ARTICLE

Depth-independent cone penetration mechanism by a discreteelement method (DEM)-based stress normalization approachSu Liu and Jianfeng Wang

Abstract: A two-dimensional discrete element model of driven piles in crushable sand was developed and validated againstlaboratory data. Numerical experiments were conducted to investigate the effects on the pile penetration behavior of initialsample porosity, particle crushability, initial stress state, ratio of pile diameter to median particle diameter, and ratio of modelwidth to pile diameter. A new stress normalization method is adopted to synthesize the data at different driven depths from thesimulations. The normalized vertical and horizontal stress fields surrounding the pile show an invariable pattern of stressdistribution, suggesting a unique penetration mechanism independent of the penetration depth. The validity of the discreteelement method (DEM) simulation results is verified by comparing the stress distributions to those observed from calibrationchamber tests on model pile installation in sands.

Key words: pile penetration, crushable sand, particle breakage, discrete element method (DEM) simulation, stress normalization.

Résumé : Un modèle bidimensionnel de pieux enfoncés dans du sable écrasable, basé sur la méthode des éléments distincts(MED), a été développé et validé a partir des données obtenues en laboratoire. On a effectué des expériences numériques pourétudier les effets de la porosité initiale d’échantillons, l’écrasabilité des particules, l’état de contrainte initial, le quotient dudiamètre du pieu par le diamètre moyen des particules et quotient de la largeur du modèle par le diamètre du pieu, sur lemécanisme de pénétration du pieu. Une nouvelle méthode de normalisation des contraintes a été adoptée, permettant desynthétiser les données de simulation obtenues a différentes profondeurs d’enfouissement du pieu. Les champs de contraintesnormalisées verticales et horizontales entourant le pieu montrent que les motifs de distribution des contraintes ne varient pas,ce qui met en évidence un mécanisme de pénétration unique indépendant de la profondeur de pénétration. On vérifie la validitédes résultats de la simulation basée sur la MED en comparant les distributions des contraintes a celles obtenues grâce a des essaisen chambre d’étalonnage effectués sur des pieux expérimentaux enfouis dans du sable. [Traduit par la Rédaction]

Mots-clés : pénétration d’un pieu, sable écrasable, rupture des particules, simulation basée sur la méthode des éléments distincts(MED), normalisation des contraintes.

IntroductionOver the past two decades, field and laboratory investigations

on instrumented prototype or model piles installed in sand havemade considerable contributions towards revealing the pile pen-etration mechanism and establishing more rational and reliablemethods for pile design in sand (e.g., Lehane 1992; Randolph et al.1994; Jardine and Chow 1996; Klotz and Coop 2001; Jardine et al.2013a). The stresses measured directly at the tip and along theshaft of the pile in this type of study provide first-hand informa-tion for understanding a range of complex soil behaviors arisingfrom the pile installation, including very large soil deformation(Liyanapathirana et al. 2000), localized interface shearing (Whiteand Bolton 2004), particle breakage (Yang et al. 2010), loadcycling and shaft friction degradation (White and Lehane 2004),soil creep and pile setup (Chow et al. 1998; Lim and Lehane 2014), etc.

By using highly instrumented calibration chamber tests on thedriving of model displacement piles (Jardine et al. 2013a, 2013b;Yang et al. 2014), the full-field stress distribution within the sandmass surrounding the pile could also be obtained, facilitating exam-ination of the chamber boundary effects and establishment of thelinkage between penetration resistances measured in the calibrationchambers and the field (Pournaghiazar et al. 2012). The experimentalresults of physical modeling tests provide benchmarks for theoreti-cal and numerical modeling of the pile penetration problems.

