d’addetta, ramm - 2006 - a microstructure-based simulation environment on the basis of an...

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Granular Matter (2006) 8: 159174DOI 10.1007/s10035-006-0004-4

O R I G I N A L PA P E R

Gian Antonio DAddetta Ekkehard Ramm

A microstructure-based simulation environment on the basis

of an interface enhanced particle model

Received: 22 May 2005 / Published online: 3 March 2006 Springer-Verlag 2006

Abstract The present contribution introduces enhanceddiscrete element simulation methodologies (DEM) with aspecial focus on a microstructure-based model environment.

Therewith, it is possible to represent the failure of cohe-sive granular materials like concrete, ceramics or marl ina qualitative as well as quantitative manner. Starting froma basic polygonal two-dimensional particle model for non-cohesive granular materials, more complex models for cohe-sive materials are obtained by inclusion of beam or interfaceelements between corresponding particles. In particular, wewill introduce an interface enhanced DEM methodologywherea standard ingredient of computational mechanics, namelyinterface elements, are combined with the particle method-ology contained in the DEM. The last step in the series ofincreasing complexity is the realization of a microstructure-based simulation environment which utilizes the interface

enhanced DEM methodology. With growing model complex-ity a wide variety of failure features of cohesive as well asnon-cohesive geomaterials can be represented. Finally, theplan of the paper is enriched by the validation of the newlyintroduced and re-developed discrete models with regard toqualitative and quantitative aspects.

Keywords Particle model Discrete element method Lattice model Interface model Geomaterial Cohesivefrictional material Microstructure

1 Introduction

From a physical point of view geomaterials like concrete,ceramics or marl can be considered as cemented granulatesforming a heterogeneous macroscopic solid. In order to pre-dict the remaining load capacity after reaching the ultimate

G. A. DAddetta(B) E. RammInstitute of Structural Mechanics, University of Stuttgart,Pfaffenwaldring 7, 70550 Stuttgart, GermanyE-mail: [email protected].: +49-711-6856123Fax: +49-711-6856130

load a substantiatedknowledge of the underlyingfailure mech-anisms is essential. Thereby, this knowledge may for exam-ple be based on micromechanical observations. The failure

mechanisms of these materials are characterized by complexfailure modes and, furthermore, they show a highly aniso-tropic bias due to their inhomogeneous microstructure. Thegrowth and coalescence of microcracks in cohesive geoma-terials lead to the formation of macroscopic crack patterns.Finally, this results in a fragmentation into separate particleclusters forming a solid-granulates mix. Behaving quasi-brit-tle under load these materials are characterized by a localiza-tion of deformations in narrow zones. Since localization phe-nomena like cracks or shear bands occur, the material cannotbe treated as continuous in the usual manner. If fracture andfragmentationof the solid occurs, the creation andcontinuousmotion of the evolving crack surfaces apparently represent

discontinuous phenomena and are difficult to handle. There-fore, most continuum models, and in particular those onesbased on continuum damage mechanics, cannot account forthe discrete nature of material failure in a natural way andneed some extension, confer [3,29,37]. Alternatively, dis-crete particle models like discrete element methods (DEM)have been developed. As the name DEM suggests, a solid isreplaced by a discontinuous particle composite which allowsfor a detachment of bonds between particles (if initially pres-ent) and a re-contact of open surfaces. In order to simulateand quantify the full range of geomaterials from non-cohe-sive ones like sand to cohesive ones like concrete, ceramics orrock, starting from the pioneering work by [7] different types

of discrete element models have been elaborated, see [6] fora recent review. Starting with a basic polygonal two-dimen-sional DEMmodel fornon-cohesive granular materials, morecomplex models for cohesive materials are obtained by inclu-sion of beam or interface elements between correspondingparticles. As the quantification of previously presented beamenhanced DEM models is an arduous task, see [10,11,20,21,22], the standard (non-cohesive) version of the DEM model,see [36], is augmented by interface elements between the par-ticle edges. The formulation and numerical implementationof this interface enhanced DEM model will be presented in


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160 G. A. DAddetta, E. Ramm

(a) (b)

Fig. 1 a Geometry of the contact. b Interpretation of contact force logic in terms of plasticity law

detailin this paper. Thereby, tension and compressionsimula-tions of cohesive rectangular particle composites are utilizedto validate the model. The next step is the realization of amicrostructure-based simulation environment which utilizesthe foregoing enhanced DEM models, confer [8]. The micro-

structure can be included, if different properties of the cohe-sive components (beam or interface) are assigned with re-spect to their position, i.e. inside the matrix, inside the aggre-gate and between aggregate and matrix. With growing modelcomplexity a wide variety of failure features of geomaterialscan be represented. Furthermore, the inclusion of an artificialmicrostructure which regards for stiffer aggregates embed-ded in a less stiffer matrix enables a coherent quantificationof the model.

In a parallel line, adequate homogenization techniqueshave been introduced since the 1990s. These techniques al-low to relate microscopic quantities, like the contact forcesand displacements, to corresponding macroscopic quantities,

like stresses and strains, compare [8,12,13,15] for a corre-sponding application to the present DEM model. The devel-opment and numerical implementation of adequate homoge-nization approaches by means of a micro to macro transitionfrom the particle to the macro level supplements the formaldefinition of the DEM models. Homogenization procedureshave been developed which allow for a transfer from a sim-ple Boltzmann continuum based particle model to a morecomplex continuum with microstructure, see [8] for a de-tailed overview. However, for brevity no details concerningthis topic are given here. Interested readers are referred to thenoted literature.

The paper is organized as follows: section 2 provides

a brief outline of the theoretical background of a standard(non-cohesive) form of the DEM model supplemented by adiscussion on the modeling of cohesion within particle mod-els. In section 4 the extension to an interface enhanced DEMmodel along with some representative simulation results issketched. Afterwards, section 5 presents the formulation ofa microstructure-based DEM environment which utilizes theinterface enhanced model according to section 4. Again, sim-ulation results are used to validate the proposed method. Weconclude with a summary of the attained insights and an out-look on future perspectives.

2 Standard DEM model

The starting point of our DEM model development is a two-dimensional DEM code with convex polygonal particlesbased on the work in [20,21,36]. As this standard DEMmodel has been presented elsewhere in more detail, see thereferences above or [8,10,11], the model is only briefly out-lined here.

