examining the mechanisms of sand creep using dem simulations

Granular MatterDOI 10.1007/s10035-014-0514-4

ORIGINAL PAPER

Examining the mechanisms of sand creep using DEM simulations

Yu-Hsing Wang · Yun Man Lau · Yan Gao

Received: 9 May 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract In this study, DEM simulations of triaxial creeptests on dense and loose sand samples were carried out toexamine the micromechanics involved during creep. Thesimulated creep responses reproduce qualitatively the pub-lished experimental results. During the primary creep, thecreep stress is gradually borne by the contact normal forcesinstead of contact tangential forces so that the columnar par-ticle structures can be formed. This process also leads to acontinuous decrease in the creep rate. The columnar struc-tures eventually are completely formed and the creep ratereaches a minimum. However, the structures become meta-stable and susceptible to buckling. This explains why a sandpacking does not show an extended period of secondary creepin the experiment. Buckling of the columnar structures alsogives rise to maximum dilatancy and a sharp transition of themajor fabric orientation of weak forces from horizontal tovertical. The continuous buckling process of columnar struc-tures increases the creep rate and sliding ratios of contactsduring the tertiary creep. In addition, the trend of contact tan-gential forces decreasing and contact normal forces increas-ing is reversed. Finally creep rupture occurs as the creepstress–strain line intersects the complete stress–strain curve.All the creep samples follow their original volume-change

Y.-H. Wang (B) · Y. M. LauDepartment of Civil and Environmental Engineering,The Hong Kong University of Science and Technology,Clear Water Bay, Hong Konge-mail: [email protected]

Y. M. Laue-mail: [email protected]

Y. GaoDepartment of Civil and Environmental Engineering, The Hong KongUniversity of Science and Technology, HKSAR, Chinae-mail: [email protected]

tendency to continue their dilation or contraction responseduring creep.

Keywords Creep · DEM · Contact forces · Fabric tensor ·Sand

1 Introduction

The time-dependent behavior of clayey soils has been widelydocumented for decades (e.g., [1–4]). However, relativelylittle research has been published on time effects in sand(e.g., see the review in [5]). The practical significance of timeeffects in sand is evident. For instance, driven or displacementpiles in sand often demonstrate a significant increase in theshaft resistance with time (e.g., [6–11]). This phenomenonis referred to as pile setup. Time effects can play a role in thesettlement of a footing constructed on sand (see the summaryby [12]). Creep settlements of shallow foundations underconstant loading can contribute a significant proportion tothe overall movements of the foundations (e.g., [13]). Hence,there is a strong need to understand the fundamentals of creepbehavior in sand.

Creep is the continuous deformation under constant stressover time. Laboratory triaxial creep tests have been carriedout by different researchers to examine the creep behaviorof sands. Comprehensive reviews have been done by, forexample, Tatsouka et al. [14], Augustesen et al. [15], andLade [16]. Figure 1a presents a typical experimental result.The salient findings that are relevant to this study from thesepublished results are:

(1) During primary creep, no matter whether the loadingis isotropic or anisotropic (i.e., subject to shearing), thecreep rate (or the strain rate) decreases continuously with

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Fig. 1 The creep rate-time response: a the experimental results of theToyoura sand at different creep stress q (after [17]), and b the typi-cal response of materials at different creep stage. Circles denote creeprupture

time [17–20]. In addition, the creep rate increases as theapplied deviatoric stress q (i.e., creep stress) increasesduring creep [17,19].

(2) Unlike the typical creep response of materials as shownin Fig. 1b, a sand packing, as Fig. 1a shows, does notexhibit an extended period of secondary creep where thecreep rate is a constant.

(3) If the creep stress is large enough, tertiary creep can beobserved. The creep rate continues to increase and ulti-mately creep rupture occurs [17,19].

(4) Triaxial drained and undrained tests show no signif-icant strain rate effects on the stress–strain relation-ships [16,21–23]. This is different from the response ofclayey soils, which demonstrate isotach behavior, i.e.,there exists a unique stress–strain–strain–rate relation-ship. Sand therefore is described as a non-isotach mate-rial and its time-dependent behavior cannot be fullydescribed by conventional viscous-type models as suc-cessfully used for clays [15,16,22,23].

Numerical simulations using the discrete element method(DEM) have been carried out to examine the underlyingmechanisms of the creep behavior observed in laboratory tri-axial creep tests by, e.g., Kuhn and Mitchell [24] and Kwokand Bolton [25]. In their simulations, the visco-frictional

contact model, which is based on the rate process theory,was adopted to quantify the contact tangential force duringcreep. These simulation results provided insightful expla-nations to the creep behavior based on the micromechanicsinvolved. For instance, Kuhn and Mitchell [24] found thatthe decrease in the creep rate during the primary creep canbe attributed to the decrease in the ratio between contacttangential and normal forces, owing to structural changesin the soil packing. Nevertheless, many questions related tosand creep remain unanswered. For instance, what are thedetails regarding the structural changes during creep, e.g.,the information of fabric anisotropy? Why is the secondarycreep stage not evident? How can the structural changes in asoil packing affect the response in the secondary and ter-tiary creep stages? Is there any difference between looseand dense sand regarding the creep responses and associatedunderlying mechanisms? What are the associated volumet-ric responses in both dense and loose sand during creep?How can a viscous-type of contact model be able to simu-late the non-isotach sand behavior? Finding answers to thesequestions using DEM simulations of triaxial creep tests isthe main objective of this study. Although this study mainlyfocuses on the creep behavior of sand, the explanations mightbe applied to clay if cluster–cluster interactions in clay can beconsidered analogous to the particle–particle interactions insand.

