Numerical modelling of subglacial sediment deformationAnders D. CHRISTENSEN1, David L. EGHOLM1, Jan A. PIOTROWSKI1,21 Department of Earth Sciences, Aarhus University, Aarhus, Denmark., 2 Department of Geography, University of Sheffield, Sheffield, United Kingdom.Correspondence: anders.damsgaard@geo.au.dk, http://cs.au.dk/~adc

Numerical simulation of subglacial sediment deformation on micro- to inter-mediate scale offers detailed insight into the origin of structural and texturalfeatures related to progressive shear strain. The Discrete Element Method(DEM) is used to simulate the behaviour of granular material under conditionsmimicking subglacial shear stress, normal stress and shearing velocities.

Based on a Lagrangian approach, each particle is an unbreakable spheri-cal entity with spatial and rotational degrees of freedom and inertia. Treatingthe particles as elastic bodies with friction at contact surfaces approximatesinter-particle contact forces. By discretizing time into small steps, the result-ing movement is calculated by integration of Newton’s second law of motion.The particle assemblage is subjected to gravity and forced by a set of fixedboundary conditions, moving in a shearing fashion with periodic lateral bound-aries, allowing high strain simulations with a limited number of total particles.The numerical approach offers complete user control of material properties,simulation parameters and boundary conditions while yielding a very highcontinuous data output readily available for high-precision statistical analy-sis.

We use laboratory ring-shear experiments on granular material as bench-mark experiments for validating the numerical model. Comparisons betweenlaboratory- and computational experiments are presented. Also presented arepreliminary results from a research project concerning pressure-distributingforce chain patterns, parameter dependence of the mechanical rheology of thematerial, and analysis of particle advection and intergranular mixing duringhigh-strain shear experiments.

The future work plan involves code modification to include porewater flowfor investigating mechanical and thermal feedbacks between the granular flowand porewater flow (e.g. during three-dimensional plowing movements). Thehigher computational requirements are supported by acceleration availablefrom general-purpose graphics processing units (GPGPUs).

Keywords: subglacial deformation, numerical modelling, discrete elementmethod, ring-shear

AbstractThe goal of this project is to quantify the impact of glaciers overriding soft, deformablesediments, to test the dynamics of such sediments under glacial conditions, and toexamine the role of the effective pressure towards the mode of deformation. A novelapproach involving a combination of experimental work and numerical modelling isapplied to account for the complexity of the subglacial system, thus aiming to improveour understanding of the movement mechanisms of large ice sheets and glaciers whenoverriding soft, unconsolidated sediments.

The starting point for the numerical modeling is the simple premise that subglacialsediment is a granular material, which is known to have an inherent ability to changephase in a mechanical sense, from viscous behavior under low pressures, and converselyacting as a brittle solid with strong pressure-carrying grain bridges when drained(Jaeger and others, 1996).

A series of laboratory experiments (see box 2. Laboratory Experiments) are con-ducted in order to determine the geotechnical properties of granular sediment, par-ticularly under shearing conditions. By establishing both the kinematic and dynamicbehavior of real sediments under subglacial conditions, the laboratory experiments mayalso act as validating benchmark experiments for the computational models (see box 3.Discrete Element Method).

The work plan of the project is structured, so that the granular material is examinedin experiments of increasing complexity. Initial experiments were performed on spher-ical glass beads, now quartz sand and in the future sampled tills. The wider range ofproperties of the material (grain-size distributions, grain surface morphology, particleelongation and strength, physico–chemical effects of clay minerals) will be graduallyincluded to obtain a realistic computational model for subglacial sediment. The inter-action with porewater will be introduced to examine the microstructural-, thermal- andgeotechnical behavior and feedbacks upon diurnal and annual fluctuations of porewaterpressure by variability of the subglacial meltwater supply.

