Micromorphic continuum stress measures calculated fromthree-dimensional ellipsoidal discrete element simulationson granular media

R.A. Regueiro, B. Zhang, & F. ShahabiDepartment of Civil, Environmental, and Architectural EngineeringUniversity of Colorado, Boulder, United States of America

ABSTRACT: The objective of the paper is to demonstrate the calculation of micromorphic continuum stressmeasures based directly on three-dimensional (3D) ellipsoidal discrete element (DE) simulations over represen-tative volume elements (RVE) within the DE domain representing granular media. We will demonstrate thesecalculations for two simulations: (1) pile penetration, and (2) cavity expansion. All calculations are assumedto be performed in the current configuration with respect to alarge deformation micromorphic continuum me-chanics analysis. The micromorphic continuum stresses include the unsymmetric third order couple stress tensormijk, symmetric second order micro-stress tensorsij, and unsymmetric second order Cauchy stress tensorσij .The DE continuum stress measure due to Christoffersen et al.(1981) and Rothenburg & Selvadurai (1981), isused as the symmetric “micro-element” stress tensorσ′

ij for micromorphic continuum mechanics (Eringen &Suhubi 1964). The balance equations solved are the balance of linear and angular momenta for the DE sim-ulations, whereas the micromorphic continuum stress measures are only used to interpret the DE simulationresults, at the moment.

1 INTRODUCTION

To date, homogenized continuum modeling of gran-ular materials has considered primarily traditionalcontinuum mechanics as the overlaying framework(Christoffersen et al. 1981, Rothenburg and Selvadu-rai 1981, Chang et al. 1992, Borja and Wren 1995,Muhlhaus et al. 2000, Peters 2005, Andrade and Tu2009) or higher order methods like micropolar con-tinuum mechanics or enhanced continua with straingradients (Chang and Gao 1995, Walsh and Torde-sillas 2004, Pasternak and Muhlhaus 2005, Gardinerand Tordesillas 2006). These methods involve at-tempts to either replace the underlying discrete par-ticle mechanics with an equivalent continuum con-stitutive model, or provide a continuum frameworkto which to up-scale the discrete particle mechanicsthrough underlying Discrete Element Method (DEM)simulations (Cundall & Strack 1979) via hierarchicalup-scaling procedures. We are interested in both ap-proaches, but follow here the spirit of the latter classof methods (hierarchical up-scaling), and we use mi-cromorphic continuum mechanics as the overlayingcontinuum mechanics framework. In the paper, we fo-cus only on the micromorphic continuum stress mea-sures, and consider micromorphic deformation and

strain measures for future work. We focus on the mi-cromorphic theory because of the additional kinemat-ics it provides, through not only the micro-rotation ofa micro-element volume (e.g., the Representative Vol-ume Element (RVE) of a granular material), but alsomicro-stretch and micro-shear. The micropolar theo-ries were primarily motivated for granular materialsbecause of the additional rotational degrees of free-dom of the individual particles, but these theories missthe ability to model clusters of particles that also shearand compact/dilate.

1.1 Brief background on the kinematics of amicromorphic continuum at finite strain

Figure 1 presents the mapping from the reference con-figurationB0 to the current configurationB throughthe deformation gradientFkK (dxk = FkKdXK) andmicro-deformation tensorχkK (ξk = χkKΞK). ΞK isthe relative position vector between the centroidCof the macro-element differential volumedV and thecentroidC ′ of the micro-element differential volumedV ′ in the reference configurationB0. Similarly, ξkis the relative position vector between the centroidcof the macro-element differential volumedv and thecentroidc′ of the micro-element differential volume

P (X,Ξ)p(x,ξ, t)

X1

X2

X3

Cc

C ′

c′

Ξξ

XxX ′

x′

F , χ

dVdv

dV ′dv′

B

B0

Figure 1: Kinematics of a micromorphic continuum at finitestrain.

dv′ in the current configurationB. Position vectors ofthese centroids are indicated in Fig.1. From these ba-sic kinematics, new micromorphic balance equations(including the balance of first moment of momen-tum, which we use below) can be derived (Eringen& Suhubi 1964, Eringen 1999, Regueiro 2011).