This paper begins with giving the simulation details,including the contact models used, and the simulation pro-cedures. Then, simulation results and discussions follow insuch a way that the macroscopic creep responses are pre-sented first and then the associated micromechanics in theprimary, secondary, and tertiary creep stages are individuallyexplored. Towards the end, a summary of the salient findingsin this study, including the features of the creep response ineach creep stage and associated micromechanical properties,is offered.

2 Simulation details

The DEM software, Particle Flow Code in three Dimensions(PFC3D version 3.1, Itasca Company, USA) was utilized inthis study. User-written C++ code was used to modify thecontact model between particles to consider also the stress–dependent rolling resistance and contact creep. This contactmodel is similar to the one in our previous work for 2D DEMsimulations of soil creep [26] and has been used in 3D DEMsimulations to reproduce the experimental findings regardingthe aging-induced stiffness increase in sand [27,28]. Detailsof the adopted contact models and the simulation proceduresof triaxial creep tests are described in the following subsec-tions.

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2.1 Contact models

2.1.1 The Hertz–Mindlin contact model (before creepoccurs)

The nonlinear simplified Hertz–Mindlin contact model wasused to guide the particle interactions in the normal and tan-gential directions before creep occurs. This model can cap-ture the stress–dependent soil responses, such as stiffness.The contact normal stiffness, kn , and the contact tangentialstiffness, ks , are based on an approximation of the theory ofMindlin and Deresiewicz [29] and described in Cundall [30],can be described as

kn =(

2G√

2R̃

3 (1 − v)

) √U n (1)

ks =⎛⎝2 3

√3G2 (1 − v) R̃

2 − v

⎞⎠ 3

√|N | (2)

where G and ν respectively are the shear modulus and Pois-son’s ratio of the material that makes the particle; N is thecontact normal force; U n is the contact overlap; and R̃ is theequivalent sphere radius. The contact normal and tangentialforce–displacement model is given by the stiffness multipliedby displacement. Sliding occurs at the particle contact whenthe tangential force reaches the maximum resistance, i.e.,fs × N where fs is the coefficient of interparticle friction.

2.1.2 Contact creep model: Burgers model

As the creep process began, the Burgers model as illustratedin Fig. 2 was used as the tangential force model to modelvisco-frictional contact creep. As for the contact normal forcemodel, as a first approximation, only elastic behavior wasassumed. This arrangement is similar to the one suggested in[24–26,31]. The Burgers model consists of a Kelvin modeland a Maxwell model, connected in series; the creep responseof each model is described in Fig. 3. The Kelvin model has aspring and a dashpot connected in parallel. Hence, the actingtangential force T is borne by the spring and the dashpot andcan be given by

T = −Kkuk − Cku̇k (3)

where Kk and Ck respectively are the spring constant anddashpot of the Kelvin model; and uk is the displacement.From Eq. (3), uk can be written as

uk = T

Kk

(1 − e−Kk t/Ck

)(4)

This indicates that the associated shear displacement is in aform of exponential decay as shown in Fig. 3, i.e., the defor-

Fig. 2 Contact models during creep for: a the normal force and therolling resistance, and b the tangential force (Burgers model)

mation feature of primary creep. Nevertheless, the Kelvinmodel does exhibit time-independent displacement uponloading and unloading, i.e., an elastic response.

The Maxwell model is made up of a spring and a dashpotconnected in series, and the acting tangential contact force Tcan be described as

T = −Kmumk = −Cmu̇mc (5)

where Km and Cm respectively are the spring constant anddashpot of the Maxwell model; umk is the displacement ofthe spring; and umc is the displacement of the dashpot. The

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Fig. 3 Responses of the Kelvin, Maxwell and Burgers models wherethe parameters used are: Km = Kk = 1.52 × 109 N/m, Cm = 3.8 ×108 N s/m, and Ck = 3.8 × 107 N s/m

total displacement of the Maxwell model, um , is

um = umk + umc = T

Km+ T

Cmt (6)

This model represents instantaneous deformation and a con-stant deformation rate (i.e., the T/Cm term) for the secondarycreep. However, the Maxwell model does not show a decreas-ing rate of deformation under a constant stress, which is thecharacteristic of primary creep.

The constitutive equation of Burgers model can be derivedby considering the displacement response u under a constanttangential force, T ,

u = um + uk

= T

Km+ T

Cmt + T

Kk

(1 − e−Kk t/Ck

). (7)

Indeed, Burgers model compensates for the limitations ofthe individual Maxwell and Kelvin models by combining thetwo models together. In addition, it is interesting to note thatin Burgers model a greater tangential force can give rise to ahigher creep rate:

u̇ = T

Cm+ T

Cke−Kk t/Ck (8)

2.1.3 The model of the rolling resistance

Previous studies on contact mechanics between two spher-ical particles reveal that a contact moment M (i.e., rollingresistance) can be generated while particles roll against eachother. Such a resistance can be affected by different mecha-nisms, such as micro-slip at the contact interface (e.g., [32]),plastic deformation within the contacting bodies (e.g., [32]),surface roughness (e.g., [32,33]), and contact adhesion (e.g.,[33–35]). For soil particles, due to their low-sphericity andangularity in shapes and of non-adhesive in contacts, the rel-ative rotation between particles are predominantly restrained

by different levels of interlocking (e.g., [35–39]). Hence,properly considering the rolling resistance, including the par-ticle interlocking effect, is crucial for better modeling of soilresponses by DEM simulations, especially for those simula-tions of which spherical particles are used instead of adoptingreal particle shapes to reduce the computational cost (e.g.,[35,37–47]). Taking the DEM simulation of triaxial tests inLau [47] as an example, a more realistic strain–softeningresponse and a peak and critical–state friction angles can bemodelled after the rolling resistance including the interlock-ing effect is considered in the contact model between spheri-cal particles. The same rolling resistance model used by Lau[47] was also adopted in this study and briefly described inthe following.