In parallel with the experimental investigations, considerableadvances have also been achieved on this topic using the discreteelement method (DEM), whose advantages include full access tothe particle-scale kinetic and kinematic information and the po-tential of being a virtual testing tool substituting physical modeltests (Lobo-Guerrero and Vallejo 2005; Jiang et al. 2006; Wang et al.2007a, 2007b; Arroyo et al. 2011; Wang and Jiang 2011; Lin andWu 2012; Mcdowell et al. 2012; Butlanska et al. 2014; Falagush et al.2015). In a previous related study (Wang and Zhao 2014), the au-thors made a detailed discrete-continuum analysis of the pilepenetration behavior based on the two-dimensional (2D) DEM simu-lation results. The stress and strain data provided by the model werecompared with experimental results from instrumented model pilesby Klotz and Coop (2001) and the particle image velocimetry (PIV)measurements of soil deformation in calibration chamber tests byWhite and Bolton (2004), and were mainly used to demonstrate theeffects of in situ stress field, initial soil density, and particle crush-ability on the pile penetration behavior. The current study, on thebasis of that previous study, aims to further probe the penetrationmechanism by deepening the stress analysis of the pile and sandmass. The goal is to unravel and present a unique penetration mech-anism within a homogeneous sand mass that is independent of thepile penetration depth. The effects of two additional model-scalevariables — namely, the ratio of pile diameter to median particle

Received 20 April 2015. Accepted 12 November 2015.

S. Liu and J. Wang. Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong, China.Corresponding author: Jianfeng Wang (email: jefwang@cityu.edu.hk).

871

Can. Geotech. J. 53: 871–883 (2016) dx.doi.org/10.1139/cgj-2015-0188 Published at www.nrcresearchpress.com/cgj on 20 November 2015.

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diameter and the ratio of model width to pile diameter — wereincluded in the current study to enhance the research findings.

Numerical methodSimulations were carried out using the PFC2D program (Itasca

Consulting Group 2008). Crushable particles are simulated byagglomerates, each of which is composed of 24–30 parallel-bondedelementary disks with diameters between 0.069 and 0.278 mm(Fig. 1d). The contact between two elementary disks within an ag-glomerate consists of three parts: a linear stiffness model, a slipmodel, and a parallel-bond model (Cheng et al. 2003; Wang and Yan2013). A parallel bond breaks if the normal or shear stress acting onthe bond exceeds its corresponding bond strength. The conven-tional linear contact model with a slip failure mechanism willtake effect after a parallel bond is broken. The advantage of theparallel-bond model over the contact-bond model is that it simu-lates a finite piece of cement between two disks providing a rota-tional resistance. More details of the particle-contact model can befound in Wang and Yan (2013).

Numerical biaxial testPrior to the numerical penetration test, the behavior of the

DEM materials was first characterized by means of the numericalbiaxial test. The size of the specimen was 80 mm in height and40 mm in width. The top and bottom walls were set to be frictionless.The lateral flexible membrane boundaries (Åström et al. 2000;Alonso-Marroquin and Herrmann 2002; Wang and Leung 2008;Zhou et al. 2013) were modeled by strings of small mono-sizedparticles connected by contact bonds. During the construction ofthe models of both the biaxial and penetration tests, the granularsamples were prepared by depositing and compacting granulatesin multiple layers to achieve a homogeneous dense state. Thegranular material was composed of crushable agglomerates oruncrushable disks with diameters uniformly varying between 0.6

and 1.2 mm. The input DEM parameters of both models are givenin Table 1.

Figure 2 shows the deviatoric stress ratio �/� (� = (�1 – �3)/2,� = (�1 + �3)/2) (where � is the deviatoric stress, and �1 and �3 arethe major and minor principal stresses, respectively) and volumet-ric strain versus axial strain relationships. Each test is denotedusing a code of “b-c, x”, where b is the parallel bond strength (inN/m), c is the initial sample porosity, and x is the confining pres-sure. A maximum confining pressure of 1.0 MPa was chosen be-cause it was close to the maximum value of confining stressencountered in penetration tests. The general trends of the stress–strain behavior from Fig. 2 include (i) reduction of the peak andcritical state stress ratios with increasing confining stress for asample with fixed bond strength and porosity, (ii) an overall upwardshift of the stress–strain curve with decreasing porosity for a samplewith fixed bond strength and confining stress, and (iii) a reduction ofboth stress ratios with decreasing bond strength for a sample withfixed porosity and confining stress. The above trends reflect the re-sults of the competition between particle breakage, sample porosity,and confining stress in controlling the stress ratio behavior (Wangand Yan 2012). It is noted that trend (i) has led to a continuous hard-ening behavior (without a distinct peak) for a dense sample with aporosity of 0.13 at a confining stress of 1.0 MPa, and a correspondingvolumetric contraction behavior throughout the test, which is all

Fig. 1. (a) Model geometry of tests 1–5; (b) layout of groups at fixedpositions, with size of each group being 0.5B × 0.5B (where B is pilediameter); (c) layout of groups at relative positions to pile tip;(d) typical agglomerate composed of parallel-bonded disks.

