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PARTICLE MODELS: SIMULATION OF DAMAGE AND FRACTURE IN COMPOSITES USING ADISCRETE ELEMENT APPROACH

Falk K. Wittel1, Ferenc Kun2, and Hans J. Herrmann3

1Institute for Statics and Dynamics of Aerospace Structures, University of Stuttgart, Pfaffenwaldring 27, D-70569Stuttgart

2Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-40101 Debrecen3Institute for Computational Physics, University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart

CONTENTS

Introduction 1

Discrete Element Method (DEM) Basics 2

Discrete Elements . . . . . . . . . . . . . . 3

Model Construction - Element Organization 4

Simulation Scheme . . . . . . . . . . . . . 5

DEM for Failure in Composites 5

Transverse Ply Cracking . . . . . . . . . . 6

Anisotropic Degradation of Wood . . . . . 9

Fracture in High Performance Fiber Rein-forced Cement Composites (HPFRCC) 11

Conclusion and Outlook 15

References 16

INTRODUCTION

Composites are today’s key engineering materials invarious applications. They are characterized by stronganisotropic properties, high strength and stiffness to

mass ratios and damage tolerance together with thepossibility of an active material design. The ba-sic requirement for designing materials is knowledgeon the influence of design parameters on physicalproperties and behavior. However, when it comesto failure and strength predictions, that are of enor-mous technological importance, composites prove tobe complicated (1). Their damage evolution is a com-plex process developing simultaneously on all inter-nal length scales like micro-, meso- or macro-scale.In composites, scales are defined by spacial exten-sions and arrangements of components like shown inFigs.4,11,15. The presence of different componentsand resulting interfaces inside the material has a re-tarding influence on the damage evolution. Cracks,originating at the atomistic scale, are deflected fromtheir preferred path at interfaces on the microscale.On this scale, damage processes are dominated bythe material disorder. If such micro cracks join, theyget the potential to break through the scale barrier tobecome relevant for the higher meso-scale. On themacro-scale, structural failure occurs as the result ofpropagating cracks, whose properties are controlledby processes on the small scales. The underlying ba-sic damage mechanisms can be found in artificial andnatural composites as well (2). The important roleof the large spectrum of Nondestructive Evaluation(NDE) methods in the discovery and study of micromechanics of damage is discussed in the first part ofthis book.Today’s composite structures are designed, using fail-ure criteria that are micro mechanically justified, butstill phenomenological (3). This is mainly since clas-sical theories like fracture or damage mechanics cannot cope with the observed complexity of the dam-

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age evolution (4). For materials with a small num-ber of major voids and relatively little disorder, frac-ture mechanics has been very successful in predictingfailure. The basic idea is to embed forming cracks,that are discontinuities in the displacement field, inthe continuum e.g. by ficticious representations or bysophisticated element formulations (compare Chapter??). Since cracks initiate and grow in the perfect con-tinuum such approaches are referred to as top-downapproach (5). For highly disordered materials, failureis distributed all over the volume of the material, al-lowing for continuum damage mechanics approachesto describe the damage evolution (4; 6). Unfortu-nately, these can rarely be extended to macroscopicfailure, nor do they make any statement on the rela-tionship between material micro structure and frac-ture behavior and all relevant damage mechanismsmust be known in advance (7).Damage simulations with particle models follow acompletely different strategy, that can be understoodas a bottom-up strategy (8). Elementary particlesare organized and assembled to form a system withmacroscopic continuum behavior determined by thedynamic interaction of all elements. These ap-proaches are often referred to as Discrete or DistinctElement Method (DEM) and they are capable to dealwith the complexity of fracture events due to the dy-namic interaction of particle assemblies. DEM can beunderstood as a powerful set of numerical methodsspecially designed to solve problems of applied me-chanics with strong discontinuities in material or geo-metric behavior (9; 10). DEM provides the time evo-lution of systems by solving the equation of motionof the elements on a micro level. Due to the naturalrepresentation of the material micro structure, dam-age phenomena originated in the interaction of geom-etry and physics can be described realistically withDE models. Only the behavior of particles and theirinteraction needs to be defined on the modeling scale.Damage mechanisms and their interaction emanatesnaturally during the simulation, which proceeds sim-ilar to a molecular dynamical simulation. Failure is ahistory dependent process in the sense that it mattersfor the outcome, in what order microscopic damageoccurs, that strongly depends on the inherent materialdisorder on representative length scales (7; 11). Dy-namic crack growth effects and multiple cracking upto fragmentation for high input energies can be ob-served, up to the extreme when systems decomposecompletely down to their single particles (12; 13; 14).It is important to note, that top-down approaches relyon strength criteria for the initiation and energy cri-teria for the propagation of damage, while bottom-upapproaches only introduce microscopic failure due tolocal over-stressing.