Bardet and Huang [41] derived the rolling resistancebetween normally compressed elastic cylinders as

M = kθ · θ = 2RN Jn · θr (9)

where kθ is the rotational stiffness which is proportional to thecontact normal force N and the radius of the cylinder R; Jn isa constant (i.e., the rolling coefficient); and θr is the rotationalangle. Based on Eq. (9), Li [43] derived the equation of rollingresistance between two cylinders of different sizes at contactsas

�M1 = −2Jn N(R1ω

′1 − R2ω

′2

)�t = −�M2 (10)

where M1 and M2 are the moments generated in cylinders1 and 2 in contact; ω′

1 and ω′2 are the associated relative

rotational velocities; and Δt is the time interval to move tothe next new contact point. As suggested by Iwashita andOda [42], this maximum rolling resistance is associated withthe contact normal force, N , and the contact width, 2B asfollows:

|Mmax| = ηB N (11)

where η is the coefficient of rotational sliding and the effectsof interlocking can be implicitly embedded in this coefficient[42]. Li [43] further modified the maximum rolling resistanceequation to Eq. (12) as

|Mmax| = η min(R1, R2)N (12)

There is in fact no analytic solution to quantify the rollingresistance, including the interlocking effect, in the three-dimensional case. Hence, Eqs. (10) and (12) were simplyextended to a 3D case and used in this study. That is, themoments of three directions were calculated based on theapproach of Eq. (10) and the magnitude of the moment fromthe three directions was used to assess whether rotationalsliding will occur or not. Note that using Eqs. (10) and (12)

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Table 1 Sample properties and the parameters used in the contact mod-els

Properties or parameters Value

The sample

Height of container 14 cm

Diameter of container 7 cm

Particle size range 0.22–0.70 mm

Dense samples

Particle numbers 12,598

Void ratio 0.633

Loose samples

Particle numbers 8,713

Void ratio 0.815

The Hertz–Mindlin contact model

Shear modulus (G) 3 × 109 Pa

Poisson’s ratio (ν) 0.2

Friction coefficient betweenwall–particle contacts

0.0

Friction coefficient betweenparticle–particle contacts

0.5

Burgers model

Viscosity of the dashpot in theMaxwell model (Cm)

3.8 × 108 (N s/m)

Viscosity of the dashpot in theKelvin model (Ck)

3.8 × 107 (N s/m)

The rolling resistance model

Rolling coefficient (Jn) 0.8

Coefficient of rotational sliding (η) 0.15

to quantify the rolling resistance, i.e., equivalent to use arolling spring and slider (to set a limit of rotational resis-tance), is analogous to the way to establish the tangentialforce–displacement model, i.e., using a shear spring (Eq. 2)and shear slider (to set a frictional limit). In addition, sucha treatment is also a common way to add rolling resistanceon the spherical particles in the DEM simulations (e.g., see[48,49]).

2.2 Simulation procedures of triaxial creep tests

2.2.1 Numerical samples

Table 1 summarizes the sample properties and the parametersused in the particle contact models. Particle with diametersranging from 0.22 to 0.70 mm, based on the ratio suggestedby Kuhn and Bagi [50], were used to form the triaxial sam-ple 7 cm in diameter and 14 cm in height (Fig. 4). The totalnumber of particles used in the simulation was 12,598 fora dense sample and 8,713 for a loose sample. The radiusexpansion method was used for the assembly generation.First, balls (i.e., particles) with small radius were generatedand positioned randomly in a cylindrical vessel. The radius

Fig. 4 The sample packing

of each ball was then multiplied by a factor to achieve thedesignated porosity. To minimize the boundary effect, a fric-tionless rigid-wall boundary was employed and the rollingresistance at each ball–wall contact was neglected.

2.2.2 Three different stages

All of the dense and loose samples underwent three differentstages: (1) isotropic consolidation; (2) triaxial shearing; and(3) creep under a constant deviatoric stress, q. A confiningpressure of 400 kPa was applied in the consolidation stage.During triaxial shearing, a constant velocity of 0.025 mm/swas assigned to move the top and bottom plates towards eachother to increase the axial loading (i.e., a strain-controlledtest) while the confining pressure is kept as 400 kPa. Theaxial loading stopped when the target deviatoric stress q wasachieved. Then, creep started under this target creep stress,q. If the applied creep stress q is above the correspondingcritical–state value qcs obtained from the complete stress–strain response, the sample was allowed to creep until thecreep stress no longer remained constant. This is when creeprupture or rupture failure occurs.