The individual particles can be considered as rigid bodies.They are notbreakable andnot deformable, but they can over-lap when pressed against each other. Three degrees of free-dom, two translational and one rotational one, are assignedto each particle center, confer Figure 1a. The local deforma-tional behavior of the particles is approximated by an elasticrepulsive force related to the overlapping area of contact-ing particles, grey shaded in Figure 1a. The contact force isdecomposed with respect to the local coordinate system ofthe contact zone

fc = fcn n + fc

t t . (1)

The normal and tangential components of the contact forcevector fc are defined by

fcn = EpAp

dc meffn vrel,n

fct = min

mefftvrel,t , fcn . (2)

The coefficients N and T refer to the viscous dissipativedamping and is chosen according to Coulombs frictionlaw. Thereby, for > 0 the present scheme allows for a non-associated plasticity law, as schematically shown in figure 1bfor three characteristic situations. If the grey shaded flow sur-

face is left as for force point 3, the force state is reflectedback in vertical direction onto the surface confined by theCoulomb criterion fcn . Ep denotes the elastic modulus ofthe particles and Ap the overlapping area of two contact-ing polygons. This overlap represents up to some extent thelocal (reversible and non-reversible) deformation of the poly-gons, comparable with the Hertzian contact law for sphericalparticles. Please note that this representation of the Coulombfriction is a simplification in that the quasistatic limit may notbe appropriately captured, since the tangential force tends tozero as the velocity decreases. Alternatively, the tangential


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A microstructure-based simulation environment on the basis of an interface enhanced particle model 161

force could be handled via a CundallStrack spring, compare[7]. This version has been widely used in the literature, espe-cially in the context of circular particles, because it allowsto carry load even if a particle sample, e. g. a heap, is inrest [26]. Since in the present paper the focus lies on polygo-nal shaped particles that, furthermore, are mostly cohesivelybonded together, we believe that the corresponding effect

is not so dominant to take it into account. However, future,careful quantitative studies are needed in order to test thevalidity of the above simplification by comparing simula-tions with and without friction. A more extensive discussionon the choice of the tangential forces in the context of DEMsimulations may for example be found in [4,23]. The relativevelocity at the contact zone and the effective mass of twocontacting particles i and j are given by

vrel = vj vi , meff =mi mj

mi + mj. (3)

Since our focus is more on cohesive granular materials thannon-cohesive ones, we believe that the rotations of the parti-

cles play a subordinate role and, therefore, the angular veloc-ities in equation (3)1 are neglected. The characteristic lengthof the contact region is defined by 1/dc = 1/di + 1/dj ,where di and dj define the diameters of circles of equiva-lent areas as the polygons i and j . The forces and resultingmoments are inserted into the equations of motion, which aresolved numerically for each particle with the aid of the GearPredictorCorrector time integration scheme, confer [1]. Thedynamical basis of the DEM implies the solution of the equa-tion of motion at discrete time steps t+ t for each particle iof all particles N within the sample. Upon definition offi forall interaction forces between two contacting particles andgi for the gravitational forces of a particle i , the generalized

equation of motion holds for each individual particle i

Mi xi = fi + gi with

Mi =

mi 0 00 mi 0

0 0 i

and fi = np

j =1

fpi j .

(4)

Mi represents the diagonal generalized mass matrix and xithe generalized acceleration of particle i . mi describes themass of polygon i (translational degrees of freedom) andi describes the mass moment of inertia of polygon i (rota-tional degree of freedom). In the case of a simple (non-cohe-sive) particle contact fi is expressed by equation (4)3, i.e. thegeneralized interaction force vector fi contains solely con-

tact forces. Thus, in the present case it is represented by thegeneralized particle force vector fp which is expressed by

fp =

fc mcT

and includes the moment mc of the con-tact forces fc with respect to the center of mass. Thereby,np denotes the number of particle contacts of a particle i .Cohesive forces, as introduced in the coming sections willbe simply added on the right hand side of equation (4)3.

The application of this basic (non-cohesive) DEM modelhas proven to be capable to qualitatively picture the behav-ior of cohesionless granular materials like sand. The locali-zation of shear bands along with the formation of complex

failure pattern was studied by means of dense and poroussamples and reflected experimental observations in an aston-ishing manner. More information on the applied model canbe found in [10,11] and descriptive simulation results oncohesionless particle samples can be found in [8,13,15].

3 Modelling of cohesion in particle models

The model presented so far allows for the application to non-cohesive granular materials like sand, stone or rock heaps.Since most geomaterials are more or less cohesive, a modelis demanded for which is capable to represent cohesion. Inother words, attractive forces between particles are neces-sary to bond these particles together to some extent. Usually,the physical effect of cohesion is limited for tension. Severalways of incorporating cohesion into particle models are fea-sible. The implementation strongly depends on the chosenparticle shape and problem class. The easiest way in the case

of circular particles is the introduction of attractive poten-tials like in Molecular Dynamics (MD). In this regard, themost prominent law that accounts for attractive action be-tween particles is the LennardJones potential adopted fromphysical chemistry, see the description in [1]. Another way,only reasonable forcircular particles, is based on an extensionof the Hertz contact law for negative overlaps. An imple-mentation is rather straightforward as the primary geometricvariable defining the contactof circlesis the distance betweenthe particle centroids, compare [14,24,30,40]. Following thisline in [25] an alternative model was proposed where an ini-tial overlap of circular particles accounted for the attractingforce law. In contrast to easy geometries like circles, thegeometric definition of polygonal particles in contact is byfar more complex. This is the case as more than one geo-metric variable describing the contact and the position of theindividual bonded edges is involved. The distinction betweena point contact (circles) and line contact (polygons) must bedrawn. Therefore, alternative options have to be considered.Two different approaches that account for cohesive bondsbetween two polygon edges have been implemented in thepresent DEM model: In the first approach beam elements be-tween the centers of mass of neighboring particles are intro-duced and constitute a beam enhanced DEM model. Thisapproach will not be given in detail here as it was extensivelydiscussed in the past, see [10,11,20,21,22]. The simulationresults with a beam enhanced DEM model show that thegeneral failure behavior of cohesive frictional materials likeconcrete is qualitatively very well represented. In particular,the beam enhanced DEM model is capable to represent typ-ical inherent failure mechanisms of cohesive geomaterials.Furthermore, it has proven to be very well suited for a visu-alization of these mechanisms which are technically difficultto observe in experiments, [8,10,11]. However, a realisticsoftening behavior cannot be obtained in simulations withbeam enhanced DEM models, due to the presently includedbreaking law of the beam elements, i.e. the failure appearsmostly brittle in the context of normalized loaddisplace-


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(a) (b)

Fig. 2 Idealization of interface in a undeformed stage and in b deformed stage

ment diagrams. Summarizing, the qualitative picture of thebeam enhanced model is satisfactory, but the quantificationremains still a demanding task. As an alternative, an inter-face enhanced DEM model was introduced where continuousinterfaces along common particle edges represent the bond-ing component. The next section is dedicated to this topic.