Basically one can distinguish two types of problems:Problems where the discreteness of the model cor-responds to the micro structure of the material, andproblems where the characteristic length scale of thesystem does not match that of the material. Type twocan simply be interpreted as a discretization of a con-tinuum, while the main focus of this article will beon type one. We shall discuss this issue by study-ing damage and fracture in three different examplesof fiber reinforced materials like uni-directional lam-inates (15; 16), spruce wood as a representative for acellular material (17) and short fiber reinforced con-crete. These examples are in the order of increas-ing intricacy from the point of modeling. First webriefly review the varieties of DE models and simu-lation, before we address specific questions of rep-resenting composite failure in the following section.Via the tree studies on transverse ply cracking, on theanisotropic degradation of wood and on the fractureof high performance fiber reinforced cement compos-ites (HPFRCC) we demonstrate the capabilities andpotential of DEM for failure in composites. Overallconclusions and an outlook on science trends are pre-sented in the closing section.

DISCRETE ELEMENT METHOD (DEM) BA-SICS

Generally spoken DEM deals with rigid or de-formable particles, their interaction and displacementin time. Therefore, the configuration of numerous-ness particles continuously changes under some ex-traneous cause as a result of interparticle interactionlaws. The simulation leads to a steady state con-figuration that can be defined e.g. by global staticequilibrium or by ceasing fracture activity. For thesake of a definition we refer to CUNDALL(9) who as-sociated the term DEM to any computational mod-eling framework that (i) allows finite displacementsand rotations of discrete bodies, including completedetachment, (ii) recognizing new contacts automat-ically during the simulation. This widish definitionapplies to a large number of methods, methodologiesand procedures, whose consideration is beyond thescope of this article. Nevertheless an interested readercan find an overview of the DEM by BICANIC(18),D’ADDETTA(19), WITTEL(20) with applications ofDEM to diverse engineering problems, materials andmodeling scales.While the behavior of granular materials can besolely described by the computational modeling ofmulti-body contacts, materials with cohesive behav-ior call for an improved discrete element simulation

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scheme. For this purpose we introduce spring orbeam elements that bond particles together to forma spring/beam-particle model with dual structure ofthe spring/beam network and the particle conglomer-ation. Over a large part of the simulation the systembehavior is determined by the cohesive behavior ofthe spring/beam network, but it can develop towards agranulate by gradual failure of spring/beam elements.The model is presented by first addressing cohesionelements and particles in contact before their organi-zation to a system is discussed, followed by the gen-eral simulation scheme.

Discrete Elements

Since DEM stands for a rather large class of mod-els, we restrict this article to elements and models,that are relevant and broadly used in the context dam-age simulation of composite materials. The interac-tion on the microscopic level either from cohesive el-ements or particles in contact, determine the elasticbehavior of the solid. By the right choice of elementsand efficient formulations, system sizes can be in-creased enormously to gain higher explanatory power.We first address spring elements for central force net-works, that have a long tradition in representing con-tinua, dating back to WIEGHARDT(21). For manypurposes, central force networks are sufficient. Nev-ertheless, beam networks built for example out of thepresented TIMOSHENKO truss-beam elements give abetter representation of the local stress state. Espe-cially contact calculations consume enormous com-putational time, so only simple particle shapes likethe presented spherical or polygonal particles with an-alytically solvable contact laws should be used.

Central Force To bond particle assemblies to-gether, it is necessary to introduce cohesive forcesbetween neighboring elements, in the simplest caseof central force networks via spring elements. Thelongitudinal force acting at site i on the center of theparticle i can be calculated with the normal flexibilityai j = li j/(EAi j) as

F ix =

1ai j (u

jx −ui

x). (1)

E is the YOUNGS’s modulus of the central force ele-ment of length li j, Ai j its cross section and ui

x,ujx the

elongation components of the displacement vector inlocal ∆x−∆y coordinates of the element.

Figure 1. Beam deformation.

Beam Bending To increase the quality of represen-tation of continua, additional degrees of freedom needincluded. This can be done either by using a largernumber of central force elements or by using higherorder elements like beam-truss elements. Using theshear and bending flexibilities bi j = li j/(GAi j) and

ci j = li j3/(EIi j), the acting shear force at site i fol-

lows as

F iy = βi j(u j

y −uiy)−

βi jli j

2(θi +θ j), (2)

and the flexural torque

Miz =

βi jli j

2(u j

y −uiy + li jθ j)+δi jli j2(θ j −θi), (3)

with αi j = 1/ai j, βi j = 1/(βi j + 1/12ci j) and δi j =βi j(bi j/ci j + 1/3). Here G is the shear modulus ofthe beam, Ii j its Moment of inertia for flexion, θi,θ j

the bending angle and bi j is chosen as bi j = 2ai j,leading to a POISSON’s ratio of ν = 0 for the beams(see Fig.1). This formulation resembles the TIMO-SHENKO beam theory, leading to a simplified equa-tions of micro-polar continuum elasticity (7). The useof the TIMOSHENKO beam theory is justified, sincethe elements are short and thick, therefore shear de-formations have to be taken into account.