1.2 Motivation in the context of overlap couplingmethods

Ongoing work involves direct overlap coupling be-tween large deformation micromorphic continuum fi-nite elements (FEs) with underlying DE particles fora concurrent multiscale computational modeling ap-proach for simulating large deformation analysis ofgranular materials in regions of interest, such as theinterface with a tire, penetrometer, or buried explosivesimulation (see Fig.2). The purpose of this multiscaleframework is to capture as much of the underlyingDE particle kinematics and kinetics within the micro-morphic continuum FE formulation, such that the DEregion can be minimized for computational efficiency,and also to minimize or negate any artificial boundaryeffects due to the overlap coupling of DE with micro-morphic FE close to the interfacial mechanics region(e.g., tire tread, penetrometer surface, etc).

2 HIERARCHICAL UP-SCALING FORMICROMORPHIC CONTINUUM STRESSCALCULATIONS FROM 3D DEMSIMULATIONS

To start, we recall the symmetric part of the granu-lar stress definition (we call this stress, in the contextof a micromorphic continuum, the “micro-element”stress (Eringen & Suhubi 1964)) as (Christoffersenet al. 1981, Rothenburg and Selvadurai 1981)

σ′

kl =1

2vRV E

M∑

ǫ=1

(f ǫkb

ǫl + bǫlf

ǫk) (1)

continuum region (FE)

finite element nodes whose motion is prescribed

by underlying particles

finite element nodes whose motion is unprescribed

free particles

ghost particles (particles whose motion is prescribed

by continuum displacement and rotation fields)

overlap region

between particle

and continuum

particle region (DE)Q

Q

D

D

Bh

Bh

Bh

BDE

Figure 2: Two-dimensional illustration of the coupling betweenparticle and continuum regions. The purple background denotesthe FE overlap regionBh with underlying ghost particles, aquablue the FE continuum regionBh with no underlying particles,and white background (with brown particles) the free particleregionBh ∪BDE .

wheref ǫk is the interparticle force at contactǫ, andbǫl

is the branch vector connecting the two particle cen-troids with contactǫ. We assume there areM totalnumber of contacts within the RVE volumevRV E atthe instant in time when the micro-element stressσ′

kl

is calculated. These RVE domains (with surface areaaRV E and unit normalsnRV E) comprise an averagingdomainΩavg

α with surfaceΓavgα (see Fig.3). The center

of Ωavgα is denoted by pointcα, and the center of the

RVEs by pointsc′. The relative position vectorξm ex-tends fromcα to c′ in the current configuration of thebodyB (see Figs.1,3). Essentially, we substitute dis-crete averaging domainΩavg

α for macroscopic differ-ential volumedv, and discrete RVE domainvRV E formicro-element differential volumedv′. It is assumedthat the macroscopic micromorphic continuum mea-sures are calculated at pointcα over discrete averagingdomainΩavg

α , for α= 1 . . . navg, wherenavg is the num-ber of averaging domains over the discrete continuumbody Bh in the current configuration, whereh is adiscretization parameter representing a finite element(FE), finite difference (FD), or meshfree approxima-tion to solve the micromorphic continuum governingequations. The advantage of this approach is to uti-lize an already well-established micromorphic contin-uum theory at finite strain (Eringen & Suhubi 1964,Suhubi & Eringen 1964, Eringen 1999), and to focuson an apparent criticism of the theory which is howto determine parameters for the higher order consti-tutive models (Forest & Sievert 2003, Regueiro 2009,

Ωavgα

Γavgα

vRV E

aRV E

nRV E

cα

c′ξ

B

∂B

Figure 3: Illustration of averaging domainΩavgα for micromor-

phic stress calculation, and smaller RVE volumevRV E for sym-metric ‘micro-element’ granular stress calculationσ′.

Regueiro 2010).To continue in this direction, we define, based on

the previous discussion, a volume average for thesymmetric micro-stressskl and unsymmetric couplestressmklm as

skldef=

1

Ωavgα

∫

Ωavgα

σ′

kldv′ (2)

mklmdef=

1

Ωavgα

∫

Ωavgα

σ′

klξmdv′ (3)

Before moving on, recall that balance equations arederived by localizing an integral over the bodyBin the current configuration. We consider discreteapproximations to these integrals (e.g., an eventualGaussian quadrature rule), such that