4 Interface enhanced DEM model

In order to represent the cohesion between particles in a DEMmodel, an approach borrowed from the Finite Element Meth-odology (FEM) is adopted. It makes use of so-called inter-face elements that are directly defined at the particle edges.In that, the model is made more complex: Instead of simpleelastic beams with an oversimplified brittle failure law as inearlier versions of this DEM model, see the references in theprevious section, more elaborate interface elements with apopular constitutive law on the basis of the plasticity the-ory are used here. The realization of this interface enhancedmodel is influenced by a former work of [39] and follows theplasticity formulation introduced therein. The motivation touse this material law in the context of the DEM follows theprinciple that a proven and established macroscopic materiallaw may be at least much that successful in modeling thefailure on a smaller scale as in a continuum model (used typ-ically for a higher scale). Apparently, this approach picturesthe real physics involved in the debonding process of bondedgranulates far better than the beam enhancing approach.

A variety of models for the representation of the interac-tion between the constituents of two- and more-phase par-ticle composites via interface elements have been proposedsince the late 1960s. The works by [17] and [31] have beenpioneering as they were among the first ones to introduce dis-continuities into numerical models in the context of the FEMeither as continuous or lumped interface elements. On the

one hand, lumped interface elements evaluate a forcerela-tive-displacement constitutive relation at a nodal point. Inso-far, to a certain extent they exert a similar behavior as simplesprings. No further assumptions with regard to tractions andrelative displacement distributions across the interfaces aremade. On the other hand, the formulation of continuous inter-face elements (line, plane or shell type) implies a continuousrelative displacement field and a traction-relative displace-ment constitutive relation.

Following these lines interface elements have been thefocus of intensive studies in the context of FEM in the past.

4.1 Basic idea

The interface layer is regarded as infinitely thin in the ini-tial stage.Since the particles are undeformable by definition,use is made of the key assumption that the deformation ofthe interface is constrained to be linear. In analogy to thepreviously introduced beam model this fact represents theBernoulli-hypothesis of a planar cross-section to some ex-tent. The interface can be discretized and is represented ina lumped sense by a fixed number of normal and tangen-tial spring sets with stiffnesses kn and kt. These springs areattached along the common edge of two initially bonded par-ticles i and j , as shown in Figure 2a. The current locationof start and end points of the springs relative to the bond areinitially fixed and evaluated after each time step. It is impor-tant to note that the interface shown in Figure 2a has a visibleinitial thickness for visualization reasons only. Ongoing rel-ative motion of the particles leads to a finite value of relativedeformation for each spring, i.e. overlapping of particles and,thus, contraction of springs (negative relative deformation)is also allowed. The relative deformation of the bonds is rep-resented by the extension and contraction of the springs, asindicated in Figure 2b.


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As further assumption the numerical integration of theconstitutive lawof the spring sets, which are noted integrationpoints in the sequel, yields an approximated stepwise con-stant stress distribution over the edge. This procedure com-pares that for the computation of stress resultants in beams.The constitutive law used throughout this chapter is a non-associated MohrCoulomb type softening plasticity model

defined by two yield surfaces according to [39] and is dis-cussed in more detail in section 4.2. A failure, or more pre-cisely detachment, of the bonds is achieved, if all integrationpoints of the interface layer are completelysoftened. It shouldbe kept in mind that no extra contact force according to equa-tion (1) is created, as long as a bond between two particlesexists. This means that once the bond is completely detached,a standard contact according to section 2 is assumed to con-trol the interaction between two particles. The forces evalu-ated in the integration points canbe combined to a bond forcevector and transferred to the mass centers of the involved par-ticles, thus, giving rise to a moment. The bond force vectorthen takes the form fbond = [ fbondx f

bondy m

bond]T and enters

the equation of motion in (4). The generalized interactionforce vector fi then takes the form

fi =

npj =1

fpi j +

nbondk=1

fbondik . (5)

Additionally to np, noting the number of particle contactsof a particle i , nbond denotes the overall number of existingbonds of a particle i .

4.2 MohrCoulomb plasticity model

The principal ingredients of a plasticity formulation are theyield condition, theflow rule andthe hardeninglaw (ifneeded),e.g. compare [19]. In the present context the basic idea of astrain-driven formulation from classical plasticity is straight-forwardly transferred to a relative displacement-driven for-mulation. The actual state of the spring set is determinedby the total relative displacement u, the plastic relative dis-placement up and the softening variable . Here, the range ofvalidity is restricted to small strains and, following the abovelogic, small relative displacements. Hence, an additive elas-ticplastic split of the relative deformation u of one springset is admissible

u = ue + up with u = unut . (6)

The stress-relative deformation law for the elastic part yields

tr = 0 + Kbond u with Kbond =

kn 00 kt

, (7)

where tr is the trial stress state, that may lie outside theflow surface f. kn and kt represent the normal and tangentialspring stiffnesses. The flow rule is denoted by

up = g(, ) (8)

with the plastic multiplier . g(, ) represents the directionof the plastic flow, which in the present case is assumed tobe non-associated, i.e. the direction of plastic flow is pre-scribed by the gradient of the plastic potential g in the formg(, ) = g/. In general, plastic flow occurs if the yieldfunction f(, ) and its derivative both vanish: f = 0 and

f = 0. The consistency condition f = 0 then yields f

h = 0 with h =

f

up

g

, (9)

where h expresses the hardening parameter. Further elabora-tion, including a combination of equations (6) to (9) results ina relation between stress and relative displacement rates and,finally, in the corrected stress state along with the plasticmultiplier

= tr f(tr, )

f

up

g

+

f

Kbond

g

Kbond

g

. (10)

The procedure in equation (10) operates as a one-step returnmapping algorithm, so that no iteration is needed. However,a simple one-step algorithm that yields an explicit solutionfor the corrected stress state is only possible if linear yieldsurfaces, linear plastic potentials and linear softening evolu-tion laws are chosen. The bracket terms in equation (10) areeither of scalar or vectorial order.

The initial behavior of the spring sets is assumed to belinear elastic and to depend on the spring constants kn andkt. Therefore, no coupling of the springs in the elastic regime

is assumed. The softening of a spring set starts when thestress state = [n t]

T reaches the yield surface that inthe present context could be viewed as a failure surface. Theinitial failure surface in the biaxial stress plane is given inFigure 3a by graph 1 and is symmetric with respect to thenormal stress. Since the failure mechanisms dominated byshear and tension differ substantially in geomaterials, it isadvisable to model them separately even on the grain scale.This implies the bounding of the elastic domain by two dis-tinct failure surfaces, usually termed two- or in the generalcase multi-surface plasticity. Indeed, this does not substan-tially change the treatment presented above for one singlesurface, except the singularities in the failure surface at the

segue of different failure surface segments, i. e. a unique flowdirection cannot be specified. The solution of this problemhas been treated by [18] or [2], confer [8] for more details.In analogy to equation (8) the flow rule

up = 1g1

+ 2

g2

(11)

is adapted to capture the flow directions defined by the twoplastic potentials along with the essential constraints 1 0and 2 0.