1. Introduction

GI Ring-Shear Laboratory Reference SheetAnders Damsgaard Christensen, anders.damsgaard@geo.au.dkLast revision: January 25, 2011 Deptartment of Earth Sciences (Geologisk Institut), Aarhus University

1 Schematic

! 2006 Geological Society of America. For permission to copy, contact Copyright Permissions, GSA, or editing@geosociety.org.Geology; October 2006; v. 34; no. 10; p. 889–892; doi: 10.1130/G22629.1; 4 figures.. 889

Microstructures and microshears as proxy for strain in subglacialdiamicts: Implications for basal till formation

Nicolaj K. LarsenJan A. PiotrowskiFrits Christiansen

Department of Earth Sciences, University of Aarhus, C.F. Møllers Alle 120, DK-8000 Arhus, Denmark

ABSTRACTRing-shear experiments were used to study subglacial sediment deformation and the

development of S-matrix microstructures and microshears to constrain glacially induceddeformation in diamicts. The experiments approximated subglacial conditions, and thediamict was sheared and sampled incrementally to strains of 0, 7, 18, 33, 57, and 107. Atstrains between 7 and 18, a steady association of S-matrix microstructures developed,which did not evolve further despite continued shearing. The IL-index, a measure ofmicroshear-orientation strength parallel to the shearing direction correlates log-linearlywith strain, which indicates its potential use as strain proxy. The ring-shear data werecompared with the characteristics of Quaternary basal tills from Denmark, Poland, andSvalbard. Based on IL-index values, we conclude that these tills only experienced strainof !101, which is orders of magnitude lower than expected for deformation tills. Thissuggests that pervasive subglacial deformation and sediment advection in the mobile layerat the ice-bed interface may be less significant than previously assumed.

Keywords: till, micromorphology, subglacial environment, strain.

Figure 1. Ring-shear apparatus setup.

INTRODUCTIONRing-shear experiments have been used to

study subglacial sediment deformation andhave increased our understanding of micro-shears (Thomason and Iverson, 2006), till fab-ric (Hooyer and Iverson, 2000a), sedimentrheology (Iverson et al., 1998), and granulardiffusive mixing processes (Hooyer and Iver-son, 2000b). Soft, deformable sediments underice sheets influence glacier dynamics (Bamberet al., 2006), landform formation (Boulton andClark, 1990), and possibly climate change(Clark et al., 1999). Subglacial deformationhas been suggested as a mechanism of sedi-ment redistribution on a continental scale(Alley, 1991), although the thickness and spa-tial continuity of the deforming layer are con-tentious (Piotrowski et al., 2001). Differentopinions exist on whether past ice sheets rest-ed on widespread, thick, pervasively deform-ing beds (Boulton and Hindmarsh, 1987; vander Meer et al., 2003) or whether this defor-mation was spatially and temporally confinedand discontinuous, with strain accumulatingtime-transgressively in a thin zone of coevallyaccreting basal till (Larsen et al., 2004; Pio-trowski et al., 2004).

Both ideas explain adequately the origin ofmacroscopically massive tills, which are ubiq-uitous in past and modern glaciated areas. Thechallenge is to find a method that effectivelydiscriminates between the two possibilities.However, there is no consensus at present as

to what methods can be applied and how toevaluate and interpret the results. A promisingapproach is the analysis of strain indicatorsbecause deformation tills have typically beensubjected to strains of 103–105 (Hooyer andIverson, 2000a; van der Wateren et al., 2000),which is orders of magnitude higher thanstrains predicted by the time-transgressivemodel. Several attempts to develop experi-mentally calibrated strain proxies have givenmixed results based on till fabric (Hooyer andIverson, 2000a), micromorphology (Hiemstraand Rijsdijk, 2003), microshears (Thomasonand Iverson, 2006), and the degree of diatomcomminution (Scherer et al., 2004). Here weexamine S-matrix microstructures and micro-shears and document how they develop underprogressively higher strain within till shearedin a ring-shear device. We then use the micro-morphological signature from all laboratoryexperiments as a reference to examine someQuaternary subglacial tills and evaluate themode and scale of sediment deformation be-neath ice sheets.