∫

B

(•)dv =

navg∑

α=1

∫

Ωavgα

(•)′dv′ ≈

navg∑

α=1

(•)αWavgΩavg

α

∫

Ωavgα

(•)′dv′ ≈

∑

RV E

(•)′WRV EvRV E (4)

whereW avg andWRV E are the weights of integra-tion, which for now we assumeW avg = WRV E = 1(later, these will take appropriate values in the contextof a FE, FD, or meshfree numerical implementation).For example, the micromorphic continuum symmetricmicro-stresssα at pointcα within averaging domainΩavg

α can be calculated as,

(skl)α =1

Ωavgα

∑

RV E

σ′

klvRV E (5)

and likewise for the micromorphic continuum unsym-metric couple stressmα as

(mklm)α =1

Ωavgα

∑

RV E

σ′

klξmvRV E (6)

The question then becomes how to calculate themicromorphic continuum unsymmetric Cauchy stressσα at pointcα. Recall from the micromorphic contin-uum theory (Eringen & Suhubi 1964) that the stressesare defined as follows,

smkdvdef=

∫

dv

σ′

mkdv′ (7)

σklnldadef=

∫

da

σ′

lkn′

lda′ (8)

mlkmnldadef=

∫

da

σ′

lkξmn′

lda′ (9)

Thus, the volume average definition fors in (2) and(5) is in line with micromorphic continuum theory in(7), whereas the volume average definition form in(3) and (6) is an assumption when compared to (9).The unsymmetric Cauchy stressσ, however, cannotbe uniquely determined from (8), nor can it be esti-mated by a volume average ofσ′ because this is thedefinition fors. Thus, we resort to the balance of firstmoment of momentum to calculateσ.

The balance of first moment of momentum of themicromorphic theory, integrated over the bodyB inthe current configuration, is written as (Eringen &Suhubi 1964, Eringen 1999),∫

B

[σmk − smk +mlkm,l + ρ(ℓkm − ωkm)]dv = 0 (10)

whereρ is the mass density,ℓkm the body force coupleper unit mass, andωkm is the micro-spin inertia. Wecan analyze each term separately, with in mind thatwe want to solve forσα at pointcα:

∫

B

σmkdv ≈

navg∑

α=1

(σmk)αΩavgα (11)

∫

B

smkdv ≈

navg∑

α=1

(smk)αΩavgα =

navg∑

α=1

∑

RV E

σ′

klvRV E

∫

B

mlkm,ldv =

∫

∂B

mlkmnlda

≈

navg∑

α=1

∑

RV E

σ′

lkξmnRV El aRV E

∫

B

ρℓkmdv ≈

navg∑

α=1

∑

RV E

ρ′gkξmvRV E

∫

B

ρωkmdv ≈

navg∑

α=1

∑

RV E

ρ′aξkξmv

RV E (12)

Table 1: Parameters for quartz sand/gravel. If there is a secondnumber, it is associated with the cavity expansion simulation.

Young’s modulusE (Pa) 2.9× 1010

Poisson’s ratioν 0.25specific gravityGs 2.65inter-particle coef. of frictionµ 0.5inter-particle contact damping ratio 5%,30%particle radii (m) 0.001∼ 0.0025viscous damping ratio 2.0/∆t, 0time step∆t (sec) 5.0× 10−6,5.0× 10−7

wheregk is the gravity acceleration vector,ρ′ is themass density of particles invRV E , andaξk = ξk is theacceleration of the relative position vector which isimportant for dynamics problems, as will be seen forthe cavity expansion problem in the Numerical Ex-amples. We calculateaξk based on the motion of therelative position vectorξk, which tracks the centroidsof the RVEsc′ with respect to the centroid of theaveraging domaincα. In this case, the RVEs are as-sumed to move with the particles, as for the cavityexpansion example in Section 3.2. Then, substitut-ing (11)-(12) into (10), we can solve for the micro-morphic continuum unsymmetric Cauchy stressσα

at point cα. Note that asξ → 0, σα = sα, and theCauchy stress is symmetric (i.e., it is the same asthe symmetric micro-stress, and vice versa). Thus,for certain continuum mechanics problems of interestto geotechnical and geological engineers, the granu-lar material can be treated as a classical continuumsuch that the assumptionξ → 0 is valid. For specificproblems of interest, however, the discrete nature ofthe granular material from the perspective of a pen-etrometer in sand, or tire rolling through gravel,ξ isfinite, and thus a generalized continuum theory likemicromorphic continuum mechanics may provide ad-ditional insight into the mechanics of granular media.At the very least, it will be relevant for overlap cou-pling procedures, as shown in Fig.2, where the degreeof freedom (dof) mismatch between DEM and clas-sical FEM in the overlap region is apparent, and theadditional dof in the micromorphic continuum shouldimprove such procedures.