In the present case the failure surface resembles a two-surface MohrCoulomb type plasticity criterion. The tensile


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(a) (b)

Fig. 3 a Failure surface. b plastic potential of interface constitutive law in biaxial stress plane

failure is governed by maxn > 0 and an angle which definethe yield surface f1. Whereas the shear failure is controlledby a classical MohrCoulomb failure envelope with cohesionmaxt > 0 and friction angle > 0 which define the yieldsurface f2. In the compressive regime (for negative n ) thetrapezoidal or triangle (in the final state) formed by the flowsurfaces is open, as no failure is possible in pure compression.With ongoing softening the failure surface shrinks to an inter-mediate stage ( 2 in Figure 3a) and ends up on the classicalMohrCoulomb failure surface without any cohesion t = 0and tension limit n = 0, expressed by 3 in Figure 3a. Thefunctional representation of the failure surfaces is given by

f1 = n + t tan (1 )maxn 0 ,

f2 = n tan + t (1 )maxt 0 .

(12)

In order to describe a non-associated plasticity along the linesof the two-surface formulation, two plastic potential surfacesare introduced:

g1 = n (1 )maxn = 0 ,g2 = n tan + t (1 )

maxt = 0 .

(13)

The plastic potential surfaces are alsosymmetric with respectto the normal stress, as visualized in Figure 3b.

The softening of the spring set is described by the param-eter which represents the actual state of damage: rangesfrom 0 in the undamaged state to 1 in the fully damaged state.The softening behavior is driven by the plastic deformationu

pn and u

pt of the spring sets in form of predefined fracture

energies G f,n and G f,t. In the case of a two-surface plastic-ity this involves two different softening evolutions for tensionand shear either, see Figure 4a and b. Pure tensile softeningends after a fracture energy G

f,nhas been released and pure

shear softening ends after a fracture energy G f,t has beenreleased. In the decoupled case, i.e. if only one mechanism isactive, the softening variable for the tensile and shear loadingis defined by

=1

umaxn un

upn , =1

umaxt ut

upt , (14)

with un = maxn /kn and u

t =

maxt /kt, see Figure 4. The

plastic part of the deformation starts at un and ut .

These evolutions are coupled in that, both tensile strengthmaxn and cohesion

maxt decrease at the same time and at the

same rate. This yieldsan isotropicshrinkage of thefailure sur-faces, as shown in Figure3a. A simultaneoustensile andshearsoftening is treated as linear combination of both, as depictedin Figure 4c. Therefore, the definitions in equation (14) areformally coupled

=1

umaxn un

upn +1

umaxt ut

upt 1 . (15)

By definition, in combined softening upn is solely determined

by the plastic deformation due to tensile loading (surface 1 f1 and g1) and not by the normal part of the plastic sheardeformation. For this reason, the dilatant behavior with shearloading does not influence the softening since it is physicallynot justified in combined softening. The predefined fractureenergies for mode I and II softening enter the above equationsthrough the integral expression of the graphs in Figure 4aand b. This formalism may be interpreted in the sense of the

cohesive crack concept, see e.g. [27], where the crack open-ing is replaced by the components of the plastic relative dis-placement. The evolution laws of the normal and tangentialstresses

n (upn ) =

maxn

1

upn

umaxn un

,

t(upt ) =

maxt

1

upt

umaxt ut

,

(16)

are inserted into the conditional equations of the fractureenergies

G f,n =

umaxn un

0 n (u

p

n ) dup

n =

1

2 umaxn un maxn ,G f,t =

umaxt ut

0

t(upt ) du

pt =

1

2

umaxt u

t

maxt .

(17)

4.3 Numerical realization

The numerical realization of the plastic interface model in thecontext of the DEM model is a straightforward task. Eachspring set is described by the material law and along each


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(a) (b) (c)

Fig. 4 a Evolution law for tensile, b shear and c combined softening

common edge a fixed number of spring sets, termed integra-tion points, are chosen. From there, it is important to definea reference edge where the corresponding relative displace-ment-force law is evaluated. Different possibilities are feasi-ble, confer [8,9].

Here, the way displayed in Figure 5 is followed: Onebond is chosen as reference edge and the local coordinatesystem with the normal vector n is fixed along this bond.Particle i can be considered as the master and the other oneas the slave particle. For this case, the relative deformationof the start and end point of the reference edge are split upinto a normal and tangential part. The normal and tangentialrelative deformation distribution between the start and endpoint of the reference edge are linearly approximated. Theactual complete relative deformation u is evaluated at eachintegration point and is processed in an incremental formatu by subtracting the complete relative deformations of twosuccessive time steps. The straightforward implementationof the elastic predictor step for the calculation of the trialstress state in equation (7) implies the determination of thespring forces at each integration point k of the bond

ftrn,k( ) =1

ni p

foldn,k( ) + knun,k( )

,

ftrt,k( ) =1

ni p

foldt,k ( ) + ktut,k( )

.

(18)

ni p denotes the number of integration points along thereference edge and is fixed for all polygons bonds. Theposition of the integration point k is determined in the lo-cal coordinate system by the normalized coordinate rang-

Fig. 5 Numerical integration Definition of reference edge

ing from [1, 1]. In order to evaluate the yield conditionsf1/2(

tr(), old()) 0 in equation (12) the local stressesat each integration point are computed

trn ( ) =ftrn ( )

h/ni p, trt ( ) =

ftrt ( )

h/ni p. (19)

If the yield surfaces f1/2 are left, a plastic corrector stepis necessary. Therefore, the type of back projection mode(determination which yield surface/s is/are active) has to beevaluated. Equation (10) yields the corrected stress state atthe integration point. Afterwards, an a posteriori check isperformed, because it cannot completely be excluded thaterroneously a corner regime is predicted, although this isactually not the case. This behavior may appear, as the finalposition of the yield surface is not known a priori, see [2].In particular, for stress states in the influence region of thecorner a transition from a tensile to a shear state may beobtained. Thereafter, the corresponding forces at the integra-tion points are calculated via an inversion of equation (19).

The new forces, the actual relative deformation and the stateof softening are saved as history variables for the next timestep. All forces along a bond are summed up and transferredto the center of the master particle givingrise to a moment. Aforce and moment with the same absolute values are appliedto the slave particle. An interface is detached or eliminated,if at all integration points of this interface the softening iscompleted, i.e. = 1 according to

=1

h

11

() d =1

ni p

nipk=1

(k) . (20)

When the softening at all integration points has reached its

limit, the bond is detached. For a detailed chart of the inter-face plasticity law algorithm for one interface with k integra-tion points interested readers are referred to [8]. A standardcontact as described in section 2 is assumed whenever twoinitially bonded particles with broken bond come again intocontact.