RING-SHEAR APPARATUS ANDEXPERIMENTAL PROCEDURE

The ring-shear apparatus used in this studyshears 14,480 cm3 of sediment at a constantrate and normal stress. Water is allowed to en-ter and exit the sediment chamber through fil-ters in the lower and upper platen and throughthe gap between the platens, keeping pore-

water pressure essentially atmospheric. Theapplied normal stress on the sediment of 85kPa is in the range of effective normal pres-sures beneath glaciers and ice sheets (Pater-son, 1994). Sediment shearing is achieved byrotating the lower platen under the rotationallyfixed upper platen while ribs prohibit slidingbetween the sediment and the platens (Fig. 1).A steady shearing rate of 363 m yr"1 com-

Schematic of GI Ring-shear apparatus setup. From Larsen and others(2006). Chamber volume: 14 480 cm3

2 Adjustable parameters

2.1 Shearing rate

The rate of ice movement varies enormously from one glacier to another, butgenerally lies in the range 3 to 300 m yr−1. A steady shearing rate of 363m yr−1 = 1.1503 × 10−5 m s−1 is comparable to fast-moving ice sheets andglaciers (Paterson, 2000). Surging glaciers move at speeds of 4–12 km yr−1,e.g. Jacobshavn ice stream: ∼9 km yr−1. Average flow velocity for the Scan-dinavian Ice Sheet during the Weichselian Glaciation: ∼100–150 m yr−1.During the Elsterian Glaciation in East Germany the ice reached velocities of∼800 m yr−1 due to basal decoupling.

2.2 Normal stress

As long as the subglacial porewater pressure (pw) is smaller than the ice over-burden pressure (pi), the substratum will be exposed to an effective normalpressure (pe):

pi = ρi · g · h , pe = pi − pw (1)

ρi: ice density, g: gravitational acceleration, h: ice thickness. The effectivepressure is equivalent to the applied normal stress σ0.

2.3 Shear stress

τs = ρi · g · h · sinα (2)

τs: shear stress, ρi: ice density, g: gravitational acceleration, h: ice thickness,α: ice surface slope. If α = 0 ⇒ τs = 0.

3 Slip surface angles

Orientation of micro shear angles (”faults”) in granular medium:

High angles ⇐⇒ Low shear strainLow angles ⇐⇒ High shear strain

4 Rheology and fabric

✻Shear stress (τ )

✲

Shear strain rate (γ)✑✑✑✑

✑✑

✑✑✑✑

✑✑

✑✑✑✑✑✑✑ Viscous def.

Plastic def.

In Boulton and Hindmarsh (1987) and Boulton and Dobbie (1998) an appli-cation of the Bingham model for investigations of till rheology was proposed:

γ = A(τ − τC)n

pme

(3)

where γ is the strain rate, (τ − τC) is the excess stress (the Mohr-Coulombyield strength (τs) is often used for τC), A, m and n are empirical constants.In this model, high n values (> 10) approximate plastic behavior, and smalln values (< 10) a viscous rheology.

Bagnold (1954) concluded that granular matter can behave like a fluid whenloosely packed (like sand in a hourglass), where the viscosity η is proportionalto the shear velocity. At rest, under the influence of a force like gravity, gran-ular matter forms a stable packing and behaves like a solid, and the samematerial can take a range of packing densities, dependent on the way of depo-sition and stress history (Herrmann, 2002). When subjected to deformation,the Mohr-Coulomb constitutive model of friction is an analogy to describethe upper limit for shear stress acting along slip planes in an ideal Coulombmaterial :

|τs| ≤ C + σ� tanφ� (4)

i.e. shear stress ≤ shear strength. σ� is the effective stress (ice overburdenstress minus porewater pressure). This upper limit of |τs| is the shear stressacting along a shear plane, which inside ideal Coulomb materials is made upof a narrow band (∼10 grain diameters) who are deforming plastically, i.e.the shear stress on the plane independent of either the extent or rate of thedeformation (Nedderman, 1992).

