Report for course “Soil * deformation: Measurements, Modelling, Visualization” Anders Damsgaard Christensen [email protected], http://cs.au.dk/ ~ adc Last revision: November 8, 2011 Contents 1 Introduction 1 2 Lab measurements 2 2.1 Precompression stress ........ 2 2.2 Macromechanical properties ..... 2 2.3 Air permeability during progressive strain ................. 3 3 Pseudoanalytical model 5 3.1 S¨ ohne’s summation procedure .... 5 3.2 SoilFlex usage ............. 5 3.3 Stress-strain models ......... 5 4 Finite Element Modelling 7 4.1 Problem description ......... 7 4.2 COMSOL setup ............ 7 4.3 Results ................. 8 5 Discrete Element Modelling 9 5.1 Theory ................. 9 5.2 Implementation ............ 10 5.3 Results ................. 10 6 Outlook 11 References 12 * Soil is sexy 1 Introduction This report is the final assignment for the course “Soil Deformation: Modelling, Measurements, and Visualization”, held at Research Centre Foulum September 26th to October 1st, 2011. The course homepage is: https://djfextranet.agrsci.dk/sites/ deformation/public/Pages/front.aspx The report is divided into four parts, correspond- ing to the overall subjects of the course. Each part is based on a self-chosen assignment, and is inde- pendent. Since my background is in glacial geol- ogy, I will try to highlight any specific interest in the subjects, and formulate and test hypotheses of direct relation to the field. With regards to formatting, bold notation in mathematical formulas denotes vectors, and dots denote time-derivatives. References to external lit- terature are found at the end of the paper. 1
This report is the final assignment for the course“Soil Deformation: Modelling, Measurements, andVisualization”, held at Research Centre FoulumSeptember 26th to October 1st, 2011. The coursehomepage is:
https://djfextranet.agrsci.dk/sites/
deformation/public/Pages/front.aspx
The report is divided into four parts, correspond-ing to the overall subjects of the course. Each partis based on a self-chosen assignment, and is inde-pendent. Since my background is in glacial geol-ogy, I will try to highlight any specific interest inthe subjects, and formulate and test hypotheses ofdirect relation to the field.
With regards to formatting, bold notation inmathematical formulas denotes vectors, and dotsdenote time-derivatives. References to external lit-terature are found at the end of the paper.
In theory, the soil deformation during e.g. com-pression takes place in an initial, reversible, elasticphase, before shifting to irreversible plastic defor-mation at higher strains. When determining theprecompression stress, the prior maximal value ofstress is found by applying stress until the plasticphase is reached.
We recommend fitting the Gompertz function to measured soil compression characteristic test data, and to define Cc
objectively as the modulus of the slope of the tangent at the inflection point, providing this lies within the measured data range,and s0p as the point of maximum curvature as defined by k.# 2005 Elsevier B.V. All rights reserved.
Keywords: Compression characteristic; Compression index (Cc); Precompression stress !s0p"; Polynomial; Sigmoidal; Curvature function (k)
1. Introduction
The soil compression characteristic is a funda-mental mechanical property of the soil that relates theeffect of compressive stress on a soil volumetricparameter. Typically, the characteristic is illustratedby plotting a logarithm (usually to base 10) of thenormal compressive stress (s0) against soil void ratio(e) or vertical strain (e) (Fig. 1). When expressed inthis way, the characteristic has two distinct regions—an elastic rebound curve at low stress and a linearvirgin compression curve at a higher stress. Themodulus of the slope of the virgin compression curveis commonly called the compression index (Cc). Thetransition point between the elastic rebound curve andthe virgin compression curve is known as the soilprecompression or preconsolidation stress !s0p".Together, the Cc and s0p are the two defining featuresof the compression characteristic.
The Cc parameter is an indicator of the suscept-ibility of a soil to damage by compaction (Baumgartland Kock, 2004; Imhoff et al., 2004) and s0p gives a
guide to the maximum stress that has previouslyloaded the soil (Mitchell, 1993; Horn, 2004). Thevalue of s0p has considerable significance for soilmanagement because it gives an estimate of themaximum stress that a soil can withstand before itdeforms irreversibly along the virgin compressioncurve (Whitlow, 1990; Alexandrou and Earl, 1995;Van den Akker and Schjønning, 2004). When soil issubjected to stresses smaller than s0p it will deform andrecover elastically along the elastic rebound curve.Cetin (2004) linked s0p with a change in the orientationof soil voids during soil compression.