Fig. 2. Evolution of (a) stress ratio and (b) volume strain of biaxialtest samples. [Colour online.]

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characteristic of a loose sand due to massive particle breakage occur-ring during the test. In contrast, trend (iii) does not show a pro-nounced effect when the confining stress is too low (e.g., 0.1 MPa),making only a slight difference in the stress–strain curves of samples“2e7-0.13, 0.1 MPa” and “1e7-0.13, 0.1 MPa”, which is now due to theinsignificant particle breakage induced. For all the cases, the criticalstate friction angle varies between 25o and 27o as the confining stressis varied.

DEM modelling of pile penetration in sand

Model constructionThe DEM model of the penetration test is made up of a rectan-

gular container filled with a well-compacted, polydispersed as-sembly of round particles and a model pile with a triangular tip(two inclined planes each making an angle of 60o with the hori-zontal) pushed gradually into the granular foundation. The fric-tion coefficient of the model pile was set to 0.5. As illustrated inFig. 1a, only the right half of the model is used, taking advantageof the axial symmetry of the problem. All the boundary walls arefixed. The bottom and left walls are set to be frictionless, while theright wall is frictional with a friction coefficient of 0.9 to minimizeany relative slip between the particles and the wall. The full modelhas the same value of width 2W and height, H, of 240 mm, and apile diameter, B, of 8 mm, giving an aspect ratio H/W of 2 and aratio of model to pile width (2W/B) of 30 in the regular-sized model(i.e., tests 1–5). Test 6 has a model width twice that of the regular-sized model, with a total number of disks about 25% higher thanthe regular-sized model; while test 7 has the same model dimen-sions except with a particle size doubling that of the regular-sizedmodel, with a total number of disks approximately 4 times that ofthe regular-sized model.

The granular foundation consists of two zones: a crushablezone surrounding the pile and an uncrushable zone surroundingthe crushable zone (Fig. 1a). The dimension of the crushable zoneis 16 mm × 216 mm in tests 1– 6 and 32 mm × 432 mm in test 7. Thegranular material in the uncrushable zone is composed of rigiddisks with diameters varying uniformly between 0.6 and 1.2 mm.The agglomerates in the crushable zone are the same material asused in the biaxial test. To achieve the same initial stress distribu-tion as in the crushable zone, the density of disks is set to

2200 kg/m3 in the uncrushable zone. More details about the modelgeneration can be found in Wang and Zhao (2014).

Simulation programSeven numerical penetration tests, as summarized in Table 2,

were conducted. Tests were grouped to examine different issuesthat influence the penetration results, such as (a) lateral boundary(tests 3 and 6), (b) pile size (tests 3 and 7), (c) initial vertical stress(tests 1, 2, and 3), (d) initial porosity (tests 3 and 4), and (e) particlecrushability (tests 4 and 5). Specifically, the boundary effect andpile size effect were studied by varying the value of W/B in test 6and B/d50 (where d50 is the median particle size) in test 7, respec-tively. The varying initial vertical stress field was achieved byapplying artificially raised gravitational acceleration, and thesand crushability was defined by different parallel bond strengthvalues (pb_s). The model pile is pushed monotonically into thegranular foundation at a constant rate of 0.1 mm/s, which is slowenough to maintain the quasi-static loading condition throughoutthe test. The maximum penetration depth is 176 mm in tests 1–6and 352 mm in test 7. The computational time using an Intel Corei7–4770K CPU at 3.50 GHz and with 16 Gb RAM was about 3 daysfor the regular-sized model and 28 days for the double-sized model(i.e., test 7).