✻τ

✲

σ

B✑✑✑✑

✑✑

✑✑

✑✑✑✑

✑✑

✑✑✑✑

✑✑

✑✑

✑✑✑

S

C

◗◗◗◗

◗◗

◗◗

◗◗◗◗

◗◗

◗◗◗◗

◗◗

◗◗

◗◗◗

S �

IYL: |τs| = C + σ tanφ

❏❏

❏❏

❏❏

❏❏

❏❏

IYL: |τs| = C + σ tanφ

✡✡

✡✡

✡✡

✡✡

✡✡

|τs| < C + σ tanφ

|τs| < C + σ tanφ

A φφ

90 − φ

σ1σ3

✲✛

C cotφ

Above: graphical representation of the Mohr-Coulomb failure criterion. Notethat angles in Mohr’s circle represent twice the rotation in real space. IYL:Internal Yield Locus, where deformation in slip band takes place. Modifiedfrom Nedderman (1992). The Mohr-Coulomb failure criterion predicts thatthe slip planes are inclined at 90 − φ relative to each other (BS � BS�), and±(45 − φ/2) to the minor principal plane.

Hindmarsh, 1987). This strain localization is irrelevant tothe problem at hand, however, because the particles andmicroshears studied were both within the shear zone and

much smaller than the shear zone; till outside this zone wasnot studied. Removal of the coarsest particles from the till(o 3.4% of the total volume) may have resulted in slightly

ARTICLE IN PRESS

Fig. 7. Photomicrographs of Douglas till showing microshear development at shear strains of (a) 5, (b) 10, (c) 39 and (d) 108. Photographs were takenunder crossed-polarized light. Arrows point to R1 and R2 shears, and IL is the index of low-angle microshear development given by Eq. (2). (e) Schematicof Riedel shear orientations and steady-state V1 directions.

J.F. Thomason, N.R. Iverson / Quaternary Science Reviews 25 (2006) 1027–1038 1033

Schematic of Riedel shear orientations and steady-state V1 directions, fromThomason and Iverson (2006).

4.1 Fabric development

The theory of fabric development in till is linked to the flow rheology: Jefferyrotation (Jeffery, 1922) describes the continuous rotation of elongate parti-cles in a viscous flow, which results in a low fabric strength (eigenvalue (ameasure of clustering) S1 ≈ 0.6). In contrast, March rotation (March, 1932)takes place in plastic flow, where elongate particles are rotated into a stableposition, and experience slip on the surface and striation from the surroundingmaterial. This results in strong particle fabric, even at low strains (Iversonand others (2008): 0.78 < S1 < 0.94).

If till rheology (like granular matter rheology) is bimodal, where the de-formation mode is dependent on the forcing conditions, the fabric strength(eigenvalue, S1) will be dependent on both the past rheology of the material,and the accumulated unidirectional strain (γ).

CLAST-FABRIC DEVELOPMENT, SUBGLACIAL TILL AND FAULT GOUGE

Geological Society of America Bulletin, May 2000 689

experiments by Ildefonse and Fernandez (1988)and Arberet et al. (1996) have shown that inter-acting clasts maintain a metastable orientationclose to the direction of shearing longer than iso-lated clasts, resulting in a stronger fabric. Unlikethe putty experiments during which clasts couldbe continuously observed, clasts in the till wereonly visible at the end of a given test. However,dissection of the specimen at the ends of tests in-dicated that ~75% of clasts had no nearest neigh-bor closer than 25 mm, as might be expectedfrom the large initial clast spacing. Therefore,clast interaction was apparently minimal.

A final potential source of uncertainty is thetransverse shear strain that occurred due to thethinning of the shear zone toward the walls of thesample chamber (Fig. 7). Such flow is not ex-pected subglacially or along fault zones and wasnot considered by Jeffery. Transverse shear strainshould cause clasts to rotate toward the verticalflow plane (x-z plane), potentially strengtheningthe fabric. For clasts located at the specimen cen-ter, the transverse shear strains caused by thesymmetric thinning of the shear zone toward theinner and outer walls are equal, resulting in no ro-tation. However, some clasts located initially atthe specimen center moved as much as 15 mmfrom the centerline during shearing. Despite thiseffect, however, calculations using the measureddisplacement field indicate, as noted earlier, thatnone of the clasts underwent a transverse shearstrain that was greater than 2% of the longitudi-nal strain. Furthermore, clasts at the centerlinewere not oriented differently than those displacedlaterally. Therefore, transverse shear strain likelyhad a negligible influence on the fabric results.This conclusion is supported by the results of theputty experiments, in which there were also smalltransverse strains but good agreement with Jef-fery’s theory (Figs. 5 and 8).