The first calculations of s0p were described byCasagrande (1936) who proposed that it is the stress atthe intersect of the linear virgin compression curveand the bisect of a horizontal and tangent drawn at thepoint on the characteristic of maximum curvature.This remains a standard definition (Arvidsson andKeller, 2004; Baumgartl and Kock, 2004; Berli et al.,2004; Cetin, 2004; Horn, 2004; Keller et al., 2004), if alittle cumbersome. It is common to fit a curve to thecompression characteristic in order to estimate thepoint of maximum curvature analytically rather thanby visual inspection. Data describing the soilcompression characteristic have been fitted to poly-nomial (Arvidsson and Keller, 2004) and sigmoidal(Baumgartl and Kock, 2004) curves. Some curves canalso be used to estimate the Cc objectively. Otherwisewe rely on linear regression of an arbitrarily definedvirgin compression region of the characteristic.
The Casagrande calculation of s0p is a standarddefinition. However, a number of other possibilitiesbased on the compression characteristic have alsobeen used to define that point where reversible elasticfailure becomes irreversible plastic deformation.These include the point of maximum curvature itselfand the intercept of the linear virgin curve with ahorizontal line drawn at zero stress (Arvidsson andKeller, 2004) or the intercept with a tangent at lowstress (Alexandrou and Earl, 1995; Liu and Carter,
A.S. Gregory et al. / Soil & Tillage Research 89 (2006) 45–5746
Fig. 1. The features of the soil compression characteristic, relatingvoid ratio (e) or vertical strain (e) to an increasing compressive stress
(s0).
Figure 1: Idealized stress-strain relationship duringsoil compression. The strain is represented as eithervertical strain (ε) or the void ratio (e). From Gregoryet al. (2006).
The stress-strain relationship is visualized byplotting log the logarithmic stress values, e.g.log(p), against volumetric strain, εp. There are anumber of possible stress paths in this system ac-cording to the material rheology. The virgin com-pression line is the straight line that is followedwhen a (new) maximum value of stress is applied,where the compression index is the slope of the vir-gin compression line. The recompression line is thestraight line that the material follows in the plotwhen stress is reapplied, but at lower streses thanthe previous maximal stresses. The recompressionlines are assumed to be parallel, and the recompres-sion index is the slope of the recompression line.The precompression stress has in glaciology beenused to determine the palaeo-subglacial stresses ex-certed to the sediment. The effective subglacialstress is decomposed into porewater pressure (pw)and the ice overburden weight (σ0). The effec-
the third compression characteristic (0–200 kPa)corresponded to a known precompression stress of100 kPa. The software recorded approximately 200and 400 data points for the 0–100 and 0–200 kPacompression characteristics respectively.
2.2. Curve fitting
Three curves were evaluated for their ability todescribe the soil compression characteristic: a fourth-order polynomial curve (Eq. (1)), as used by Arvidssonand Keller (2004) and two sigmoidal curves—thelogistic (Eq. (2)), and the Gompertz (Eq. (3)):
e ! a"log10 s0#4 $ b"log10 s0#3 $ c"log10 s0#2
$ f "log10 s0# $ g (1)
e ! a$ c%1$ exp"b""log10 s0# & m##'&1 (2)
e ! a$ c exp%&exp"b""log10 s0# & m##' (3)
where a, b, c, f , g and m are fitted parameters. Thethree different functions were used to fit each of the 0–100 and 0–200 kPa compression characteristicsrecorded for the 12 soils.