Results and discussion

Tip resistance curvesFigure 3 shows the pile tip resistance, qb, registered from all

simulation tests. Results are categorized to show the effects ofmodel variables on the penetration resistance in six subfigures.Figures 3a and 3b show the effects of artificial gravity levels on thetip resistance using the equivalent prototype depth (where 1 mmof depth in the model with an artificial gravity of xg representsx mm of depth at prototype scale) and the actual model penetra-tion depth, respectively. Use of the prototype depth enables ex-amination of the variation of penetration resistance against the insitu field stress based on the model result. It is clear that the threegravity levels produce three distinct curves in the plot using themodel penetration depth (Fig. 3b), but result in one unique curvewhen plotted against the prototype depth (Fig. 3a). Note, thatthe data from test 1 are excluded from Fig. 3a due to inadequate

Table 1. Input parameters for DEM simulations.

ParameterBiaxial test; crushablezone of penetration test

Flexibleboundary

Uncrushable zoneof penetration test

Diameter of agglomerates (mm) 0.6–1.2 0.225 0.6–1.2Diameter of elementary disks (mm) 0.069–0.278 — —Density of disk (kg/m3) 2650 1000 2200Normal and shear stiffnesses of disk (N/m) 4e8 8e7 4e8

Friction coefficient of disk 0.5 0.0 0.5Normal and shear parallel-bond strengths (N/m2) 1e7, 2e7 — —Normal and shear parallel-bond stiffnesses (N/m3) 1.5e12 — —Normal and shear contact-bond strengths (N) — 1.0e100 —Ratio of parallel bond radius to disk radius 0.5 — —

Table 2. Characteristics of pile penetration tests.

Test IDDimensions,W×H (m×m)

Pile diameter,B (m)

2W/B;W/R B/d50

Gravitylevel (g) Porosity

Bond strength,pb_s (N/m)

Numberof disks

Average modelrun (days)

Test 1 0.12×0.24 0.008 30 8.9 1 0.2 2e7 174 343 3.0Test 2 0.12×0.24 0.008 30 8.9 50 0.2 2e7 174 358 3.0Test 3 0.12×0.24 0.008 30 8.9 100 0.2 2e7 174 406 3.0Test 4 0.12×0.24 0.008 30 8.9 100 0.13 2e7 190 678 3.1Test 5 0.12×0.24 0.008 30 8.9 100 0.13 1e7 190 678 3.1Test 6 0.24×0.24 0.008 60 8.9 100 0.2 2e7 213 133 3.2Test 7 0.24×0.48 0.016 30 17.8 100 0.2 2e7 695 649 28

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Fig. 3. Unit base resistance, qb, under influence of (a, b) gravity level; (c) porosity; (d) parallel bond strength; (e) boundary effect; (f) scaledparticle size effect. [Colour online.]

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Fig. 4. Normalized (a) horizontal, (b) shear, and (c) vertical stressesmeasured at r/R = 1.5 in test 7. [Colour online.]

Fig. 5. Normalized (a) horizontal, (b) shear, and (c) vertical stressesmeasured at y/R = 19.5 in test 7. [Colour online.]

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Fig. 6. Normalized stresses measured at relative positions to pile tip at various penetration depths in (a, d, g) test 7, derived from(� ′ � �0

′ )/qb; (b, e, h) test 7, derived from (� ′/qb)/(�z0′ /pA)0.13); and (c, f, i) test 3, derived from (� ′ � �0

′ )/qb). [Colour online.]

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prototype depth. Data between penetration depths of 24 and176 mm in Fig. 3b were fitted linearly, with the ratio between theslopes of the lines from tests 1, 2, and 3 roughly equal to the ratioof their gravity levels. The figure is replotted in semi-log scale tobe able to appreciate the trend of test 1. Data from the penetrationdepth less than 24 mm were not used for the linear regressionanalysis due to a different, shallow penetration mechanism (Jianget al. 2006), and exclusion of these data was done to the linearregression in all the other subfigures of Fig. 3. The above resultsindicate the clear capability of the current DEM model in reflect-ing the in situ stress effect on the tip resistance behavior. Thelinear distribution of the tip resistance suggests a consistent pen-etration mechanism independent of the penetration depth.