Implications for Field Studies

Glacial geologists have often inferred fromfield observations that subglacial deformation oftill produces a weaker clast fabric than the lodg-ment process (e.g., Dowdeswell and Sharp, 1986;Hicock, 1992; Hart, 1994; Clark, 1997). Eigen-values from such studies are shown in Figure 11,along with our laboratory results. Figure 11 illus-trates that if the inferred lodgment tills are reallya result of the lodgment process, then deforma-tion should strengthen, rather than weaken, thefabric of such tills. The strengths of fabrics gen-erated in our experiments, as represented byS1 eigenvalues, are greater than those of either thedeformation or lodgment tills inferred from thesefield studies (Fig. 11).

A possible source of uncertainty in this com-parison of laboratory and field results is that the

experimental clasts were smaller than those oftenused in field studies. If clasts are surrounded bypredominately smaller particles, however, the be-havior of clasts during shearing should be inde-pendent of their size. About 75% of the till massconsisted of particles that were more than an or-der of magnitude smaller than the experimentalclasts. Furthermore, the self-similar particle-sizedistributions of some basal tills (Hooke and Iver-son, 1995) indicate that clasts, regardless of theirsize, should be surrounded by smaller particles.

The results of Figure 11 indicate that there isgood reason to be skeptical of the field criteriaused to independently identify so-called “defor-mation tills.” In fact, the state of the art is such thattwo scientists working on the same till exposuresmay reach markedly different conclusions. Forexample, Allen et al. (1991) measured fabrics intwo till units in East Anglia, United Kingdom.The lower unit, which they thought had beensheared subglacially, has strong fabrics withS1 eigenvalues of 0.77–0.86 (as calculated by

Figure 8. Rotation of clast long axes as predicted by Jeffery (1922) and March (1932), as ob-served in the putty (open circles), and in the till (solid circles). Orientations of clasts in the puttyare based on the average orientation of five clasts presented in Figure 5B (aspect ratio = 2.0).Orientations of clasts in till are based on the average orientation of ~50 clasts (aspect ratio = 2.0)in experiments terminated at the indicated strains.

Figure 9. Fabric strength (S1 eigenvalue) as a function of shear strain in till experiments withinitially vertical clasts. The sense of shearing in the stereograms is bottom north, top south.

on July 18, 2010gsabulletin.gsapubs.orgDownloaded from

From Hooyer and Iverson (2000a); a study of clast-fabric development inring-shear experiments on different material: 1) Linear-viscous putty and 2)

Summarized data from respectively a clay-rich- and a clay-poor till.

5 Dilation – strain hardening or softening

While the lower wall is fixed in vertical position, the upper wall is free to movevertically, while exerting a normal pressure to the shearing chamber. Initiallycompacted rigig granular materials subjected to shearing stress require anincrease in volume (dilatancy) to deform (Reynolds, 1885).

Volume changes during formation of shear bands in materials with porewa-ter can have one of two effects, dependent on the degree of grain crushingand clay mineral content. In clay-rich material contraction is often seen as aneffect of rotation of clay minerals into parallel position.

Dilatant hardening: (Iverson and others, 1998) Without significant graincrushing the shear zone dilates under deformation, which decreases the lo-cal porewater pressure. This increases the shear strength of the till (eq. 4),and could lead to migration of the shear zone, if the hydraulic conductivityis sufficiently low.