The two sigmoidal curves are S-shaped withasymptotes at maximum, or initial, e (a + c) andminimum e (a). The simple logistic function issymmetrical about the inflection (m), whereas theGompertz function is asymmetrical. Both curves aredeveloped here in analogy to growth modelling. Thelogistic function describes growth on the assumptionsthat the rate of growth is proportional to the current stateof growth and to the available substrate but with theconstraint that substrate can become limiting; that is tosay, substrate plus growth are fixed. The Gompertzgrowth curve makes the same assumptions but withoutthe constraint that substrate is considered to be limiting.Both functions are routinely used to describe the declinein a quantity as well as growth. By analogy, s0 is themeans (substrate) that brings about a decline in e. Thelogistic function represents the case where either thesoil strongly resists compression or the loading frame isset to reduce load in proportion to the resistance fromthe soil. The Gompertz function is more appropriatewhere the loading frame is set to a constant load and sothe compressive forces are to all intents and purposesnon-limiting. Although the process being describedhere seems more appropriate to the Gompertz function,both sigmoidal curves were evaluated.
2.3. Compression index (Cc) estimation
Each of the three functions fitted to the 0–100 and0–200 kPa compression characteristics had a linearvirgin compression curve based on the region of thecharacteristic with a linear response to increasingcompressive stress. The modulus of the slope of thevirgin compression curve gives Cc. For the case wherethe polynomial function was used to model thecompression characteristic, the virgin compressioncurve was estimated by a separate linear regression ofthe data points lying within the arbitrarily definedbounds of 50–100 and 100–200 kPa for the 0–100 and0–200 kPa compression characteristics respectively.These regions of the compression characteristic werethose that appeared by eye to be linear. Similarregressions were made by Arvidsson and Keller(2004) and Imhoff et al. (2004). This particular Cc
estimate was therefore not defined by the polynomialfunction (Eq. (1)).
For the two sigmoidal curves, the Cc was estimatedas the modulus of the slope at the inflection point, m,
A.S. Gregory et al. / Soil & Tillage Research 89 (2006) 45–5748
Fig. 2. The soil compression characteristic data used, showing
sequential compression to 50, 100 and 200 kPa.Figure 2: Stress strain path during repeated load-ing/unloading. Note that the recompression lines arenot completely straight and parallel, however close toit. From Gregory et al. (2006).
tive stress (σ′) is the ice overburden weight withthe porewater pressure subtracted. Field measure-ments (Bartholomaus et al., 2011, e.g.) show thatthe subglacial porewater pressure is highly dynamicin response to seasonal and diurnal fluctuations inablation and changes in the glacial hydrological sys-tem. If we assume that the maximal effective stresscorresponds to the lowest porewater pressure dur-ing the relative thickest ice cover, we can estimatethe ice sheet thickness.
The past ice advances from the Scandinavian IceSheet have had decreasing lateral extent and ver-tical thickness, and glacial depositions have beencorrelated to specific ice sheet advances by lookingat the differences in precompressional stress values.
2.2 Macromechanical properties
The macromechanical properties angle of internalfriction (φ) and cohesion (C) are defined as the lin-ear representation of the value of the material shearstrength (τ∗) under a range of effective normal pres-sures (σ′). The Mohr-Coulomb criterion describesthe upper limit for shear stress acting along slipplanes in an ideal Coulomb material :
|τ | ≤ C + σ′ tanφ (1)
The parameters can be constrained from labora-tory tests in e.g. a shearing device, where annularshearing- or ring-shearing (torsional) apparatuses
minimize the end-zone effects associated with di-rect shear boxes. The used apparatus will usuallymeasure the applied effective stress, the shear stress(or the torque in rotational shearing) and the dila-tion. Figure 3 shows an idealistic version of sensorreadings when shearing a granular material. Theshearing can be subdivided into four stages (Li andAydin, 2010):
1. At the beginning, the shearing results in con-traction, caused by elastic deformation of thegrains and further reorganization of the parti-cle assemblage towards smaller volume (Tay-lor, 1948) in the new, stronger stress field (thestress is now a product of the perpendicularshear stress τ and effective stress σ′.
2. The assemblage dilates due to beginning par-ticle movement Reynolds (1885). At the endof this period, the peak shear strength will bereached (τu).
3. The shear strength of the material decreasesunto a steady state. The shear zone evolvesinto a high-porosity layer, with a minimalthickness of 5–10 grain diameters in non-cohesive granular material (de Gennes, 1999).The grain activation intensity decays exponen-tially with the distance from the center of theshear band (Herrmann, 2001).