4.4 Numerical results

The interface enhanced DEM model is involved in the sim-ulation of standard loading setups. The structured validation


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(a) (b)

Fig. 6 a Stressstrain and b stressdisplacement diagram of tension test

of this model, first, on the integration point level and, second,by simple loading setups of quadratic rows with a thicknessof one particle are presented elsewhere, see [39] and [8]. As

these preliminary test programs have been successful, largersamples with variable particle shapes are simulated in uniax-ial tension and compression.

4.4.1 Tension simulations

As mentioned earlier, a parameter identification of the beamenhanced DEM model parameters with respect to a quantifi-cation of the load-displacement behavior of geomaterials isnot convincing at all. The following examples will highlightthat the bond description via interfaces is capable to representthe softening not only in very simple tests, as shown before,but also in the context of standard experiments. The soften-ing behavior is investigated by means of a comparative studyof rectangular plates with a width of 40cm and a variableheight of 10 cm 1, 20cm 2, 30cm 3, 40cm 4, 64cm 5and 72 cm 6. Thus, the amount of included particles rangesfrom 400 to 2880. First, the results of a uniaxial tension sim-ulation are presented. The specimens are vertically loadedunder constant strain rate conditions by applying the load ina strain driven format via a constant increase of the velocityof the particles at the top and bottom of the plate. Thereby, wefollow the procedure firstly described in [33], as it ensuresthat disturbing initial dynamic effects are reduced to a tol-erable level, see the discussion in [22]. The parameters andmaterial properties of the model have been chosen in orderto obtain results that are as close as possible to experimentalresults obtained from the literature. However, only limitedinformation on the real values of the bond strength and thecorresponding softening behavior is available in the litera-ture. Therefore, no attempt was undertaken to fit the materialparameters more closely than needed, since the needed con-siderable time amount is disproportionate to the additionalinsight obtained from such a fitting!

Uniaxial tension experiments on concrete by [16] as wellas [27] are considered for a quantitative comparison. Thenormal and tangential stiffness of the interfaces are chosen

as kn = 1, 200 kN/cm2 and kt = 360 kN/cm

2. Moreover,the density was chosen as = 2.5 g/cm3 and the time stepas t = 5 107 s. The shape parameters of the plasticity

model are = 26.6, = 10 and = 0, except themaximum yield stress maxn =

maxt = 0.12kN/cm

2 and

the softening parameters G f,n = 5.94 104 kN/cm2 and

G f,t = 2.98103 kN/cm2. The yield stresses for each inter-

face are statistically distributed around 10% the averagevalue defined above. Parameters of the contact model accord-ing to section 2 like those concerning the viscous dampingand friction are set to zero (N = 0, T = 0, = 0)and the contact stiffness to Ep = 100 kN/cm

2. The stress-strain curves for the six simulations 1 to 6 are plotted inFigure 6a along with the experimental results by [16], whichare included as a grey shaded band.

It becomes clear that the stress-strain relation is a non-objective measure of the softening, i.e. the longer the speci-men the steeper is the post-peak behavior. Nevertheless, thenormalized load-displacement relations in Figure 6a gener-allyelucidate thetransition from a linear relation to a horizon-tal tangent up to a complete overall softening of the sample.The stresses are measured in the form of a normalized loadas ratio of the reaction force of the boundary particles and thespecimen width. The strain is measured via a determinationof the ratio of the actual length and the initial length of thespecimen. The mesh/length dependence of the strain is one ofthe mainreasons whydata of tension tests (ofexperimental aswell as simulation nature) is typically displayed by a stress orload versus. crack width diagram in the literature. The axial

deformation is evaluated in a comparable format as in exper-iments by a measuring device of fixed length lme = 5 cmfor all simulations. This means that not the elongation ofthe complete specimen, but the extension of a fixed area ofthe specimen containing the macroscopic crack is evaluated.The extension of this fixed area compares to the crack widthw, see the inset of Figure 6(b). Thus, if a stressw dia-gram is plotted, all curves coincide and the softening effectis the same regardless of the height of the sample, compareFigure 6b. Moreover, these results have been compared withexperimental results in the form of the stress-average-crack


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(a) (b)

Fig. 7 Simulation output in the post-peak regime: a Final stage d. b Detail of macroscopic crack of stages a to d

opening diagram of a concrete with a maximum aggregatesize dmax = 1.6 cm according to Figure 3.49 of [27]. Thenumerical results fit qualitatively as well as quantitativelywell the experimental ones. However, the height of the tailin the softening region could not completely be recast, as thestress drops below the experimental value in the later simula-tionstages. In theexperimentsstill a considerablestresstrans-fer is obtained in the later softening regime. The reason forthis is quite conceivable and can be traced back to the maxi-mumsize of aggregatescontained within a sample. Moreover,following the discussion in [27] this behavior is an indirect

effect of crack face bridging which provides a crack toughen-ing mechanism. In order to enclose the feature of crack facebridging and a corresponding stress transfer even in a latersoftening stage, one may include a distinct microstructurewhich for example regards for stiffer aggregates embeddedin a less stiffer matrix. This path will be followed in the nextsection.

An alternative concept takes into account a higher disper-sion of bond stiffness and/or yield stresses.

A cutout of the deformed sample 4 in figure 7 at fourdifferent time steps a, b, c and d according to the stressstrain curve in Figure 6a emphasizes the distinct macroscopiccrack opening as well as the evolutionof softening. Thebonds

are represented by a line between the respective particles.The scale defines the transition from a non-softening stageto a fully softened stage just before elimination of the bond(black). The state of softening of the bonds in Figure 7bis computed based on equation (20) as the average of thesoftening of all springs (i.e. integration points) representingthis bond. One can see that at stage d the macroscopic crackhas completely formed. However, three interfaces positionedperpendicular to the primary load transfer direction (markedby circles) sustain the load and yield a non-vanishing stressof curve 4.

4.4.2 Compression simulations

Next, uniaxial compression tests of plateswith differentgeom-etries using the same parameter set as for the tension sim-ulation are carried out. Therefore, a similar load setup asdiscussed before for the tension case but with interchangedloading direction is applied and additionally different bound-ary conditions are studied. Here, we will discuss the result ofa specimen with an unrestricted boundary which representsa situation with very low friction at the boundary. Therefore,the particles forming the upper and lower boundary of the

specimen are completely free to move, e.g. like a loading byteflon platens or platens with brushes. For brevity, the stress-strain diagram and a comparison with experimental result isnot given, compare [8] for more details. The failure evolu-tion of the unrestricted compression simulation is visualizedby means of the output at two time stages 1 and 2 inFigure 8, which capture the situation around peak load. Onceagain, the bonds between the particles are represented by aline between them. Time stage 1 refers to a situation justbefore the peak and stage 2 to a situation after formation ofa failure pattern. The gradual development of inclined local-ization zones, as shown in Figure 8b has started long beforethe peak value is reached and yields a sudden breakdown

of the particle composite. This result agrees quite well withthe experimental results on cubical specimens by [38,39].However, the softening behavior is less pronounced in com-pression with respect to the experimental results by Vonk.The simulation output of a restricted sample (high frictioncase, not given here) resembles the same hourglass failuremode coupled with en-enchelon cracks as observed in simu-lations with the beam enhanced DEM model in [8,10,11] aswell as in experiments.