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2.2.3 Contact models involved in different stages

In the first two stages creep did not occur and the con-tact behavior between particles was governed by the Hertz–Mindlin model. Before the creep model was activated inthe third stage, i.e., creep under a constant q, some con-versions between the contact models were needed. First, inthe tangential direction of all the active ball–ball contacts,the Hertz–Mindlin model was converted into Burgers Modelwith inactivated dashpots. The tangential spring constant inthe Maxwell and Kelvin models, i.e., Km and Kk in Fig. 2b,was twice that obtained from the Hertz–Mindlin model atthe end of the stage two. Thus, the equivalent spring constantremained unchanged before and after changing the contactmodel. The sample equilibrium is not affected by the pro-cedure of changing the contact model. The creep processwas then started under a constant q by activating the dash-pot. Note that the spring constants in Burgers model werequantified by the Hertz–Mindlin model and therefore theyare not the same among particle contacts in the DEM simu-lations of this study. Once again, this arrangement is impor-tant to capture the stress–dependent soil responses, such asstiffness.

Burgers model is a phenomenological model so in princi-ple the viscosities of the dashpots, i.e., Cm and Ck in Fig. 2,can be chosen to be any numbers to fit the creep responseof the material under study. Different values of Cm and Ck

mainly affect the time needed to reach the same amount ofcreep but not the underlying mechanisms that govern differ-ent creep behavior. Hence, in this study, the specific values ofCm and Ck were selected to make computation time afford-able. That is, the creep process was accelerated. The wallvelocity was assigned an upper bound of 0.1 mm/s to ensurethe quasi-static state is achieved in every step, even during thecreep process. The rolling resistance model is always addedin all three stages.

3 Simulation results and discussion: macroscopicresponses

3.1 Stress–strain and volumetric responses

Figures 5 and 6 present the stress–strain and volumetricresponses of both dense and loose samples after creep.For comparison, their complete stress–strain and volumet-ric responses, sheared up to 30 % of axial strain, are alsoincluded in the figures. Both the dense and loose samples,based on their complete stress–strain curves, have the samecritical–state friction angle, φcs ≈ 27.7◦. For the dense sam-ple, the corresponding peak friction angle φp is about 38.0◦.As seen in Figs. 5b and 6b, loose samples tend to contractmore while dense samples dilate less after being subjected to

Fig. 5 Simulated results of triaxial creep tests on the dense samples: athe stress strain response, b the volumetric response, and c an enlargedview of the volumetric response

creep, if compared with the response of their complete vol-umetric response obtained from the sample without creep.Still, the samples follow their original volume-change ten-dency to continue their dilation or contraction during creep.This behavior is more evident as shown in Fig. 5c, if thecreep starts at the stage that the sample is still under con-traction, e.g., curve 1©; the q = 400 kPa case, the creepvolumetric response will contract first and then dilate. Ifthe creep starts at the stage when the sample is about todilate, e.g., curve 4©; q = 700 kPa case, the creep volu-metric response will dilate right away. Experimental obser-vations made by Colliat-Dangus et al. [18] and Bowman andSoga [8,51] confirm such volume change behavior during soilcreep.

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Fig. 6 Simulated results of triaxial creep tests on the loose samples: athe stress–strain response, and b the volumetric response

3.2 Creep rate responses

Figure 7a presents the creep rate-time response of the densesamples, i.e., the deviatoric strain rate, ε̇q , versus the creeptime, t , in a log–log scale. The response is similar to theexperimental findings as shown in Fig. 1 and other publishedresults in sand (e.g., [18–20]) and in clay (e.g., [52,53]) aswell as the results from DEM simulations (e.g., [24,25]). Thesoil starts with the primary creep, followed by the secondaryand tertiary creep if the creep stress is greater than a certainvalue.

At the primary creep stage, the creep rates decay in a linearmanner and all the lines are approximately parallel to eachother (including the response of loose samples as shown inFig. 8). In other words, all the samples in this study, regardlessof the packing density, i.e., loose or dense, and the applieddeviatoric stress during creep, exhibit a similar creep ratechange, i.e., the same slope m where m is defined by Singhand Mitchell [54] as

m = −� log ε̇q

� log t(13)

In addition, a larger creep stress, i.e., a higher deviatoricstress, q, applied during creep, gives a greater creep rate.This is because a higher creep stress gives rise to larger con-

Fig. 7 The creep rate-time response of the dense samples in a a log–logscale, and b a log-linear scale. Circles denote creep rupture

Fig. 8 The creep rate-time response of the loose samples. Theresponses from the dense samples are included for comparisons

tact tangential forces. As indicated in Eq. (8), the creep rateincreases as the tangential forces increase. The sample withan isotropic creep stress has the lowest creep rate.

The stage of secondary creep, the same as the experimentalfindings (e.g., [17] for sand, and [52] for clay), is difficult tobe distinguished in Fig. 7a in those cases where creep stressq ≥ 850 kPa. Note that dense samples with a creep stressq ≤ 700 kPa, and loose samples (see Fig. 8) only undergoprimary creep. The constant, minimum creep-rate period, i.e.,the secondary creep, can be better seen if the time axis ofFig. 7a is changed to a linear scale as illustrated in Fig. 7b. Inorder to quantify this period, the corresponding axial strain,

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Fig. 9 The corresponding axial strains at the maximum dilatancy, theminimum creep rate ε̇q,min, creep rupture, and shear failure (i.e., theintersection point between the constant q line and the complete stress–strain curve) as a function of creep stress q. The dotted line representsthe axial strain, εa = 4.1 %, at the maximum dilatancy obtained fromthe complete volumetric response curve. The range bar covers the rangeof the minimum creep rate, ε̇q,min, plus and minus its 0.5 % deviations

which covers the range of the minimum creep rate, ε̇q,min,plus and minus its 0.5 % deviations, i.e., the range for ε̇q,min±0.5 % × ε̇q,min, are presented in Fig. 9. As the creep stresslowers, a larger range bar can be seen in the figure. Thissuggests that secondary creep can last for a longer creep timeor greater axial strain changes by creep as the applied creepstress q is lowered. As q increases, the secondary creep onlycan appears for a very short time. Further discussions on thisaspect from the micromechanics point of view are continuedin the next section.