3 NUMERICAL EXAMPLES

We demonstrate the hierarchical up-scaling proce-dure in Section 2 for calculating the three micromor-phic continuum stresses (σij , sij ,mijk) by conductingthree-dimensional (3D) ellipsoidal DE simulations fortwo examples: (1) pile penetration, and (2) cavity ex-pansion. Particle properties are indicated in Table 1.

For comparison purposes in the results, we alsoconsider the unsymmetric granular stress definition

σΩ

kl =1

Ωavgα

N∑

ǫ=1

(bǫlfǫk) (13)

whereN is the total number of contacts inΩavgα .

Figure 4: Side view of pile penetration simulation, showingaver-aging domainΩavg

α drawn by box in the initial penetration zone.

3.1 Pile penetration

This example is run quasi-statically, with the ‘pile’represented as a large ellipsoidal particle for now(see Fig.4). Further details are found in Yan et al.(2010). The averaging domainΩavg

α is drawn by abox in the initial penetration zone, and is divided into2× 2× 2 = 8 RVEs. These RVEs are fixed. For com-parison purposes, the mean couple stress and devia-toric couple stress are defined as follows,

(pcouplem )α

def=

1

3(maam)α (14)

=⇒ pcoupleα =

√p

coupleα · p

coupleα

dev(mlkm)αdef= (mlkm)α − (pcouple

m )αδlk (15)

=⇒ τoct,coupleα =

√1

3

√(devmα)

...(devmα)

The plots of mean stress and octahedral shear stressare shown in Fig.5, calculated over the averaging do-mainΩavg

α as discussed in Section 2. They compare thefollowing six stress results: (i) symmetric micro-stresssα, (ii) unsymmetric Cauchy stressσα, (iii) unsym-metric Cauchy stressσα without micro-spin inertiaρω term in (10), (iv) unsymmetric Cauchy stressσα

without micro-spin inertiaρω and body force coupleρℓ terms, (v) unsymmetric couple stressmα, and (vi)unsymmetric granular stressσΩ. It is observed thatfor each mean stress history in Fig.5, as the pile pen-etrates through the averaging domainΩavg

α (see box inFig.4), the mean stress increases in compression (neg-ative), and then begins to decrease as the pile passesthroughΩavg

α (the depth ofΩavgα is 0.03m). Similarly,

for octahedral shear stress in Fig.5, the shear stressincreases as the particles inΩavg

α shear with respect toeach other during pile penetration. Now let us com-pare the various stress measures. For both mean and

0 0.02 0.04 0.06 0.08 0.1−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

depth of end of pile (m)

mean stress (kPa)

couple stress (normalized)

Cauchy stress

Cauchy stress, no accel

Cauchy stress, no accel and grav

symmetric micro−stress

unsymmetric granular stress

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

depth of end of pile (m)

octa

he

dra

l sh

ea

r str

ess (

kP

a)

unsymmetric granular stresssymmetric micro−stresscouple stress (normalized)Cauchy stressCauchy stress, no accelCauchy stress, no accel and grav

Figure 5: Mean and octahedral shear stress versus time for thevarious stress measures of the pile penetration example.

octahedral shear stress, the symmetric micro-stresssαand unsymmetric granular stressσΩ are not equal, in-dicating that the granular stress is unsymmetric. Theunsymmetric Cauchy stressσα, as a result of account-ing for the couple terms in the balance of first mo-ment of momentum in (10), (11)-(12), shows a re-duced value when compared tosα andσΩ in Fig.5,and the effect of acceleration and gravity force cou-ple termsρω andρℓ are negligible. Recall the unitsof the couple stress, which are Pa.m. Thus, to com-pare the couple stress values on the same plot, themean and octahedral shear couple stresses are nor-malized by the relative position vectorξ, which hasunit of length; also,−pcouple

α is plotted. Note, the maincomparison is betweensα andσα to see the effect ofthe length scale (i.e., the relative position vectorξ)through the couple stress divergence term in the bal-ance of first moment of momentum. Further work willconsider various-sized averaging domains and RVEsto better understand this assumption of length scale(i.e.,ξ), and its influence on the stresses (and in turn,constitutive equations).