Note that the additional complexity of the bond descrip-tion compared to the beam one is accompanied by a higher


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Fig. 8 Failure evolution of compression simulation

sensitivity of the parameter choice with respect to the post-peak failure. This implies that the more or less controlledfailure in compression as obtained with the beam enhancedDEM model, e.g. compare the results in [8,10,11], is notreproducible with the interface enhanced DEM model in thepresent form. For example, a continuation of the loading afterformation of a failure pattern like that presented in Figure 8yields a complete disintegration of the initially bonded par-ticle assembly. Thus, no conical rest pieces as in the exper-iments, e.g. in the form of fragments composed by bondedparticles, remain after complete cracking. In the context ofthis destabilizing effect the following two points are worthto be further investigated: First, the formulation of the inter-face model has to be enhanced. For example an inclusionof a physical coherent damping or comparable stabilizingcontribution to the interface forces may provide for a sta-ble simulation path in the later post-peak regime. Further-more, a second, probable reason for the found behavior isthe missing rotational resistance of two bonded granulates inthe interface enhanced model compared to a consideration ofit in the beam enhanced model. There, the rotational resis-tance is considered via the corresponding stiffness entry inthe Timoshenko stiffness matrix and the corresponding rota-tional breaking parameter. The effect of the rotational resis-tance inherent within the beam enhanced DEM model hasalready been emphasized as a basic microdeformation mech-anism for the simulation of geomaterials by [32].

4.5 Discussion of cohesion modeling

Two approaches that account for cohesion between two poly-gon edges have been implemented in the context of the DEMmodel according to section 2. In the first, rather simplistic ap-proach (not presented in detail here) beam elements betweenthe centers of mass of neighboring particles have been intro-duced, confer [10,11,20,21,22]. As an alternative, a modelbased on continuous interfaces along common particle edgeswas proposed in the past section.

The approach based on the introduction of beam elementshas shown to be capable to represent most inherent fracturemechanisms of cohesive frictional materials, see the simula-tion resultsin [8,10,11]. Thesimulations fit qualitatively wellexperimental observations. However,the quantification of thecorresponding parameters withrespect to the output of exper-iments like stress-strain curves is still unsatisfactory, conferthe discussion in [8]. This is primary due to the choice of thetwo beam breaking parameters. Although physically plausi-ble from the micromechanical point of view, these param-eters cannot be identified with any known (micro) materialparameters of geomaterials. Hence, at the present stage, apure parameter search would end up in a curve fitting with-out winning any new knowledge on the physics behind it.Anyhow, it should be kept in mind that this model is verywell suited for a visualization of typical failure mechanismsappearing in the testing of cohesive geomaterials. The inter-face enhanced DEM model shows the advantage to picturethe real physics involved in the debonding process of bondedgranulates far better than the beam enhanced one. However,this is only possible at the expense of a higher computationalcost which is required due to the increased complexity of themodel and a higher sensitivity of the simulation results withrespect to the chosen parameters. The inclusion of the morecomplex bond representation favors the quantification of themodel to experimental results, i.e. the post-peak softening isbetter reproducible and quantifiable than in the case of thebeam enhanced DEM model. Though, still some features like

the softening behavior at a later simulation stage in tensionas well as a less pronounced softening in compression are tobe clarified. Therefore, the complexity of the enhanced DEMmodel is further enhanced by defining a microstructure-basedsimulation environment.

5 Microstructure-based DEM model

The inclusion of a microstructure is thought to remedy thementioned problem points. This section comprises simula-


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(a) (b)

Fig. 9 a Definition of particle properties (cohesive components suppressed). b Definition of properties of cohesive component

tions with the interface enhanced DEM model where the par-ticle mesh represents an artificial microstructure. Therefore,stiff macro particles that are composed by an agglomerationof a finite number of particles are embedded in a soft matrix,which is also represented by a composition of particles. Inso-far, this so-called microstructure-based DEM model may beseen as an extension of the previously introduced interfaceenhanced DEM model. However, a realization of a micro-structure-based simulation environment on the basis of themuch simpler beam enhaced DEM method is also conceiv-able. In order to include a real microstructure correspond-ing digital image processing software has to be connected tothe mesh generation module of the DEM program. However,this way is not followed here, since at the present stage aneasily realizable implementation is sought. Instead, an alter-native, approximative implementation is carried out by creat-ing an artificial microstructure, see [8] for more details. Inthat, the procedures proposed in the framework of the FEMe.g. by [5,35,39] are principally paralleled, as these publi-cations concern the creation of artificial two-phase materialsusing polygonal aggregate shapes. Furthermore, the presentapproach is related to that presented in [34], see also [27],where circular aggregate particles are generated based on astatistical distribution which is related to grading curves ofconcrete. Thus, the composition of a real microstructure ofconcrete-type materials is treated in an approximate and arti-ficial way here. It is not thought that the simulation resultsbased on a real and an artificial two-phase microstructurediffer that much though. First, the procedure applied for thecreation of an artificial microstructure is given. Afterwards,we will present simulation results of uniaxial compression

and tension load setups and a concluding discussion of theattained insights.

5.1 Generation of a microstructure

The generation of an artificial microstructure is based on theformulation of macro particles, i.e.one macro particle is com-posed by an accumulation of a finite number of polygonalparticles which are denoted as micro particles. An initiallycreated polygon mesh is overlaid by a mesh composed of

larger particles. Certainly, the geometry of a known micro-structure, e.g. by means of electron microscopy images, couldbe included as an overlaying mesh. This option is not con-sidered here, however, is in the realm of possibility in futureimplementations. After generating a corresponding particleassembly (dense or porous), the particles are scaled downand rotated in a statistical fashion. The scaling down of these(macro) particles by a variable factor is controlled by thedesired proportion of aggregate to matrix volume. In orderto obtain the overlaying mesh of larger particles the aforementioned mesh is scaled up. This scaling up depends onthe favored size of aggregates. Finally, this mesh is laid overthe (underlying) dense particle mesh and represents the pat-tern for the definition of the macro particles. Based on thisinformation the (micro) particles of the underlying mesh areflagged as being inside or outside the macro particles. Themacro particles represent the stiff aggregates and the remain-ing particles define the matrix. As example the contiguousparticles colored in light grey in Figure 9a are identified asthe macro particles and the particles colored in dark grey asthe matrix material. The corresponding cohesive componentsare classified as follows, compare alsothe sketchin Figure9b:Inside the aggregate (a) (middle grey), inside the matrix (m)(light grey) and defining the interface layer between aggre-gate and matrix (i ) (black). This interface layer shouldnot be confused with the interface elements used within theinterface enhanced DEM model. The first one describes areal material interface layer (between aggregate and matrix),while the last one represents a model interface in form of acomponent of the enhanced DEM model.

