cycles. Actually, in termsof computational cost, it is obvious that the higher theratio h/k, the better it is. It is therefore advantageous toincrease h with the total number of iteration. In this study,the parameter h can be varied with respect to the number ofiterations, since the error decreases when the total numberof calculated cycles increases.

5.2. Convergence study

In order to validate the suggested procedure, theobtained overall residual displacements are compared to

0 100 200 300 4000.0385

0.039

0.0395

0.04

0.0405

0.041

0.0415

0.042

0.0425

Number of cycles

Load

ed s

urfa

ce la

titud

e, z

max

(m

)

Convergence study for ki=h=10

k=10

k=15k=20k=30

k=40MD

a

Fig. 11. Convergence of the procedure with the simulation parameters k (a)

full molecular dynamics calculations. The simulationsparameters used for the convergence study are detailed inTable 1. Fig. 11 shows the settlement path of the granularsample using the molecular dynamics (MD) as well as thesettlement paths calculated with the suggested procedurefor different parameters. As a first step, the initial numberof cycles calculated before the sequential procedure is takenki = 10 and the length of the estimation interval is taken ash = 10, while varying k. It can be noticed in Fig. 11a thatthe error between the full molecular dynamics and theobtained results decreases with the number of calculatedcycles k. Similarly, the effect of the estimation intervallength on the accuracy of the procedure can be tested bytaking a constant initial number of cycles ki = 10 and aconstant number of calculated cycles k = 40. Fig. 11ashows that the error decreases when h decreases, as canbe expected. More interestingly, it can be seen that theerror stabilizes when the length of the estimation intervaltends to 0.

The presented procedure can be used either in a compac-tion process or for settlement calculation, as stated in theintroduction. If used in a compaction process, the proce-dure is self-consistent since the objective is to obtain adense granular sample for other studies. In this case, theparameters ki, k and h are selected in such a way that thedense state is obtained with the minimum number of itera-tions. However, in case of settlement calculation, the objec-tive is to study the behavior of granular materials underrepeated loading, in terms of residual displacements.Therefore, the accuracy of the procedure has to be estab-lished carefully, by choosing the suitable parameters ki, k

and h, in order to ensure acceptable results.

5.3. Application and scope

The suggested procedure is developed to study themechanism of long term granular materials settlementunder cyclic loading. Experimental studies conducted ongranular materials [2–4] showed that the permanent

0 500 1000 1500 2000 25000.039

0.0395

0.04

0.0405

0.041

0.0415

0.042

0.0425

Number of Cycles

Load

ed s

urfa

ce la

titud

e, z

max

(m

)

Convergence study for ki = 10 and k = 30

h=30h=25h=20MDh=10

b

and h (b) and comparison with the full molecular dynamics calculations.

92 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

displacements depend on several parameters such as theapplied force, frequency and material properties. In addi-tion, the mode of loading is expected to have a crucial effecton the amount of settlement when the considered granularsample is periodically excited. In this section, the effect ofthe gained time on the precision of the whole procedurewill be evaluated. Moreover, different modes of loading willbe investigated in order to show the limits of the suggestedprocedure.

The simulations under consideration are performedusing the dimensions and material properties presented inTable 1, however, the periodic excitation presented inFig. 2 will be varied according to the mode of loading.The stress amplitude of the signal applied to the fully con-fined sample is Dr = 14.15 kPa and the excitation fre-quency is f = 50 Hz. In general, the procedure worksproperly when the residual displacements are relativelysmall as compared to the particle size. When the granularmaterial is fully confined the settlement mechanism ismainly related to the particle rearrangement when the exci-tation take place and consequently, the displacement is rel-atively small. In this case, it can be seen in Figs. 11 and 12athat the procedure is accurate enough to predict the evolu-tion of the sample with respect to the number of cycles, interms of residual displacement and coordination number.The precision can also be evaluated in terms of overall

0 200 400 600 800 10002.5

3

3.5

4

4.5

5

5.5

Number of cycles

Coo

rdin

atio

n nu

mbe

r

a

c

h = 00h = 10h = 20h = 30

0 50 100 150 2000

0.002

0.004

0.006

0.008

0.01

0.012

Gained calculation time (%)

Rel

ativ

e E

rror

Fig. 12. (a) Variation of the coordination number with respect to the numberpositions, and (d) overall residual displacement.

residual displacement, coordination number, and configu-ration as can be seen in Fig. 12. It can be noticed thatfor all the above mentioned criteria, the relative errorincreases with the gained time. For instance, it can benoticed that the coordination number presents a relativeerror of about 12% when 180% of gained calculation timeis achieved. Moreover, the relative error calculated in termsof configuration precision (using Eq. (22)) is of about1% for 180% of gained calculation time. The residualdisplacement is evaluated in a more accurate manner sincethe relative error does not exceed 0.5% when the samegained time level is achieved.