4. In the final, critical state, the shear willdisplay no low-frequency volumetric changes,only high-frequency fluctuations caused by thegranularity of the material, and the reorgani-zation of the force-bearing system within (Liand Aydin, 2010).
If the material is not preconsolidated, the theshear stress and dilation will increase to the criticalstate without passing through a maximum (Ned-derman, 1992).
The macromechanical properties can be furtherdivided into apparent (peak) values, φa and Ca, andeffective values, φ′ and C ′. The apparent (peak)values are determined as the maximum value mea-sured during shear, denoted as τu. The effectivevalues are derived from the critical state shearstrength of the material. In saturated granularmaterials, the effective strength has the value ofthe normal stress with the porewater pressure sub-tracted; σ′ = σ0 − pw.
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
Figure 3: A four-stage shearing model for preconsoli-dated granular materials in direct shear. From Li andAydin (2010).
The shear experiment is repeated on preferablynew material, otherwise homogenized material, un-der a different effective stress. The peak- and ef-fective shear stress values are recorded along withapplied normal stress values. Figure 4 shows an ex-ample of a linear regression to constrain macrome-chanical properties of a number of shear experi-ments.
2.3 Air permeability during progres-sive strain
The air permeability is a direct measurement of theporespace network, and is a function of the size ofthe pore spaces and the degree of interconnection ofthe voids. It can be measured during compressionor shear by recording the air flow rate required tomaintain a steady value of the pressure gradientbetween two opposite sample boundaries.
The airflow is assumed to be laminar, and anintegrated form of Darcy’s law is calculated with:
Q =ka∆pasηLs
(2)
where Q is the measured volumetric flow rate, ∆pis the pressure difference, as is the cross-sectionalarea, ka is the air permeability, and Ls is theheight of the sample. The air viscosity is ηair =1.827× 10−5 kg m−1 s−1.
During compression, interaggregate pores areless stable than cylindrical macropores, and pores
0 50 100 150−5
0
5
10
15
20
25
30
35
40
45
Normal stress, σ´ [kPa]
Ultim
ate
shear
str
ength
, τ u
[kP
a]
Sample: GB, φa = 39.7125
o, φ´ = 28.2315
o, C
a = 0.42608 kPa, C´ = −0.6214 kPa
Linear Fit, Pre failure
RS Data, Pre failure
Linear Fir, Post failure
RS Data, Post failure
Figure 4: (σ′, τu)- and (σ′, τ ′)-plot and linear regres-sion of ring-shear data, performed on a well-sorted gran-ular material. Constrained macromechanical propertiesare φa = 39.7◦ and Ca = −0.4 kPa and φ′ = 28.2◦ andC′ = −0.6 kPa. It should be noted that the true cohe-sion cannot be negative and thus should be assumed tobe zero.
parallel to the maximal stress field gradient are me-chanically more stable than oblique pores (Schafferet al., 2008).
The dynamics of the porespace ratio is very inter-esting in the context of progressive shear strain, asthey have a high potential impact on the mechani-cal properties of shear zones in saturated materialswith low hydraulic conductivity. During deforma-tion of a saturated granular material, the void ratiowill increase during plastic deformation (Reynolds,1885), which leads to a decrease in the porewaterpressure in the shear zone. As the effective pres-sure is increased (recall that σ′ = σ0 − pw), theMohr Coulomb criterion predicts an increase of theshear strength (Iverson, 2010). If the relationshipbetween the shear velocity and the hydraulic con-ductivity is sufficiently high, the hydraulic gradientwill not have sufficient time to reestablish pressureequilibrium. The net effect is a strain hardening.The shear zone would thus tend to relocate to aweaker part. If the shear is driven not by con-stant shear velocity, but constant shear stress, theresulting shear displacement will be periodic, dis-crete events (Iverson, 2010).
These considerations however treat grain crush-ing as neglectible, and it should be kept in mindthat it can have a high potential effect onto the
grain size distribution in the shear zone, demon-strated in e.g. Sassa et al. (2004). The grain crush-ing will reduce the volume/thickness of the shearzone, resulting in an opposite increase in the pore-water pressure, i.e. a possible strain softening. Forthis reason, it is necessary to consider the mechan-ical durability of the shear zone grains with regardto the magnitude of the force chains in the grainstress network.