Finally, the values of the stiffnesses of the interface ele-

ment (kn , kt), the corresponding fracture law (maxn , maxt ) orboth stiffness and fracture lawparameters, are chosenaccord-ingly. In order to consider the microstructural properties of aconcrete-type material like concrete as best as possible, ben-efit was made from the extensive knowledge of the groupof van Mier. This knowledge comprises the simulation ofmicrostructures via pure beam lattice models starting fromthe first publication by [34], compare also [27]. The respec-tive parameter ratios between (a), (m) and (i ) are estimatedbased on the corresponding ratios introduced in the context oflattice models. In that, an analogous approach of calibrating


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(a) (b)

Fig. 10 a Stressstrain diagram of simulations and Vonkss experiments. b Eliminated bonds at stage 2

the stiffnesses and fracture law parameters noted in [28] ispartially followed. In order to prevent aggregate (i.e. macroparticle) cracking at the present stage the ratios noted in [28]are partly adopted. The following choices were made:

k(i)n

k(m)n

= 0.1,k

(i )t

k(m)t

= 0.4,

max,(i)n,t

max,(m)n,t

= 0.25,

k(a)n

k(m)n

= 2.8,k

(a)t

k(m)t

= 2.8,

max,(a)n,t

max,(m)n,t

= 100.(21)

The matrix values ()m are used as input values and theother values, ()a and ()i , result from equation (21). Keepin mind that the choice of parameters is only influenced bythe fitting of the softening behavior and without consider-ation of a detailed, because unknown,knowledge of the realmicromechanical parameters. Despite the advanced measur-ing devices and setups available in laboratories, the realparameters of the phases (a), (m) and (i ) actually remain inthe dark, i.e. no precise and experimentally verified valuesof the phases stiffness, yield strength and softening law areavailable. In order to fit experimental results of concrete-typematerials these parameters may be estimated on the basis ofa macroscopic view of the problem: One supposes that theknown macroscopic parameters of concrete are identical tothe microscopic ones. For example, the fracture energy of theinterface elements is identified as the experimentally mea-sured, macroscopic one and so forth. An alternative and, inthe authors opinion, more promising way is the estimationof these parameters on the basis of the insights of section 4and utilizing the experience of the group of van Mier in thecontext of the micromechanical simulations of concrete-typematerials.

5.2 Numerical results

It is a moot point whether the inclusion of a material descrip-tion based on an artificial microstructure as introduced in theprevious section instead of the model material may help toovercome the deficiencies of thebeam andinterface enhanced

DEM models. Recall that the model material in section 4included nearly identical stiffnesses and failure laws of thecohesive components. Compression and tension simulationsof an artificial microstructure based on the interface enhancedDEM model presented in section 4 have been carried out. Themodel parameters have been fit to the compression experi-ments by [38] under consideration of equation (21).

5.2.1 Compression simulations

The load setup and measuring procedure was already dis-cussed in the context of the tension simulation in section 4.4.Note that the parameter choice is the result of a parame-ter fitting to experimental results on concrete in absence of

more detailed information on the micromechanical parame-ters. The following matrix parameters have been used: Thenormal and tangential stiffnesses are chosen askn

(m) = 2,000 kN/cm2, kt(m) = 600 kN/cm2 and the frac-

ture energies as G f,n(m) = 4.996 104 kN/cm2, G f,t

(m)

= 5.988 103 kN/cm2. The yield stresses are statisticallydistributed for each interface about 10% the average value.On average one gets n

max,(m) = 0.04kN/cm2, tmax,(m)

= 0.12kN/cm2. Corresponding values for the aggregateand the interface layer between aggregate and matrix areobtained via equation (21). The shape parameters are unal-tered with respect to the choice in section 4 and are noted = 26.6, = 10 and = 0. The general parameters of

the DEM model are also unchanged: The density was cho-sen as = 2.5 g/cm3, the time step as t = 5 107 s,the viscous damping and friction are set to zero (N = 0,T = 0, = 0) and the contact stiffness amounts to Ep= 100 kN/cm2.

Seven compression simulations with the same amountof aggregates and almost identical aggregate/matrix ratioshave been carried out. The only difference between thesesimulation sets is the starting number of the random num-ber generator for the generation of the underlying particlemesh. The quadratic 75 75cm2 sample used in simulation


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Fig. 11 Graphical output of compression simulation

series si m1 is composed by 5501 particles and a total of16,489 interface elements between all particles within the

sample. The microstructure is created by use of 46 macroparticles. The composition of the sample includes a totalof 2,896 particles representing the aggregates and a total of2,605 particles representing the matrix. Thus, the fractionof aggregate volume to complete volume is 0.525. The sim-ulations are confronted with the experiments of [38,39] bymeans of thenominal stressvs. nominal straindiagram and, inparticular, the associated softening branch. The stress-straincurves of all simulation series si mi with i = 1, . . . , 7 areaveraged via a superposition of the corresponding data files.This average curve is symbolically denoted by si mi . InFigure10a thecomparison ofsi mi , simulation seriessi m1

and Vonks tests [38] on concrete with different boundaryconditions is presented. After linearly increasing in the firstpart of the loading program the average stressstrain curvesi mi turns into a non-linear regime up to the peak. After-wards, in contrast to the simulation results presented in theprevious chapters the softening regime is less pronounced,i.e. it follows the predicted experimental results quite well.The continuous failure of interface elements yields a de-crease of the stresses with increasing strains. The averagesimulation result si mi lies near the limit curves of thelow friction boundary cases (teflon/brushes) in the post-peakregime. In summary, the inclusion of artificial microstruc-tures has proven to be an important feature for a realisticrepresentation of the post-peak softening behavior in termsof the stress-strain relation.