The influence of the initial porosity of the sand is shown inFig. 3c, where a reduction of the porosity from 0.2 to 0.13 leads toa 70% increase of the tip resistance at the final driven depth. Thedecrease of sand crushability via the increase of parallel bondstrength results in a similarly clear increase of the tip resistance(Fig. 3d). In both cases, little difference of the tip resistance is ob-served for a prototype depth less than 11 m, above which the confin-ing pressure is relatively low. It is worth noting that, under a lowconfining stress, the volumetric strains of the granular materialsused in tests 3–5 are very close to each other, while under a highconfining pressure, the material in test 4 contracts much less thanthe others (Fig. 2b).

The influences of the two model-scale variables W/R (where R isthe pile radius) and B/d50, are given in Figs. 3e and 3f, respectively.It is found that doubling the model width causes little change ofthe tip resistance curve (Fig. 3e), indicating that the regular-sizedmodel does not have a significant lateral boundary effect and themodel width of 120 mm is enough to produce the physically soundand realistic penetration behavior. Interestingly, doubling theB/d50 ratio produces a tip resistance curve (in test 7) that extends toa prototype depth twice that of a regular-sized test (e.g., test 3) andhas a bilinear profile, whose first linear segment almost overlapswith that of test 3 and a second linear segment that has a slightlylower slope (Fig. 3f). This result has two implications: firstly, usinga larger pile width does not change the tip resistance significantlywithin the low to moderate range of prototype depth (0–18 m),and the original pile width of 8 mm is enough for modeling pur-poses; secondly, for the larger penetration range of 18–36 m, the tipresistance curve exhibits a lower rate of increase, mainly due to thehigher amount of particle breakage occurring around the pile tip.

Pile penetration mechanism via stressnormalization approach

In the following section of the paper, it is endeavored to reveala unique pile penetration mechanism by applying a stress normal-ization approach to the homogenized stress data obtained fromthe DEM simulations. This is motivated by the observed linear effectof in situ vertical stress on the tip resistance behavior and the at-tempt to establishing a unique linkage between the tip resistanceand penetration-induced stresses within the surrounding soil massby removing the in situ stress effect. Such a goal is accomplished bypresenting the stress analysis data in two parts: (i) normalized stressdata measured on particle groups in this section and (ii) normalizedfull-field stress distribution in the next section.

Group-based stress measurementAs illustrated in Fig. 1b, a total number of 1800 particle groups,

with a size of 0.5B × 0.5B for each group, are identified beforepenetration in tests 1–5 and test 7. Each particle group containsabout 100 crushable agglomerates or uncrushable disks in test 7and 25 in tests 1–5. The average stress in a group is found using asimilar averaging procedure based on the measurement logic inPFC2D (Itasca Consulting Group 2008) and the position of each groupis represented by the average coordinates of all particles within the

group. It should be pointed out that the stress measurements weremade on particle groups identified on the undeformed configurationprior to the pile penetration in this study, which is different from thestress measurements based on sampling windows defined on thedeformed configuration in the previous study (Wang and Zhao 2014).This is because the current way of stress measurement allows a directcomparison with the experimentally measured stress data from thelaboratory model pile test (Jardine et al. 2013b), in which the stresssensors were embedded within the sand mass before pile penetra-tion.

Synthesis of group-based stress measurementStresses in the sand subjected to pile penetration consist of the

initial stress and the stress change caused by penetration. Theinitial vertical stress in the field or centrifuge tests increases withdepth, while remaining constant in calibration chamber tests withconstant stress boundary conditions. The initial stress near the piletip is relatively small compared with tip resistance and the impact ofpenetration decreases rapidly with increasing distance from thepile tip. A number of different methods (Jardine et al. 2013b; Yanget al. 2014) can be found in the literature to normalize the effectivestresses (� ′) developed in sand during installation. If the stressesare normalized by the tip resistance (qb) after eliminating theinfluence of initial stress, they can be expressed by a 2D functionof the normalized vertical and horizontal distances from the piletip as (� ′ � �0

′ )/qb = �� ′/qb = f(h/R, r/R), where �0′ is the initial

effective stress, r is the relative offset from the pile axis, h is therelative height from pile tip. So, r/R is the normalized relativeoffset from the pile axis, and h/R is the normalized relative heightfrom pile tip. A negative value of h/R means a position below thepile tip. In this way, the stress field within the sand mass becomesa function of the relative distance from the tip of a “stationary”pile, independent of the actual penetration depth of the tip.