Dilatant softening: (Wafid and others, 2004) High amount of grain-crushing → negative dilatancy (contraction). This leads to a local increaseof porewater pressure, and thus a shear strength decrease (eq. 4). The graincrushing leads to a decrease in hydraulic conductivity by production of finermaterial, and thus lowers the permeability and excess porewater pressuredissipation rate (Okada and others, 2004). This tendency of strain soften-ing sustains the shear zone in constant place. The dilatancy is however atransient effect, constrained to the initial stages of deformation.

5.1 Dilatancy- and stress fluctuations

stress occurs the strains tend to localize in the end zones of thepotential shear zone. This stage, named as ‘end zone deformation’(Fig. 2), ceases at the lowest point of volumetric strain where granularinterlocking prevents further contraction.

In the second stage, ‘particle interlocking’ (Fig. 2), particles,especially those in the shear zone, have to overcome the interlockingas they approach each other, leading to local dilation to dominate andtherefore volumetric strain to increase. The peak stress occurs whenthe particle interlocking within the potential shear surface reaches itsclimax. This particle interlocking re!ects the extent to which particlesprevent each other from relative movement (which may includerolling, sliding and rotating) and is determined by the amount ofparticles (and their shapes and surface roughness in natural soils)involved in the particle interlocking and interparticle friction. Taylor(1948) noted that the greatest gradient of vertical displacement !y/!xoccurs at the same time when peak stress appears. He attributed thisfast increase in volume to particle interlocking in the potential shearzone. Taylor (1948) further divided peak shear stress ratio ("/#') intotwo components, particle interlocking (!y/!x) and friction $, that is "/#'=(!y/!x)+$. With this model, he found that at the peak point, thestrength arising from resistance to volume increase (dilation) was asmuch as 26% of the total shear stress of 1.94 tons/ft2, for a sandspecimen.

When the shearing process enters into the third stage, ‘shear zoneformation’, the potential shear zone becomesmore activewith relativemovement of particles, resulting in a relatively looser layer (Oda andKunishi 1974; Fukuoka et al. 2006).Within this layer, smaller particlestend to be pushed into nearby voids, while larger particles tend torotate or roll to let the shear zone adjust its structure to reduce theresistance. Local dilation caused by coarse particle interlocking andlocal contraction caused by dynamic compression work to moderateeach other. With the increase of shear displacement, a dynamic butrelatively steady structure is generally formed. The mechanism of thisprocess termed as shear-induced packing is discussed in detail later inthis paper in relation to its controlling factors, such as particle sizedistribution, normal stress and shearing rate.

At point B, a steady shear zone forms as a relatively regular butuneven layer with neighboring resistant coarse particles entrapped in

or drawn into the matrix of this layer, and thus not causing signi"cantdilatancy. Hereafter, the inner structure of the specimen remainssteady and the upper block moves forward with constant waveamplitude regardless of shear displacement. From the macroscopicpoint of view, the main style of movement becomes sliding along awavy surface with a dynamic equilibrium of local dilation andcontraction, though the particle interlocking still plays an importantrole in the magnitude and pattern of shear stress.

3.2. Fluctuations at residual state

Recurrent jagged !uctuations in the vertical position of theupper half of the shear box during residual state re!ect relativelystable, rough morphology of the shear surface. An example of theseforced !uctuations in the measured vertical displacement and thecorresponding stress ratio is given in Fig. 3. Note that !uctuationsin the stress ratio and the vertical displacement appear to besynchronized with very similar wavelengths. This implies that!uctuations in stress ratio (or more precisely !uctuations inshear force) occur to overcome dilation component of the shearresistance.

The value of basic (or residual) friction coef"cient that corre-sponds to the mean stress ratio (as de"ned by Coulomb) may beestimated by a simple statistical analysis of the whole digital data or ofa suf"cient number of peak and trough values. A more explicitapproach to predict the actual value of residual friction coef"cientmay be possible if a) the characteristic amplitude and wavelength ofthe !uctuations can be linked to the dilation component (as presentedbelow) and b) these wave characteristics can be determined (aspresented in the following section).

The standard function for vertical displacement y of a particleriding a wave is given by:

y = A sin kx!wt + !! " !1"

where A is the wave amplitude, k (=2%/&) the angular wave number,w (=2%f) the angular frequency, f the frequency and ' the initial

Fig. 2. A four-stage shearing model for granular materials.