3 Pseudoanalytical model
The pseudoanalytical model presented in the fol-lowing, is used for investigating static problems ofstress and strain in soil mechanics. Boussinesq’sproblem deals with a point load on the surface ofan elastic half space (Boussinesq, 1885; Johnson,1985), where it is possible to calculate the theoret-ical stress at any point in the half space. If thestress is integrated over a lateral infinite plane par-allel to the surface, placed at a certain depth, thevalue will always equal the point load, independentof the depth. The stress is however more widelyspatially distributed with great depth.
3.1 Sohne’s summation procedure
SoilFlex is a pseudoanalytical model, in the waythat it uses Sohne’s summation procedure (se be-low) to superposition a number of results from theanalytical Boussinesq’ equation to determine thestress state in an elastic half space, when the up-per boundary is stressed by an irregular load areadiscretized into a number of point loads. Sohne’ssummation procedure consists of the following:
• The contact area of a non-point load is dividedinto a number of point loads.
• It is solved by applying the analytical solutionfor each of these points.
• The model correctly reproduces impacts ofload and contact stress
3.2 SoilFlex usage
SoilFlex is an interactive Excel algorithm, backedby Visual Basic macros, aimed at agricultural ap-plications where compaction of the substratum byagricultural vehicles is of interest. The work flow isdivided into three steps:
1. Formulation of the upper boundary condition:Stress field normal to the tyre-soil interface.FRIDA model predicts the tyre stress in (x, y)-space based on values for the tyre air pressure,the tyre size, the wheel load and the soil prop-erties. The load can be applied in increments.
2. Sohne’s summation procedure in the soil basedon the analytical Boussinesq equation (1885):
The contact plane is divided into point loads,where each is treated as a Boussinesq problem.
3. The soil strain response to the stress (stress-strain relationship) is chosen from:
• Gupta and Larson (1982)
• Bailey and Johnson (1989)
• O’Sullivan and Robertson (1996)
In relation to field measurements, the stresses areusually underestimated in the top soil, and over-estimated in the lower parts. The reason for thiscan be that the upper model boundary condition isinaccurate, the model for stress propagation is in-sufficient, or that the soil stress measurements donein the field are inaccurate.
SoilFlex has capabilities of handling verticalstrain in two ways; 1) Uniaxial inferred strain,where the vertical strain equals the volumetricstrain constrained from e.g. triaxial tests, and 2)plane strain, where the vertical strain is a functionof both the volumetric strain and the shear strain.Plane strain is closer to the real conditions, butneeds additional input parameters which are oftennot available.
3.3 Stress-strain models
In the following, the variations in the predictedbulk density between two different stress-strainmodels is investigated.
Two identical single wheels with wheel load3000 kg, a tyre inflation pressure of 724 kPa, a tyrewidth of 0.50 m, and a diameter of 1.13 m are sim-ulated. The resulting contact area and pressure isshown in figure 5. The mesh resolution is 0.05 m.
The surface-parallel shear stress are neglected,and only the vertical stresses are calculated (σz).Figure 6 shows the vertical stress with depth alongthe (x = 0, y = 0, z) line.
The substratum is chosen to consist of a singlesoil layer with a thickness of 1.50 m. The soil cohe-sion is set to C = 60 kPa and the angle of internalfriction is φ = 20◦. These and the soil mechanicalproperties were held constant between the differentstress-strain models.
The input data to the O’Sullivan and Robertson(1996) stress strain model is based on precompres-sional compaction parameters, where the slope of
Figure 5: Contact area of wheel and theoretical load,calculated by SoilFlex.
Figure 6: Vertical stress with depth, calculated bySoilFlex.
Figure 7: Compaction model by O’Sullivan andRobertson (1996).
Figure 8: Compaction model by Gupta and Larson(1982).
the virgin compressional line and the recompres-sional line among other parameters are set. Se fig-ure 7 for variations of the bulk density with depth.This model predicts an initial increase in the den-sity in the uppermost 0.3 m of the soil. After thesecond wheel has passed, the density is increasedeven further. This implies that this model predictsplastic deformation, even when the second load isidentical to the first one.