The point of minimum creep rate, ε̇q,min is also definedas the onset of tertiary creep herein after which the creeprate accelerates rapidly and the sample eventually undergoescreep rupture or creep failure. Campanella and Vaid [52],based on the test results on the same clay samples, found thatthe corresponding axial strains at the minimum creep rateare all in the range of 2.5–3% axial strains regardless of theapplied creep stress. This strain level is also very close to theaxial strain at the peak deviatoric stress, 2.5–3 %, as reportedin Vaid and Campanella [53]. Kwok and Bolton [25], basedon DEM simulations, also suggested that the axial strain atthe onset of tertiary creep is close to the strains at the peakstress and the maximum dilatancy. In our study, as shown inFig. 9, the axial strains at the minimum creep rate and theaxial strains at the maximum dilatancy are also very similar.However, there exists a distinguishable difference betweenthese two axial strains when q ≤ 950 kPa and this differencebecomes larger at a lower q. Indeed, if the experimental datapresented by Vaid and Campanella [53] and DEM simulationresults in Kwok and Bolton [25] are carefully reexamined,a similar trend of such an axial strain difference can also be

found. The underlying mechanism accounting for this dif-ference is also discussed in the next section. It is interestingto note that the axial strains εa at the maximum dilatancy ofthe creep samples regardless of the applied creep stress andthat of the sample subjected only to triaxial shearing (withoutcreep) are similar, i.e., εa = 4.1 % (see Fig. 9).

3.3 Creep rupture

As shown in Fig. 5a for those cases with q ≥ 850 kPa,the creep stress cannot be maintained constant and insteaddecreases after the intersection with the complete stress–strain curve and therefore creep rupture occurs at this inter-section point. Figure 9 also summarizes this behavior fordense samples with different creep stresses where the shearfailure point is the point of intersection between the con-stant q line and the complete stress–strain curve as shownin Fig. 5. Rock samples show similar behavior. If the creepstress is above a critical level and close to the peak stress,creep turns into rupture as the creep stress–strain responsecurve intersects with the complete stress–strain curve (e.g.,see [55]).

4 Simulation results and discussion: micromechanicalinsights

A further examination of the micromechanics involved dur-ing creep was carried out using seven samples. They were:(1) dense samples with creep stresses of 1,050, 900, and850 kPa, i.e., samples A, B and C; (2) dense samples withcreep stresses of 500 and 400 kPa, i.e., samples D and E; and(3) loose samples with creep stresses of 500 and 425 kPa, i.e.,samples F and G. In dense samples, samples A, B, C showedcreep rupture at the end but samples D and E only underwentprimary creep. In loose samples F and G, only primary creepwas found.

4.1 Dense samples

The contributions of contact normal and tangential forcesto the stress tensor can be calculated using Eq. (14) (e.g., see[56]),

σi j = 1

V

Nc∑k=1

Rk N knki nk

j + 1

V

Nc∑k=1

Rk T knki tk

j (14)

where σi j is the average stress tensor; V is the packing vol-ume; Nc is the total number of contacts; Rknk

i defines theposition vector connecting the center of particles in kth con-tact; N k and T k are the normal and tangential contact forcesin the kth contact; and nk

i and tkj are the unit vectors perpen-

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Fig. 10 The contributions of the contact normal and tangential forces to the applied deviatoric stress q during creep for dense samples: a sampleA, q = 1,050 kPa, b sample B, q = 900 kPa, c sample C, q = 850 kPa, d sample D, q = 500 kPa, and e sample E, q = 400 kPa

dicular to and parallel to the kth contact plane respectively.Based on Eq. (14), the contributions of contact normal andtangential forces to the applied deviatoric stress (q = σ1−σ3)

during creep can be quantified. Figure 10 presents the resultsfor dense samples and reveals that the deviatoric stress q(i.e., the solid line) is gradually borne by the contact normalforces (i.e., the dash line) instead of contact tangential forces(i.e., the dotted line) after creep under a constant q initiates.This is the same as the findings made by Kuhn and Mitchell[24]. However, this trend is reversed after around the onsetof tertiary creep in samples A, B, and C. These changes incontact forces can be more clearly seen in Fig. 11 where the

percentage change in the figure is calculated with respect tothe initial value while creep starts. The onset of tertiary creepis at the point of minimum creep rate ε̇q,min as indicated bythe triangle symbol and the trend of the percentage changesof the contact–force contribution to q (for samples A, B, andC) is reversed around this point.

To better understand how the contact force changes affectthe creep behavior, the associated changes in direction haveto be identified. Figures 12 and 13 present the evolution ofcontact normal distributions of strong and weak forces forsamples B and D on the x–z plane where z is the verticaldirection. Following the definition in Radjaï et al. [57], if

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Fig. 11 Percentage changes of the contact–force contribution to theapplied deviatoric stress q during creep for: a the contact normalforce, and b the contact tangential force where the negative sign meansdecreasing

the contact normal force, N , which is greater than its mean〈N 〉 (i.e., N/〈N 〉 > 1), it belongs to the strong forces (orthe strong force network); otherwise it is counted in weakforces (or the weak force network). The evolution of contactnormal distributions is similar among samples A, B, and C,and almost identical between samples D and E. Hence, onlythe results of samples B and D are presented.