Unlike the fully confined mode of loading, where the set-tlement is mainly due to granular materials rearrangement,the partially confined mode of loading results in particlesrearrangement as well as particles migration under periodicexcitation. In fact, the particles move toward the regionswith lower loading level. Therefore, the residual displace-ment largely undergo the settlement produced in case ofequivalent fully confined samples. This behavior makes itdifficult to obtain more accurate displacement estimations,thereby inducing anomalous configurations in terms ofexcessive penetration. For instance, the partially confinedloading case of interest in this section is characterized bya stress amplitude of Dr = 5.87 kPa, an excitation fre-quency of f = 10 Hz, and a confinement level of 80%.

b

d

0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

Gained calculation time (%)

Rel

ativ

e E

rror

0 50 100 150 2000

1

2

3

4

5x 10

Gained calculation time (%)

Rel

ativ

e E

rror

of cycles, (b) precision in terms of final coordination number, (c) particle

0 50 100 150 200 250 3004.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

Number of cycles

Coo

rdin

atio

n nu

mbe

r

ki = 10 and k = 10

G = 0%G = 30%G = 60%G = 90%

0 50 100 150 200 250 3000.0408

0.041

0.0412

0.0414

0.0416

0.0418

0.042

0.0422

0.0424

Number of cycles

Load

ed s

urfa

ce la

titud

e, z

max

(m

)

ki = 10 and k = 10

MD

h = 06

h = 09h =12

0 20 40 60 80 100 1200

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Gained calculation time (%)

Rel

ativ

e E

rror

0 20 40 60 80 100 120 1400

0.002

0.004

0.006

0.008

0.01

Gained calculation time (%)

Rel

ativ

e E

rror

a b

c d

Fig. 13. (a) Coordination number with respect to the number of cycles, (b) Settlement with respect to the number of cycles, relative error in terms of (c)Coordination number and (d) Settlement.

A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94 93

The material properties, sample dimensions, and dissipa-tion parameters are kept unchanged as can be seen in Table1. Fig. 13a and b shows the variation of the coordinationnumber and the loaded surface width with respect to thenumber of cycles. It can be noticed that the slope of settle-ment is higher in the case of partially confined loading ascompared to the fully confined case. However, it can beseen that the coordination number average remains in thesame range. On the other hand, Fig. 13c shows the relativeerror in terms of coordination number with respect to thegained calculation time. Note that the error is calculatedrelative to the coordination number obtained with fullmolecular dynamics calculation (since the curve crossesthe origin). The same procedure is used to calculate theerror in terms of settlement. Fig. 13d shows the variationof error in terms of gained calculation time. It can benoticed that the error in this case is about 8% when thegained time reaches 120%. This means that the errordecreases with the degree of confinement.

The examples presented above demonstrate that the sug-gested procedure works properly in the case of fully con-fined modes of loading. However, its accuracy decreasesin terms of settlement prediction when it is used for par-tially confined modes of loading. We also note that in thispaper, the procedure is implemented with a moleculardynamics scheme, however, it can be easily adopted in con-tact dynamic based algorithms.

6. Conclusion

A computational procedure for long term granularmaterial settlement calculations has been presented. Thesuggested procedure sequentially uses the moleculardynamics, a time averaging technique, and a relaxationmethod in order to estimate the granular flow under cyclicloading. The molecular dynamics represents the engine ofthe suggested method, it consists of integrating the parti-cle’s equations of motion. The Hertz–Mindlin law is usedto describe elastic particle interaction, a Coulomb law isused to describe the frictional dissipation, and a propor-tional damping law is used to describe the viscous aspectof the particles interaction. The averaging technique isbased on information collected during the moleculardynamics calculation. A comparison between calculatedand estimated configurations has been carried out andshowed satisfactory agreements between the averagingand calculated results. Finally, a relaxation method is sug-gested. It consists of optimizing the granular particles posi-tions in order to determine suitable configurations forfurther molecular dynamics calculations.

The whole procedure is implemented in an understand-able manner in order to calculate granular materials flowunder repeated loading. The numerical results from thesuggested procedure show that the necessary time for cal-culating long term settlement is reduced. In addition, the

94 A. Karrech et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 80–94

suggested procedure provides relatively accurate results.However, according to the convergence study that has beencarried out, there is a compromise between the accuracyand the time reduction.

The suggested procedure is mainly developed to studythe behavior of granular materials under a large numberof repeated cycles. In a future work, it will be applied onseveral granular samples with different exciting frequenciesand degrees of confinement in order to study the mecha-nism of long term granular settlement.

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