The Gupta and Larson (1982) model is adjustedby setting parameters based of the the virgin com-pression line, parameters related to the saturationof the soil, and others. Figure 8 shows the pre-dicted strain from this model. A major difference isthat the application of the second, identical, wheelload does not result in further plastic deformation,which fits to precompression theory.
The O’Sullivan and Robertson (1996) predictsa higher bulk density of the uppermost soil. Thedepth of the plastic deformation is close to identicalbetween the two models.
4 Finite Element Modelling
Finite Element Modelling (FEM) is a numericaltechnique for finding approximate solutions for par-tial differential equations (PDEs). FEM is a goodtechnique fpr describing any physical phenomenaof continuum bodies (soil, liquid, gas) that can becaptured mathematically with a PDE, e.g. defor-mation of a solid body due to applied stress or fluidand heat flow through porous materials.
Most PDEs cannot be solved analytically be-cause the boundary conditions are too complicated,the material properties are heterogeneous and/oranisotropic, the constitutive laws are non-linear orthe processes that the PDEs describe are coupled.
The basic concept is that a body or structuremay be discretized into smaller elements of finitedimensions and homogeneous properties called fi-nite elements. The original body or structure isthen considered as an assemblage of finite elementsas nodes in the finite element mesh. Equilibriumequations at each element ensure continuity at eachnode. The temporal domain is also discretized toobserve non-steady state systems.
4.1 Problem description
Glacial water transport can take place in a varietyof ways (figure 9). The majority of the meltwateris transported subglacially, either in a distributedmode, where it flows as groundwater flow in thesubglacial bed or in a thin film at the ice-bed in-terface, and/or in a non-distributed (discrete) modethrough channels. These channels can be carved upinto the ice (Rohtlisberger), typically when the iceis resting on stable bedrock surfaces. On soft, de-formable beds, the non-distributed flow takes placein Nye-channels (or N-channels) instead, that areeroded into the sedimentary bed. On the upperboundary of the channel, the cryogenic pressureof the slow-flowing ice will try to make the icemove into the channel, while the meltwater pres-sure and/or -heat keeps the channel open. Mean-while the meltwater erodes into the subglacial bed,and the deformable bed can creep into the channelfrom below. In these situations, the bed will tendto be softer than the glacial ice.
I will use a FEM model to simulate the situationwhere the meltwater flow in the channel has ceased,and the relative soft bed creeps into the channel.
Figure 9: Types of glacial water transport. Subglacialbed: grey, glacier: transparent, arrows: flow paths
The model geometry is displayed in figure 10.
4.2 COMSOL setup
COMSOL Multiphysics is used for model setup, nu-merical computation and data analysis. The prob-lem is formulated as a 2D “Structural Mechanics(solid)” setup. The study type is time dependentto allow continuous deformation of the sediment.
The right and left boundaries (10) are given the“roller” boundary condition (no friction). The up-per left and upper right boundaries (marked withred color) are excerting a prescribed stress down-wards of 85 kPa, a value in the range of effectivenormal pressures beneath glaciers and ice sheets(Cuffey and Paterson, 2010). The channel bound-aries are free to move. The mesh is physics-controlled (automatically generated) with an extrafine element size, resulting in 5442 elements. Thegrid is configured for adaptive mesh refinement, soareas with the highest numerical gradients recivefiner mesh nodes after automatical remeshings dur-ing the computations. The model is run from time0 s to 10 s with output every 0.1 s.
The sedimentary bed is given the behavior closeto a subglacial till. Young’s modulus was E =207× 106 Pa, Poisson’s ratio was ν = 0.3, the den-
Figure 10: Model geometry and initial, automatic mesh. The channel is 0.20 m deep.
Figure 11: Soil stress magnitude at time t = 10.0 s.
sity was ρ = 1700 kg m−1, the angle of internal fric-tion was φ = 17◦, and the cohesion was C = 60 kPa.The bed was set to deform with the COMSOL “SoilPlasticity” model, and given the Mohr-Coulombyield criterion.