In order to give a detailed view of the failure withinthe specimen snapshots of simulation series si m1 are con-sidered. The softening stadium of the interface elements ismonitored at two time steps 1 and 2 in Figure 11. Thesesimulation stages document the course of the non-linear andsoftening branch of curve si m1 in Figure 10a. The sim-plified picture of the composite structure of the sample inFigure 10bhighlights the crack propagation through the spec-imen shortly after peak load. The dark grey color denotes theaggregate particles and light grey the surrounding matrix.Bear in mind that the aggregates are represented by a finite

amount of (macro) particles. For this reason, the shape ofthe aggregates is irregular. The black lines represent the real

geometry of the interface elements that have been eliminatedin the course of the simulation, i.e. = 1. These linesconnect the start and end point of an eliminated interfaceelement and represent the corresponding particle edges. Thetensile splitting type failure, mostly along the boundaries ofthe aggregates, is obvious. Two crack zones in the left andright part of the specimen initiate the macroscopic failure ofthe particle sample. In Figure 11, the brighter backgroundarea represents the aggregates, while the darker backgrounddisplays the matrix. The scale included in this figure refersonly to the softening of the interface elements marked by theline connections between bonded particles. Please note thatif an interface element reaches a stage > 1, the corre-sponding line connection in figure 11 is eliminated. Thus, thedark grey colored matrix particles become visible in regionswhere the failure localizes, i.e. the overlaying interface ele-ment mesh has been partially dissolved in these zones. In thecourse of simulation at stage 1 two macroscopically fail-ure zones form, see Figure 11a. The continuous debondingresults in the final failure mechanism in form of a completedisintegration of the matrix material in these zones, comparestage 2, see Figure 11b. The final failure at stage 2 picturesthe reality quite well, compare the fractured samples by [39].

5.2.2 Tension simulations

The load setup and strain measurement of the tension simu-lation is accommodated according to section 4.4.

Irrespective of the different material parametersof the tar-get concretes to be compared with, the same DEMparametersas used in the compression simulations are considered in thetension simulations. In tension the experiments by [27] witha concrete with a maximum aggregate size dmax = 1.6cmare considered, compare also Figure 6b. Simulations withdifferent aggregate sizes and varying dispersions of the yieldstresses have been performed. The yield stresses have beenstatistically distributed for each interface element around10, 50 and 90% the average values. The exemplary


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(a) (b)

Fig. 12 a Stressdisplacement diagram of simulations and experiments by van Mier. b Eliminated bonds at stage 1

results detailed below are concerned with a simulation serieswith a dispersion of the yield stresses of 50% and inclu-sion of 220 aggregates. Thus, the aggregate size is smallercompared to the compression simulations. Smaller aggre-gate sizes have been considered to accommodate the smallermaximum aggregate size of the target material in mind. Inthe context of the realized implementation, the higher theamount of aggregates is chosen, the smaller is the corre-sponding aggregate size. This yields a higher volume fractionof matrix particles and, finally, a smaller effective stiffnessof the composite sample. The 75 75cm2 test sample wascomposed by 2,145 particles representing the aggregates and3,354 particles representing the matrix. Thefraction of aggre-gate volume to the complete volume amounts to 38.4% andthe ratio of aggregate to matrix volume to 0.623.

Figure 12a compares the described simulation series andexperimental results of a concrete witha maximum aggregatesize dmax =1.6cm according to [27]. The crack width wis measured as difference between the top and bottom parti-cles, since two macroscopically observable cracksappear,seeFigure 12b. The sample height is assumed as a representarycrack zone with a finite width. The simulated peak stress isless than the experimental one by a factor of 2, if the param-eter set of the compression simulation is used.

One reason for this difference is the uncertainty of theparameter choice, since the extraction of real parametersfrom laboratory tests for a definition of the model parametersis not straightforward. Another reason concerns the different

target concretes used for the comparisons in the compressionand tension simulations. Therefore, the corresponding curvesare given in a normalized format by scaling the stresses bythe respective maximum stress. Anyhow, the course of thesimulated and experimental curves agree qualitatively well.The post-peak behavior is more ductile than predicted bythe experiments. Probably, the fracture energy choice wasslightly to high. This is not astonishing at all, as the parame-ters have been pre-optimizedfor the compression simulationsand the corresponding material parameters. As the concretetested by Vonk differs from that used by van Mier with re-

gard to the different key material parameters, this behavior isquite comprehensible. The snapshot in form of the simplifiedpicture of eliminated interface elements in Figure 12b high-lights the crack propagation through the specimen. As ex-pected, only cracks in the horizontal direction and, thus, per-pendicular to the loading direction are obtained. The cracksappear mostly at the interface layer between aggregate andmatrix and fit quite well to the cracking in concrete.

6 Conclusion

Starting from a basic DEM model for non-cohesive polygo-nal particles, the complexity of the model was successivelyincreased in order to include a coherent representation ofcohesive particle assemblies which implied a qualitative aswell as quantitative reproduction of characteristic features ofcohesive geomaterials. As a quantification of beam enhancedDEM models represents an arduous task, interface elementsbetween the particles have been inserted in order to form acohesive bond. The last step in the series of increasing com-plexity was the realization of a microstructure-based sim-ulation environment which utilizes the foregoing enhancedDEM models. With growing intricacy and, therewith, flexi-bility of the models a wide varietyof typical features of cohe-sive frictional materials could be represented in a satisfactorymanner. It was shown that the microstructure-based interfaceenhanced DEM environment represents an effective approachto remedy the deficiencies of the non-microstructure envi-ronments. The inclusion of an artificial microstructure viathe definition of corresponding stiffnesses and yield stresseshas shown up to be an important feature for a realistic repre-sentation of the post-peak softening behavior in terms of thestressstrain relation. With increasing complexity all charac-teristics of a cohesive frictional material can be represented.

As future perspective, the procedures for a quantificationof the parameters should be advanced by further elaboratingthe relation between the micromechanical properties ofgeomaterials and the corresponding model parameters. This


15/16


implies a consideration of special experimental setups to gainmore knowledge on parameters like the micromechanicalyield strength or the corresponding ratios for the aggregate,matrix and interface layer components. In order to achievethis, very simple composite (model) materials should be cre-ated, e.g. regular polygonal particles embedded in a soft ma-trix with glue between matrix and particles. Due to the clear

knowledge of all micromechanical material and, therewith,model parameters, a comparison of simulation and experi-ment should get much easier. If this point is satisfactorilysolved, a real microstructure-based DEM environment maybe considered. Then, an interface module between a digi-tal image processing tool that is capable to handle electronmicroscopy images and the mesh generation module of theDEM program may be incorporated.

Acknowledgements The authorsare indebted forthe financial supportof the German Research Foundation (DFG) within the research groupFOR 326 Modellierung kohsiver Reibungsmaterialien under grant no.RA 218/20. Furthermore, the authors aregrateful forthe interestingandenlightening discussions with H. J. Herrmann from the University ofStuttgart, Germany, F. Kun from the University of Debrecen, Hungaryand S. Luding from Delft University, The Netherlands.

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