The stress evolutions of particle groups whose initial positionsare either on the same row or the same column (Fig. 1b) are tracked.Figure 4 shows the normalized stresses (�11

′ , �12′ , and �22

′ denote

Fig. 7. Friction angle measured at relative positions to pile tip atvarious penetration depths in test 7. [Colour online.]

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horizontal stress, shear stress, and vertical stress, respectively) offour groups of particles (i.e., G1, G2, G3, and G4) positioned at col-umn 2, where r/R = 1.5, against the normalized penetration depth y0/B(where y0 is the penetration depth from pile tip to the upper bound-ary) from test 7. It is seen that maxima of the normalized horizontaland vertical stresses developed, while the sign of the normalizedshear stress changed as the pile tip leveled with the center of eachgroup. The peak magnitudes of the normalized horizontal stress vary

between 14.5% and 20.7%, show a similar level of variation to those ofthe normalized vertical stress, which vary between 9.2% and 13.6%.The similar pattern of results indicates that the normalized stressesvary as function of the normalized relative height from the pile tip.

The normalized stresses measured in four groups of particles(i.e., G2, G5, G6, and G7 in Fig. 1b) positioned at the same row withy/R = 19.5 (where y is the vertical position of a particle group fromupper boundary) are shown in Fig. 5. One common feature is the

Fig. 8. Normalized (a) horizontal, (c) vertical, and (e) shear stress contours (in %) during penetration in test 7; (b) radial and (d) vertical stresscontours (in %) from experimental data (after Jardine et al. 2013b). [Colour online.]

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rapid decrease of maximum stresses with the increasing distancefrom the pile tip, indicating the normalized stresses being a func-tion of the normalized relative offset from the pile axis. The nor-malized stresses vary in the range of ±2% in “G6” and “G7”, whichare positioned at r/R = 19.5 and r/R = 28.5, respectively. This obser-vation indicates again that no severe lateral boundary effect isintroduced in this model.

At various penetration depths, the normalized stresses of nineparticle groups with initial positions fixed with respect to the piletip (Fig. 1c) from tests 3 and 7 are plotted against y0/B in Fig. 6. Atany instance of penetration depth, the positions of any two groupsare not on the same row or column so as to obtain a comprehensiveand unbiased picture of the normalized stress distribution as a func-tion of the relative distance from the particle group to the pile tip.

The relationship between �� ′/qb and y0/B (Figs. 6a, 6d, and 6gfrom test 7; Figs. 6c, 6f, and 6i from test 3) can be best representedas a constant function, which means the normalized stresses canbe expressed as 2D, axially symmetric functions of the relativeposition to the pile tip. Note this is the case for all particle groupsexcept “relG1”, which is immediately below the pile tip and notsurprisingly shows some oscillations due to the very large defor-mation of the group. The higher particle number in each group intest 7 is seen to lead to a less scattered trend than test 3. Manyauthors (e.g., Yang et al. 2014) have used the empirical relation-ship (� ′/qb)/(�z0

′ /pA)0.13 to account for the dependence of qb on �z0′ ,

where �z0′ is the initial vertical effective stress and pA is the atmo-

spheric pressure. Figures 6b, 6e, and 6h show an example of anattempt to use this approach for stress normalization in test 7. Itis clear that the normalized horizontal and vertical stresses of thegroups away from the pile tip have a tendency to increase withpenetration depth (Figs. 6b and 6h). This is because the tip resis-tance has a decreasing influence on the particle groups away fromthe pile tip and taking (�z0

′ /pA)0.13 as a normalization factor under-estimates the effect of initial vertical stress.