4.3 Results
Figure 11 shows the magnitude of the stress in thesediment at the end of the simulation. When in-creasing the surface scale factor, it becomes appar-ent that the sediment has started to creep into thevoid. The magnitude of this creep is however no-tably low. Therefore this mechanism seems unlikelyto cause the channel closure.
The sediment close to the channel floor is ex-
certed with a very low effective pressure (blue col-ors in figure 11). The sediment would be in a loosestate, since the Mohr-Coulomb criteria based shearstrength is proportional to the effective pressure.This implies that it would be very easy for themeltwater to erode the channel floor as soon as thechannel meltwater pressure drops beneath the valueof the cryogenic ice pressure. This suggests howceasing meltwater channels could act as effectiveerosional agents of the substratum, which is a veryinteresting result and to my knowledge not yet con-sidered within glacial geology. The erosive effect ofsubglacial channels is documented in the geomor-phological tunnel-valley structures, where the modeof erosion has been widely discussed (Jørgensen andSandersen, 2006).
5 Discrete Element Modelling
The Discrete Element Method (DEM, also calledthe distinct element method) (Cundall and Strack,1979) simulates the physical behavior and interac-tion of discontinuous material, and is ideal for sim-ulating the dynamic behavior of granular material.In this exercise, a capillary bond approximationwas incorporated in a three-dimensional model, andthe macroscopic behavior and stability was evalu-ated.
5.1 Theory
In this DEM formulation, each particle is a discrete,unbreakable unit with its own mass and inertia, andthe assemblage dynamics are examined under theinfluence of e.g. gravity and boundary conditionssuch as moving walls. In this case, the particlesare treated as spherical entities, which reduces thecomplexity of contact search and -dynamics. Basedon the net forces applied, the movement and rota-tion of each particle is calculated during each smalltime step (∆t) by application of Newton’s law ofmotion for entities with constant mass:
mx = F Iω = T (3)
where m is the particle mass, x is the particle po-sition vector and F is the total force vector. I isthe moment of inertia, ω is the angular velocity andT is the total rotational moment. A second-orderscheme based on the Taylor expansion was chosenas the integration scheme on the base of accuracyand simplicity (Kruggel-Emden et al., 2008).
Using the soft body contact model, the time isdiscritized into small steps (typically ∆t ≈ 10−8),and the particles are allowed to overlap (overlapdenoted δ). For a particle pair a and b:
δab = ||xa − xb|| − (ra + rb) (4)
where r represents the individual particle radius. Ifδ < 0, there is an overlap of the particles. The nor-mal vector between the two particles is calculatedfrom the vector between the particle centers:
nab =xab||xab||
(5)
From the overlap, a contact model is used to con-strain the normal- and tangential components of
the resulting repulsive force, in this case modelledwith a linear-elastic model, where the tangentialshear force is limited by the Mohr-Coulomb consti-tutive model. The linear-elastic model was chosenover Hertzian models due to the results of Di Renzoand Di Maio (2004), which reported a better fit toexperimental data with linear formulations. Thespring constant at the contact normal is given a re-duced value for the unloading of the contact thanduring the loading, to simulate energy dissipationby plastic deformation caused by the contact (Wal-ton and Braun, 1986):
fn = −knabδabnab (loading) (6)
fn = −knabαδabnab (unloading) (7)
where knab is the elastic stiffness of the contact (aglobal value of kn = 105 N m−1 was used), nab isthe normal vector between the particle position vec-tors, and α ∈]0; 1] is a dimensionless constant forincorporating the dissipative energy loss (a valueof α = 0.9 was used). The tangential shear forcecomponent is a nonlinear relation limited by fric-tion (Richefeu et al., 2006, e.g.):
fs = −min(γt||δt||, µ||fn − fc||)δt
||δt||(8)
where γt denotes the tangential viscosity of the con-tact. The shear velocity vector is calculated as:
δtab = xab − (xab · nab)nab (9)
The liquid bond force fc were introduced usingthe simple capillary bond implementation proposedby Richefeu et al. (2006), which is simple to im-plement while still providing realistic results. Thecapillary attraction is implemented as a force lawexpressing the capillary force as a function of thedistance, water volume, and particle diameters (seeRichefeu et al. (2006) for details on the algorithm).