99Y.R. Li, A. Aydin / Engineering Geology 115 (2010) 96–104

A four-stage shearing model for granular materials in direct shear. From Liand Aydin (2010).

Miller and others (1996) present measurements of stress from continuoussheared (quasi-static flow) glass spheres in an annular couette geometry appa-ratus. The recorded values of shear- and normal stress were highly fluctuating,with peak values of up to one or two orders of magnitude larger than the aver-age stress. These fluctuations where later visually connected to the dynamicsof the fluctuating force chain network by observing the shifting birefringenceof photoelastic plastic discs along the force chains (Veje and others, 1999).Li and Aydin (2010) varied different experiment parameters, to observe theeffect of particle size distribution, normal stress and shearing rate:

Parameter EffectParticle size distribution Wider variance in particle sizes: Increasing

amplitude and wavelength of the fluctuations.Normal stress Increasing normal stress: Increasing ampli-

tude and wavelength of fluctuations.Shearing rate Increasing shearing rate: Decreasing wave-

length and amplitude of fluctuations.

6 Particle diffusion

The particle diffusion law (Weertman, 1968) adapted by Piotrowski and Tu-laczyk (1999):

λ =

�8(1 − n)tγ

5πD (5)

λ: vertical particle diffusion, n: porosity, γ: shear strain rate (10–100 yr−1),t : time, D: diameter of spherical particle. Hooyer and Iverson (2000b) usedan alternate approach to describe particle diffusion in ring-shear experimentsconstructed from kinetic gas theory.

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ogy , 7(50), 161–165.

Principle sketch of Ring-Shear apparatus.From Larsen and others (2006).Chamber volume: 14 480 cm3

Ring-Shear apparatus.a: Sample chamber, b: Stationary load platen,

c: Load bearing frame, d: Rotating lower platen.Data is continuously recorded with a normal stress sensor, two

shear stress sensors, and three dilatancy sensors.

Strain marker locations in quartz sand after 200 mm sheardistance.

The laboratory work is focused on determining the macroscopic and microscopic geotech-nical properties of a granular sediment, particularly under shearing conditions. In theunique Ring-Shear apparatus at the Department of Earth Sciences, Aarhus University, thesediment samples are sheared under a range of pressurized conditions corresponding tonatural conditions under warm-based continental ice sheets and glaciers. The adjustableparameters (e.g. normal stress, shearing velocity) are varied, and their impact is analyzedand quantified. The evolution of sediment properties is monitored using the continuouslyrecorded stress dynamics.

Artificial glass beads and aeolian quartz sand were selected as suitable, simple granularmaterials, and the laboratory work also involves standard analyzing techniques of e.g.grain size distribution, grain surface morphology and -mineralogy, and micro-/and macro-mechanical properties of the sediment. In addition, the acoustic signal during shearing isrecorded to continuously monitor the degree of grain crushing during progressive shearing.These results will be used to constrain interpretation of past subglacial processes and thenumerical modelling thereof.

2. Laboratory Experiments

0 10 20 30 40 50 60 70 80−1

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Data/QS−usp6−data.txt. Ultimate shear strength: τu = 47.09 kPa

Shear displacement, ∆x [mm]

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Sample: QS, φa = 44.9375

o, φ´ = 36.8455

o, C

a = 3.55 kPa, C´ = −0.32573 kPa

Linear Fit, Pre failure

RS Data, Pre failure

Linear Fir, Post failure

RS Data, Post failure

Left: Typical plot of ring-shear sensor output. σ0 = 100 kPa.Right: (σ ′, τu)-plot and linear regression of ring-shear stress data on quartz sand.The discrete element method (or distinct element method) was initially formulated by Cundall and Strack (1979). It simulates the physical behavior and

interaction of discrete, unbreakable particles, with their own mass and inertia, under the influence of e.g. gravity and boundary conditions such as movingwalls. By discretizing time into small time steps (∆t ≈ 10−8 s) a second order integration scheme based on Taylor expansion of particle kinematics is usedfor integration of Newton’s second law of motion (mx = F and Iω = τ), and is used to predict the new position and kinematic values for each particlefrom the previous sums of forces. This Lagrangian approach is ideal for simulating discontinuous materials, such as granularities. The complexity of thecomputations is kept low by representing the particles as spheres, which keeps contact-searching algorithms simple. Still, the spherical particles may bebonded together to form larger irregular particle unities.