3.2 Cavity expansion

This example is run dynamically, where the cavityexpansion is started by a high initial outward veloc-ity of an interior cluster of particles, that then pushes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

time (s)

mean stress (kPa)

Cauchy stress, no accel and grav

Cauchy stress, no accel

Cauchy stress

couple stress (normalized)

symmetric micro−stress

unsymmetric granular stress

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time (s)

octa

he

dra

l sh

ea

r str

ess (

kP

a)

Cauchy stress

couple stress (normalized)

Cauchy stress, no accel

symmetric micro−stress

unsymmetric granular stress

Cauchy stress, no accel and grav

Figure 6: Mean stress and octahedral shear stress versus time forthe various stress measures of the cavity expansion example.

the remaining particles (initially at rest under grav-ity) upward in a box, that then settle down again bygravity (see Fig.7). The changing averaging domainΩavg

α is indicated by the box drawn at various timesin Fig.7. This domain is in turn further divided into4 × 4 × 3 = 48 RVEs. Figure 6 displays the com-parison of the stress values over the simulation timefor mean and octahedral shear stresses. It is observedthat as the cavity expands and pushes the particles up(Fig.7(b)), all mean stress values approach 0 (Fig.6)because there are few particle contacts. As the par-ticles settle by gravity, the mean stress increases incompression (negative), until they come to rest. Sim-ilarly for the octahedral shear stress, as the particlesreach their maximum height (and least number ofcontacts) at 0.025s, the octahedral shear stress ap-proaches 0 (Fig.6). Upon settling, the octahedral shearstress increases again, and then decreases to a steadystate value when the particles come to rest. Here, thesymmetric micro-stresssα and unsymmetric granularstressσΩ are equal, as there is little shearing of parti-cles as compared to the pile penetration problem. Forthis dynamic example, the significance of the micro-spin inertiaρω and gravity force coupleρℓ are evidenton the calculation of unsymmetric Cauchy stressσα.

(a) (b) (c)Figure 7: Side view of cavity expansion simulation, showingav-eraging domainΩavg

α drawn by box, moving with particles at dif-ferent simulation times: (a) 0.000s, (b) 0.075s, and (c) 0.375s.

4 CONCLUSIONS

The paper focussed on deriving the equations to cal-culate the three stresses within a micromorphic con-tinuum theory, for an averaging domainΩavg

α using 3Dellipsoidal DEM simulations: (1) symmetric micro-stresssα, (2) unsymmetric Cauchy stressσα, and (3)unsymmetric couple stressmα. The stressessα andmα could be calculated directly from volume aver-age definitions, whereasσα required evaluating thebalance of first moment of momentum. Stress resultsfor pile penetration and cavity expansion DEM simu-lations were compared. It was observed that depend-ing on the example, significant shearing can generatean unsymmetric granular stressσΩ as well as Cauchystressσα (for pile penetration), and that for dynam-ics (cavity expansion), the role of micro-spin inertiaand gravity couple force can significantly affect thecalculation of Cauchy stressσα.

Such a hierarchical multiscale framework for esti-mating micromorphic stresses, when extended to in-clude the calculation of micromorphic strains, can beused to estimate material parameters of micromor-phic constitutive models, as well as which parame-ters may be set to zero, and also how to estimate thelength scale (directly related to the relative positionvectorξ in the micromorphic theory). This relies onthe DEM model (or other underlying discrete physics-based model) to be physically relevant itself, and thereare open questions on that issue that are beyond thescope of the paper (e.g., interparticle constitutive rela-tions, particle shape/fracture, particle mineralogy andcrystallographic orientation effects, pore water and aireffects, ...).

Future work includes further investigation of thesize of the averaging domainΩavg

α and correspondingRVE sizesvRV E within Ωavg

α , their relative positionvectorsξ, and resulting stress calculations.

5 ACKNOWLEDGEMENTS

We would like to acknowledge funding for the re-search from ONR MURI grant N00014-11-1-0691.

RAR also acknowledges support from a UPS Founda-tion visiting professorship in the Department of Civiland Environmental Engineering at Stanford Univer-sity during the writing of the paper.

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