The use of contact fabric tensor, φi j , following e.g., Radjaiet al. [58] and Shaebani et al. [59], is a better way to quantifythe contact normal distribution (i.e., those rose diagrams inFigs. 12, 13),

φi j = 1

NC

NC∑k=1

nki nk

j (15)

where nki is the unit vector describing the contact normal

direction for the kth contact and the summation includes thetotal number of contacts NC in the packing. The contact fabrictensor can also be partitioned for the strong “S” and weak“W” force networks as described in Thornton and Antony[60],

φi j = (1 − Np)φWi j + Npφ

Si j (16)

Fig. 12 The evolution of contact normal distribution in dense sample B (q = 950 kPa)

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Fig. 13 The evolution of contact normal distributions in dense sample D (q = 500 kPa)

where Np is the proportion of contacts that belong to thestrong forces. Similar to the calculation of the principalstresses, the orientation of the major fabric with respect tothe horizontal plane is

θ = 1

2tan−1

(2φ13

φ11 − φ33

)(17)

The major and minor principal fabric, 1 and 3, can becalculated by the following equation,

[1

3

]= 1

2(φ11 + φ33) ±

√(φ11 − φ33

2

)2

+ φ213 (18)

The deviator fabric is defined as (1 − 3), which has beenused as an indicator of structural anisotropy [56]. Thesefabric–tensor-related parameters are also used in the follow-ing discussions for the changes in direction of the contactforces.

4.1.1 The primary creep

As found in Fig. 12, B0 → B2 and Fig. 13, D0 → D4,the strong forces, which follow the same trend in response toaxial loading, continue to align along the vertical directionduring primary creep. The weak forces also show the sametrend and therefore more weak forces gradually change fromthe horizontal direction to the vertical direction. This behav-ior can also be seen in Fig. 14 where the deviator fabricof strong forces increases, i.e., becoming more aligned inthe vertical direction, and the deviator fabric of weak forcesdecreases, i.e., those weak forces originally aligned in thehorizontal direction decreasing. Together with the contactforce changes in Figs. 10 and 11 where the deviatoric stress,q, is gradually borne by the contact normal forces instead ofcontact tangential forces, it is believed that during the pri-mary creep, the scenario is similar to that prevailing in thestrain hardening (pre-peak) process during triaxial shearing.As creep straining proceeds (indeed like a process of con-tinuous shearing under a constant q), the weak structure (orvoid cell) should be destroyed first. The void cell elongatedalong the major principal stress direction is more stable thanthat aligned along the minor principal stress [61]. Hence,

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Fig. 14 Deviator fabric as a function of axial strains in dense samplesfor: a the overall forces, b the strong forces, and c the weak forces

particle arching with an elongated void along the major prin-cipal stress direction can survive and gradually prevails dur-ing creep so that all of the forces tend to align in the verti-cal direction. This gives the formation of columnar structureand vertically elongated void as also found in the biaxial test[41,62,63]. Based on the void-shape analysis, Kang et al.[26] reported that the void space has its long axis redirectedto the major principle stress direction during creep.

The formation of columnar structure, on one hand, makesa more stable structure and therefore the sliding ratio of con-tacts gradually decreases as shown in Fig. 15. The slidingratio is defined as the ratio of the number of sliding contactsover the total number of active contacts. Experimental find-ings in Tanaka and Tanimoto [64] support this decreasing

Fig. 15 The sliding ratio as a function of axial strains in dense samplesfor: a the overall contacts, b the contacts with strong forces, and c thecontacts with weak forces

trend of sliding ratios. They reported that there were con-siderable movements of soil particles during creep based onthe recorded acoustic emission (AE) activities, and that theparticle movement decreased with increasing creep time asrecorded AE activities decreased. On the other hand, the for-mation of columnar structure leads to the decrease of thecontact tangential forces as shown in Figs. 10 and 11. This,as indicated in Eq. (8), in turn slows down the creep rate dur-ing primary creep. The same explanations were also givenby Kuhn and Mitchell [24].

Although samples D and E show behavior similar to thoseof samples A, B, and C during primary creep, it is interest-ing to note their differences shown in Fig. 11. The changes

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Fig. 16 The major fabric orientation (in dense samples) with respectto the horizontal direction by Eq. (17) for: a the strong forces, and bthe weak forces where the trend lines for the responses of samples A,B, and C are added

in the contact normal or tangential force contribution to thedeviatoric stress for samples D and E are greater than thosefor samples A, B, C. As a matter of fact, the changes incontact forces increase as the creep stress q decreases. Asnoted, shearing by increasing the axial loading produces asimilar consequence as creep. When a higher q is appliedbefore creep starts, the shearing process would have alreadyhelped adjust a great portion of the contact forces to form thecolumnar structure and therefore less contact force adjust-ments remain during creep.