The computational experiments allow processes and feedbacks to be studied at a more detailed scale when compared to the laboratory experiments,hereby offering greater insight into the origin of structural and textural features related to progressive shear strain.

Initially, a 2D formulation of the method (Egholm, 2007; Egholm and others, 2007) (see pictures below) was adapted to a ring-shear like model setup. Itshows the potential of the method by offering high data output volumes on a particle-by-particle base, readily available for further analysis. Based on thisinitial experience, a new generation of the computer code will be constructed, which simulates particles in a three-dimensional domain (see box 4. Futuregoals and prospects).

3. Discrete Element Method

2D simulation example done with SDEM software (Egholm, 2007). Left- and right boundaries are periodic.

Substantial effort is currently being put into developing a new generation of the DEMcode, where particles interact in three-dimensional space, employing the massive paral-lel computational power of graphics processing units (GPUs) and the CUDA architecture(NVIDIA, 2010). The discrete element method has a high potential data parallelism, andGPUs with support for the IEEE floating-point standard are attractive computational en-gines that allow simulation of a very large number of particles (Kirk and Hwu, 2010;Green, 2010). Modern many-core GPUs offer about 10× the floating-point throughputthan multicore CPUs.

This will enable a heightened sense of realism, both by enabling 3D movement and ahigh number of particles (106). The goals consist of expanding the model to include:• 3D simulations• Temperature budget and transport by convection and

conduction• Fluid coupling: Enabling porewater pressure- and

thermal effects and feedbacks with the EulerianNavier-Stokes formulation of inter-particle fluid.• Larger particles represented by breakable clusters of

smaller spheres• Clay mineral physico—chemical effects (Yao and

Anandarajah, 2003)The model software will be deployed to simulate subglacial conditions under both smallervalley glaciers as well as larger continental ice sheets, yielding information concerningdynamics of the geotechnical properties, microstructure development with progressive shearstrain, sediment rheology, strain distribution and -localization.

4. Future goals and prospects

ReferencesCundall, P.A and O.D.L. Strack, 1979. A discrete numerical model for granular as-

semblies, Géotechnique, 29, 47–65.

Egholm, D.L., 2007. A new strategy for discrete element numerical models. 1. Theory,J. Geophys. Res., 112(B05203), doi:10.1029/2006JB004557.

Egholm, D.L., M. Sandiford, O.R. Clausen and S.B. Nielsen, 2007. A new strategy for

discrete element numerical models. 2. Sandbox applications, J. Geophys. Res.,112(B05204), doi:10.1029/2006JB004558.

Green, S., 2010. Particle Simulation using CUDA, NVIDIA Whitepaper, November ,12 pp.

Jaeger, H.M., S.R. Nagel and R.P. Behringer, 1996. Granular solids, liquids, andgases, Reviews of Modern Physics, 68(4), 1259–1273.

Kirk, David B. and W-M. W. Hwu, 2010. Programming Massively Parallel Proces-sors, Morgan Kaufmann, Elsevier.

Larsen, N.K., J.A. Piotrowski and F. Christiansen, 2006. Microstructures and mi-croshears as proxy for strain in subglacial diamicts: Implications for basal tillformation, Geology, 34(10), 889.

NVIDIA, 2010. CUDA C Programming Guide, NVIDIA Corporation: Santa Clara,CA, USA, 3.2 ed.

Yao, M. and A. Anandarajah, 2003. Three-Dimensional Discrete Ele-ment Method of Analysis of Clays, Journal of Engineering Mechanics,129(10.1061/(ASCE)0733-9399(2003)129:6(585)).