Besides the normalized stresses, it is also interesting to examinethe trend of the mobilized friction angle of the particle groups.Figure 7 shows the variation of the mobilized friction angle againsty0/B from selected groups “relG1”, “relG4”, and “relG7” in test 7. Thefriction angle is calculated directly from arcsin((�1

′ – �3′ )/(�1

′ + �3′ )),

where �1′ and �3

′ are the major and minor principal stresses, respec-tively. The friction angle of “relG1” is seen to decrease from about 29o

to 22o within a normalized driven depth of 3 to 9, and thereafterremains at a roughly constant value of 22o, which is close to thecritical-state friction angle of DEM material “2e7-0.2” from the bi-axial simulation under the confining stress of 1.0 MPa. At “relG4” and“relG7”, the mobilized friction angles are much lower and exhibitmuch larger scatters, indicating that granular materials at these twopositions have rarely reached the critical state.

Full-field stress distributionGiven the independency of the normalized stresses from the

penetration depth seen in Fig. 6, it is now interesting to examinethe contour map of the normalized stresses around a pile. Thecontour map is generated by calculating the average values of thenormalized stresses of all particle groups at various penetrationdepths and then presenting the average full-field stress distributionaround a pile with a generic penetration depth. Typical contourmaps of the normalized horizontal, vertical, and shear stressesfrom test 7 are shown in Fig. 8. The position of each particle groupis calculated as the average coordinates of all particles within thegroup at the deformed configuration. To demonstrate the validityof the simulation results, experimental data of the model pilefrom the calibration chamber experiments by Jardine et al. (2013b)are included in Fig. 8 for comparison. Although the normalizationmethod of Jardine et al. (2013b) also takes the form of (� ′/qb)/(�z0

′ /pA)0.13,it provides the useful benchmarks for the simulation results, es-pecially when the different initial stress conditions are taken intoconsideration.

Compared with the measured data of Jardine et al. (2013b), thenormalized horizontal (Fig. 8a) and vertical stress (Fig. 8c) contourmaps from test 7 show generally comparable patterns within azone from the pile centerline to a horizontal offset about r/R = 10.Some divergence from the experimental results beyond this zone,especially near the boundaries, is observed. The difference is causedby different boundary conditions, which are the constant normalstress upper boundary with zero surcharge and fixed lower bound-ary in the numerical model, while there was a top and a bottommembrane applying a surcharge pressure of 150 kPa in the calibra-tion chamber tests of Jardine et al. (2013b). The horizontal stresscontour map shows a roughly symmetric distribution about the tiphorizon, with the stress contour bulb extending to a horizontal dis-tance of about 8R and decreasing rapidly to 3% at a vertical distanceof 2R–3R vertically. In contrary, the vertical stress contour mapshows an unsymmetric distribution about the tip horizon, withthe stress contour bulb around the tip expanding almost down-ward vertically in the simulation, or in an inclined direction be-low the tip horizon in the experiment. Due to the completeelimination of the effect of initial stress, the normalized maxi-mum horizontal and vertical stresses, which were 14% and 13%,respectively, are slightly lower than the experimental results. Thenormalized shear stress distribution displays a usual “X”pattern(Sheng et al. 2005), and can be divided into five zones, as shown inFig. 8e, to allow a comparison with the sand deformation zonesobserved in the digital image correlation (DIC) study by Arshadet al. (2014). DIC is a noncontact image analysis method, based ongrey-value digital images, which can determine the displacementand strain. In their study, the lower band of shear stress localiza-tion coincides with zone III, where the displacement field rotatesfrom the vertical direction in zone I to the radial direction in zoneIV. An additional zone V is identified between zone IV and thecrushed band based on the observed upper shear band. The bandalso indicates that there are relative displacements between zones

Fig. 8 (concluded).

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Fig. 9. Normalized stress contours (in %) during penetration in (a) test 3; (b) test 2; (c) test 4; (d) test 5. [Colour online.]

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Fig. 9 (concluded).

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IV and V and the former zone moves relatively toward the upper-right.