4.1.2 During the secondary creep for samples A, B, and C

Since only samples A, B, and C showed secondary and ter-tiary creep, the following discussion focuses only on thesethree samples. When the formation of columnar structure iscompleted, the creep rate reaches a minimum. However, thewhole structure becomes more meta-stable as described inMitchell and Soga [65]. Buckling of the columnar structureand the maximum dilatancy inevitably occurs in response tofurther creep straining. This scenario can be better describedin Fig. 16 by the major fabric orientation in weak and strongforces. As shown in Fig. 16 before maximum dilatancyoccurs, the strong forces are mainly aligned in the verticalloading direction (i.e., θ ≈ 90◦) while the weak forces are

Fig. 17 The contributions of the contact normal and tangential forcesto the applied deviatoric stress q during creep for loose samples: asample F, q = 500 kPa, and b sample G, q = 425 kPa

mainly in the horizontal direction (i.e., θ ≈ 0◦). That is,the weak forces serve as lateral support to prevent the buck-ling of load-bearing particle columns where the strong forcesare located. As the buckling of load-bearing columns andthe associated maximum dilatancy occur, the lateral support(weak forces) shows a sharp transition from the horizon-tal orientation to the vertical orientation. That is, the meta-stable columnar structure can only withstand creep strainingfor a little longer before buckling and the subsequent failureprocess occur. This is why the strain range of the secondarycreep in soil is small or even unobserved in general.

As can also be seen in Fig. 16b, the change in the major fab-ric orientation of weak forces is more distinct as q increases.Such a response is also analogous to the height of the rangebar for the minimum creep rate in Fig. 9. As creep stressq increases, the process of structure buckling is expected tooccur more suddenly and therefore the change in the majorfabric orientation of weak forces is sharper and the range bar,somewhat representing the strain range of secondary creep, isshorter. A similar explanation can be applied to the differencebetween the axial strain εa at the maximum dilatancy and εa

at the minimum creep rate as also demonstrated in Fig. 9. Asthe creep stress q gets smaller, e.g., the q = 850 kPa case,although creep straining damages the structure and enablesmaximum dilatancy to take place at the similar strain levelas that obtained from the complete stress–strain curve (i.e.,

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Fig. 18 The evolution of contact normal distributions in the loose sample F (q = 500 kPa)

at εa = 4.1 %), the whole structure can still maintain meta-stable for a further straining. This is because there is a dis-tinct difference between the applied creep stress and the peakdeviatoric stress obtained from the complete-stress curve.As the creep stress nears the peak deviatoric stress, e.g., theq = 1,050 kPa case, the buckling of the structure becomesmore sudden and the whole structure cannot remain stableenough for further straining after maximum dilatancy devel-ops. The axial strains at the minimum creep rate and at themaximum dilatancy then become the same.

4.1.3 During tertiary creep for samples A, B, and C

After the buckling of load-bearing particle columns, the fail-ure process towards creep rupture was triggered in samplesA, B, and C during tertiary creep. As shown in Figs. 10 and11, the trend of contact tangential forces decreasing and con-tact normal forces increasing is reversed. Hence, the slid-ing ratio rises as shown in Fig. 15 due to the increase inthe driving tangential forces and the decrease in the shearresistance (due to the decrease in contact normal forces). Inaddition, as shown in Fig. 12, B2 → B4, it can be observedthat the strong forces decrease rapidly in the vertical loading

direction while the weak forces become mainly aligned in thesame direction. This gives a decrease in the deviator fabric ofstrong forces and an increase in the deviator fabric of weakforces as shown in Fig. 14. Note that the shape of the contactnormal distribution of the weak forces has changed from ahorizontal ellipse to a vertical one (see Fig. 12). Figure 16also indicates that the major fabric orientation in the weakforce is changed to the vertical direction while the one forthe strong forces is slightly tilted from the vertical. All themicromechanical information, once again, suggests that thecontinuous buckling process of the columnar particle struc-ture takes place during the tertiary creep. Ultimately, as thecreep stress–strain line intersects with the complete stress–strain curve, the creep stress q cannot remain constant anddecreases. This is precisely when creep rupture occurs.

The numerical creep tests were performed under a con-stant q and therefore the velocity of the wall (boundaries) hadto be varied during creep to maintain the target stress level.The creep rate rapidly accelerated after the onset of tertiarycreep due to the continuous process of load-bearing particlecolumns buckling. Hence, the walls had to be moved fasterand faster to maintain the constant creep stress. Ultimately,the wall velocity reached the upper bound, 0.1 mm/s, and

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Fig. 19 Deviator fabric as a function of axial strains in loose samplesfor: a the overall forces, b the strong forces, and c the weak forces

could not be adjusted to compensate for the decrease in themajor principle stress due to buckling of load-bearing particlecolumns. This resulted in a decrease in q. As a matter of fact,after the wall velocity reached the limit, the creep test wasturned into a typical strain-controlled, triaxial shearing testagain. This explains why the creep stress–strain responsesfollow the complete stress–strain curve to decrease after theintersection as shown in Fig. 5a.

4.2 Loose samples (only for primary creep)

Although loose samples continue to contract (no dilation)as shown in Fig. 6a, the creep behavior and the involvedmicromechanics are similar to those of the dense samplesduring the primary creep. As shown in Fig. 17, the devia-

toric stress, q, is also gradually borne by the contact nor-mal forces instead of contact tangential forces. This change,again, explains why the creep rate decreases in loose sam-ples during primary creep as shown in Fig. 8. The evolution ofcontact normal distributions for the loose sample F as shownin Fig. 18 are also comparable with those observed for thedense sample D (Fig. 14). Both strong and weak forces tendto gradually align along the vertical loading direction duringcreep. Similarly, the changes in the deviator fabric (Fig. 19)show a similar trend to the one for the dense sample D. Thedeviator fabric of strong forces increases while the deviatorfabric of weak forces decreases.