It is based on an exponential decay of thecontractive cohesional capillary force below thedebonding distance (δdebond). θ is the surface wet-tability, Vb is the liquid bond volume, R is the ge-ometrical mean of the two particle radii, and λ isa non-static value accounting for the exponentialdecay of the bond strength with distance. Thevalues were set to θ = 0 and Vb = 1× 10−12 m3,and the tangential viscosity was given a value ofγt = 1 N s m−1.
This approach assumes that there is sufficient liq-uid for all possible bonds, and that the bond is in-stantaneously formed when the particles are withinthe debonding distance. The algorithm could pos-sibly be improved by having seperate bonding- anddebonding distances, where the bonding distanceis of smaller length. This approach would howeveralso require management of the contact history, andis thus less parallel in nature.
5.2 Implementation
This three-dimensional DEM algorithm is formu-lated for GPU computation using the C/C++CUDA API (NVIDIA, 2010a), which takes advan-tage of the parallel nature of the problem. Theshear force component formulation in equation 8was chosen over incremental shear force formula-tions to make the contact force calculations evenmore parallel, and avoiding the need for contacthistory bookkeeping. Each component of the con-tact search-, contact model-, and integration com-putations for each particle is a single-instruction,multiple-data problem, suited for the massivelyparallel structure of the GPU streaming multipro-cessors (Kirk and Hwu, 2010; NVIDIA, 2010b).The neighbor search is reduced by discretizing thespatial model domain into a uniform, cubic gridwith Thrust sorted cell lists. The algorithm checkswether the cell size is sufficient to include thedebonding distance.
Visualization of the particle assemblage is per-formed using a custom CUDA ray-tracing algo-rithm. The capillary bonds were not rendered.
Simple walls are implemented, parallel to theworld coordinate system. The wall-sphere collisionsare treated as inelastic collisions with a 10% energyloss.
5.3 Results
A number of experiments were conducted with thealgorithm, figure 12 shows an example of 5832 par-
ticles clustering together due to the cohesive capil-lary forces. Gravity was disabled to better visualizethe clustering in a static image.
Figure 12: 5832 particles in experiment with capillarybonds. The simulation domain has the dimensions 3 ×3 × 3 cm.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.5
1
1.5x 10−4 Dissipation of kinetic energy
Time [s]
Tota
l kin
etic
ene
rgy
[J]
Figure 13: Total kinetic energy.
Figure 13 shows the total kinetic energy of thesystem at time t by calculating the sum of the ki-netic energy of all N particles:
Ekin =
N∑i=1
1
2
(4
3πr3i ρi
)v2i (13)
where ρ is the particle material density (in this caseρ = 3600 kg m−3), and v is the particle velocity.The plot shows an initial increase in kinetic energydue to capillary cohesion when the general interpar-ticle distance is 0 < δ < δdebond. The energy startsto dissipate when it is lost in contact collisions (eq.6, 7 and 8) when δ < 0.
6 Outlook
What could I use from the course?As I have had it introduced from a constructionalpoint of view, I had a large interest in the new, agri-culturally related angle on geotechnical problems. Iwas pleasantly surprised how much of the materialI can directly reuse when considering the subglacialmechanics. Prior to the course, I was unaware ofthe geotechnical research activities at Foulum. Ifpossible, I am sure that your extensive laboratoryfacilities and knowledge in the field will come intohigh value later during my studies.
My main fields of interest during the course werethe review of granular material theory and Molec-ular Dynamics/Discrete Element Method, as theseare most directly related to my work, and are dif-ficult to find included in a course. Even though Ihave been involved in the subjects for some timenow, I still learned a lot of new ways of handlingspecific tasks. With regards to the laboratory work,I am sure I will find good use of the theory withregards to air permeability when I reach the de-velopment phase of the fluid-particle coupling inmy Discrete Element model. It seems like a goodbenchmark for the permeability of the material un-der different stress conditions.
While it was nice to review the main subjectsonce again during the preparation of this report,I have to admit that I think the requirements forthis final assignment were too extensive and timeconsuming, relative to other 5 ECTS PhD courses Ihave participated in. I would however not hesitateto recommend this course to other students withinmy field, if held again at a later point in time.
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