Figure 9 shows the effects of initial vertical stress field, initialporosity, and particle crushability of tests 2–5. The computedstress contours of test 2 (Fig. 9b) generally agree well with that oftest 3 (Fig. 9a), although some slight differences in the horizontaldirection are found between the two cases. This totally agreeswith the expectation of the authors, as the effect of the initialvertical stress field created by the use of artificial gravity acceler-ation is fully eliminated in such a normalized approach. The dif-ferences mainly occur in the horizontal stress and shear stresscontours, with slightly larger values of normalized stresses foundat the same offset from the pile centerline in test 3. A stronginfluence of the initial porosity is observed in Figs. 9c and 9d,where a reduction of the porosity leads to an up-moving trend(from the tip to the shoulder) of the maxima points, especially inthe normalized horizontal stress contour map. Anticlockwise (up-per shear band) and clockwise (lower shear band) rotation is alsoobserved. These changes are attributed to the denser fabric that isable to support higher deviatoric stress caused by the pile pene-tration. For the particle crushability effect, it is found that thestress contour bulbs tend to shrink towards the maxima point(Fig. 9d), suggesting more localized deformation near the pile tipin the more crushable sample (Arshad et al. 2014).

Finally, the quantitative relationship of the stress decay withthe horizontal distance from the pile centerline is examined. Theaveraged ��11

′ /qb at the “maximum”level (h/R = 0.25 in tests 2, 3,and 7; h/R = 1.25 in tests 4 and 5) are plotted against r/R on a log–logdiagram in Fig. 10. Consistent linear patterns are found in allcases. The slope of the correlation ranges from −0.88 to −1.11, andshows dependence on the friction angle of the DEM material. Theslopes agree well with the measured gradient by Jardine et al.(2013b). This range of slopes lies between the ranges of the “spher-ical case” (−1.18 to −1.36) and “cylindrical case” (−0.59 to −0.68)predicted by the cavity expansion theory (Yu and Houlsby 1991),suggesting the actual penetration mechanism is a mixed type ofthe two theoretical cases.

ConclusionThis study endeavors to unravel a unique pile penetration

mechanism in a homogeneous sand mass through an in-depthanalysis of the stress field surrounding the pile from 2D DEMsimulations. An invariable penetration mechanism revealed bythe normalized stress fields eliminating the initial vertical stresseffect was shown to be independent of the penetration depth. Anew stress normalization method was adopted to synthesize theDEM-based stress data at different driven depths. The validity ofthe normalization method was verified by comparing the DEMsimulation results with experimental data from calibration cham-ber tests, adopting an empirical normalization method. It wasshown that the normalized horizontal and vertical stress contourmaps from DEM simulations have generally comparable patternsto those from the experiments within a zone from the pile cen-terline to a horizontal offset about r/R = 10. Due to the differentboundary conditions adopted in the simulations and experiments,some difference is observed near the lateral boundaries.

The normalized shear stress field showed incremental displace-ment zones similar to those observed in a previous DIC study. Theidentification of different shear stress zones provides insightsinto the general failure mechanism during the penetration pro-cess. Results from the parametric simulation study show that alower initial porosity led to an up-moving trend of the maximapoint of the stress contour and had a larger influence on the normal-ized shear stress field than other factors. The averaged ��11

′ /qb calcu-lated at the “maximum” level from all the simulations showed aconsistent linear pattern. The slopes of the linear correlations agreewell with the measured gradients by Jardine et al. (2013b), but fallbetween the ranges of the “spherical case” and “cylindrical case”predicted by the cavity expansion theory, suggesting the actual pen-etration mechanism is a mixed type of the two theoretical cases.

Finally, it needs to be mentioned that the current 2D model hassome limitations, such as the absence of information about thecircumferential stress and its influence on the pile tip resistance,and possibly larger lateral boundary effects, etc. These issues willbe overcome in a three-dimensional simulation study that is cur-rently underway using an open-source DEM code.

AcknowledgementsThe study presented in this article was supported by the Gen-

eral Research Fund No. CityU 122813 from the Research GrantCouncil of the Hong Kong SAR, Research Grant No. 51379180 fromthe National Science Foundation of China, and Shenzhen (China)Basic Research Project No. GJHS20120702140546194.

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