It is interesting to note that although both the loose sampleF and the dense sample D are under the same creep stress,q = 500 kPa, the loose sample F has a greater creep ratethan the dense sample D as shown in Fig. 8. This is becausethe loose sample has fewer contacts and therefore a greateraverage contact tangential force to give a higher creep ratethan what the dense sample can give.

4.3 Summary and a final thought

Figure 20 summarizes the creep behavior and the underly-ing mechanisms based on the associated micromechanicalproperties. The structural changes of the soil packing indeeddominate the whole creep response. For instance, the forma-tion of columnar structures enables the process of contactforce transfer so that the creep stress is gradually borne bythe contact normal forces instead of contact tangential forces.Such structural and associated contact–force changes, notthe contact-creep model itself alone, give the response ofthe primary creep, i.e., creep rate decreasing with time. Thisstatement can be further supported by the following evidence.Kuhn and Mitchell [24] used the Maxwell model, which onlygives the response of secondary creep under a constant force.However, the primary creep response can still be given intheir simulation. In our study, the contact creep model itselfis not able to accommodate the response of tertiary creep (seeFig. 3). However, tertiary creep can still be modeled.

The same logic can be applied to justify why a viscouscontact model can be used to simulate time-dependent sandbehavior, which has been considered as non-isotach, i.e., nosignificant strain rate effects on the stress–strain relationships[15,16,22,23]. The viscous contact model is used to quan-tify the properties of the material that makes the sand particle.The time-dependent behavior of a sand packing however isstrongly influenced by the structural changes of the packingas demonstrated in this study and in Kang et al. [26]. This phe-nomenon is analogous to a comparison between the concretematerial and a concrete column where the structural changesof the column, e.g., buckling of the column, is significantlyaffected by the column geometry.

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peercyadnoceSpeercyramirPegatspeerC Tertiary creep Creep rate responses Decreasing with time A constant, minimum creep rate Increasing with time

Structural changes The formation of columnar structures. Meta-stable columnar structures

(onset of tertiary creep).

Continuous process of columnar structures buckling

(creep rupture occurs at the end). Micromechanical properties

• The creep stress is gradually borne by the contact normal forces instead of contact tangential forces.

• Sliding ratios continue to decrease. • Deviator fabric in the strong forces increases

(i.e., become more aligned in the vertical loading direction).

• Deviator fabric in the weak forces decreases (i.e., the portion originally aligned in the horizontal direction decreases but the major fabric orientation is still in the horizontal direction).

• Maximum dilatancy occurs. • The sliding ratio is the lowest. • The major fabric orientation of the weak

forces is about to change from horizontal to vertical.

• The trend of contact tangential forces decreasing and contact normal forces increasing is reversed.

• Sliding ratios continue to increases • Deviator fabric in the strong forces

decreases (i.e., becomes less aligned in the vertical direction and the major fabric orientation finally is slightly tilted from the vertical direction).

• Deviator fabric in weak forces increases (i.e., the major fabric orientation is changed from horizontal to vertical).

Fig. 20 Summary of the creep behavior and the associated micromechanical properties

5 Conclusion

In this study, DEM simulations of triaxial creep tests werecarried out to explore the underlying mechanisms and themicromechanical properties involved during creep in sand.Burgers model was used to simulate the visco-frictional con-tact creep and the rolling resistance between particles wasconsidered. The simulated creep responses are consistentwith the published experimental results. For instance, in bothloose and dense samples, the creep rate (i.e., the deviatoricstrain rate) decreases continuously during the primary creep,and the creep rate increases with increasing creep stress q.The stage of secondary creep is not evident, especially whenthe applied creep stress is large. During the tertiary creep,the creep rate rapidly accelerates and finally creep ruptureoccurs if the creep stress is large enough in the dense sam-ples. In addition, all of the creep samples follow their orig-inal volume-change tendency to continue their dilation orcontraction response during creep.

The structural changes play an important role to governthe creep response. During the primary creep in both looseand dense samples, the contact forces are gradually changedin a way that the tangential forces continue to decrease but thenormal forces continue to increases to form columnar parti-cle structures. The deviator fabric of strong forces increaseswhile the one for weak forces decreases because forces tendto align along the vertical direction. These, on one hand,make a more stable structure and therefore the sliding ratios

of contacts gradually decrease. On the other hand, these leadto a decrease in the creep rate. The columnar structures even-tually are completely formed and the creep rate attains aminimum. Nevertheless, the structures become meta-stableand vulnerable to buckling. This is why a sand packing ingeneral does not show a distinct period of secondary creep.The higher the creep stress applied, the shorter the periodof secondary creep is. As the columnar structures buckledand maximum dilatancy occur, the major fabric orientationof the weak forces display a sharp transition from the hori-zontal to the vertical directions, especially for the cases witha higher creep stress. The nonstop process of columnar struc-tures buckling accelerates the creep rate and raises the contactsliding ratios during the tertiary creep. Apart from that, thetendency of contact tangential forces decreasing and contactnormal forces increasing becomes the opposite. Finally, themajor fabric orientation in strong forces is slightly tilted fromthe vertical direction.

Acknowledgments This research was supported by the Hong KongResearch Grants Council (GRF 620310). The writers are grateful to thereviewers for valuable comments.

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