Particles in Contact Particles are considered tobe rigid, undeformable, unbreakable bodies that canoverlap when they are in contact. This representsto some extent the local deformation of particles.The simplest, analytically describable particles arespheres (see Fig.2(a)), and the HERTZ contact lawprovides the force fc

n between two particles with theYOUNG’s, shear modulus Ei,E j, Gi,G j and POIS-SON’s ratio νi,ν j as a function of the overlap dis-tance δ = ri j − (r1 + r j) and the constant f1 depend-ing on the elastic and geometric properties of the

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spheres (22; 23). For two spheres with radii ri,r j (seeFig.2(a))

fcn = f1δ

32 ni j, with f1 =

3E∗

4

(

1ri

+1r j

)− 12

,

1/E∗ = (1−ν2i )/Ei +(1−ν2

j)/E j (4)

with the contact modulus E∗. Other particle shapesare computationally much more costly and it is notsurprising, that a large amount of literature aboutDEM is concerned with mandatory approximationmethods for contact force calculations. Overlapping

Figure 2. Contact of spheres (a) and polygons (b)with overlapping volume/area A in dark grey with theglobal x-y coordinate system.

polygonal particles i, j (see Fig.2(b)) have two inter-section points P1,P2 defining the contact line (withtangential and normal unit vector t and n), that is usedfor decomposing the total contact force fc into normaland tangential force Fn Ft , i.e. fc

i j = Fni j · n + F t

i j · t.The magnitude of Fn and F t for overlapping polygonswith E = Ei = E j, including damping and friction ac-cording to COULOMB’s law with parameter µ can be

written as

Fni j = −EA

Lc−mi j

e f f · γn · vnrel , (5)

Fti j = min(−mi j

e f f γt |vtrel |,µ|Fn

i j|). (6)

Therefore the repulsive normal force Fn contains anelastic and a damping term (with the damping coef-ficient γn, the relative velocity vrel = v j − vi and theeffective mass mi j

e f f = mi ·m j/(mi +m j)), while F t isresponsible for the friction described via µ and a vis-cous damping coefficient γt . The elastic part of Eq.5is proportional to the overlapping area A divided by acharacteristic length Lc of the interacting pair of poly-gons. We define Lc by 1/Lc = 1/2 (1/ri +1/r j) withri,r j being radii of circles with identical area as theircorresponding polygons.

Local vs. Non-Local Interaction In the DEMframework, short range interactions are representedrather well, due to local neighbor-neighbor interac-tions, however leaving to problematic representationsof long range interactions. One way around this prob-lem is the introduction of rougher length scales to dealwith long wave interactions similar to time-space fi-nite element methods (24). In this work non-local in-teractions are included only for the case of HPFRCC,where additional elements with a larger characteristiclength than the system length have a micro structuralcounterpart in randomly distributed steel fibers (seeFig.18).

Model Construction - Element Organization

The construction of an initial system configurationis a crucial point in representing specific continuumproperties and realistic failure behavior as well. Thisautomatically raises the question of the role of dis-order in the system. One can distinguish betweenquenched and annealed disorder. In contrast to an-nealed disorder, the first case of disorder is chosenand fixed once and for all before the simulation starts.In regular arrangements, disorder can be introducedin several ways, either by the elastic constants, inthe threshold values or in the presence (dilution) ofa bond. Regular lattices can be square, triangularor hexagonal. One needs to be aware, that POIS-SON’s ratio for square lattices are anisotropic andν = 0 for load in lattice directions, while for trian-gular lattices ν = 0.33 is isotropic. For square lat-tices a large anisotropy is observed, while triangu-lar lattices are capable to describe isotropic contin-uum properties very well. Unfortunately this is not

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true for fracture, where cracks are observed to growpreferably in the lattice orientations. To overcomethis problem, disorder needs to be introduced not onlyin the presence, but in the position of particle centers.For this purpose a so called vectorizable random lat-tice, proposed by MOUKARZEL&HERRMANN(25) isused, which is a VORONOI construction with slightlyreduced disorder. Starting from a regular square orhexagonal lattice with the characteristic lattice spac-ing Lc, points are randomly and independently thrownon squares or hexagons of the side length a with0 ≤ a ≤ Lc, centered on the lattice sites. The advan-tage of this construction is, that the connectivity isdetermined by the underlying lattice, and the random-ness of the tessellation can be controlled by the valueof the continuous parameter a (see Fig.3). With the

Figure 3. Model construction on a hexagonal lattice.

new positions and the connectivity, the correspondingVORONOI polygons are determined. Next the widthhi j and cross section Ai j of the element with length li j,connecting polygons i and j needs to be calculated,based on the represented area F i j

v with the relationshipF i j

v = li j · hi j = ai j, entailing disorder in the elementmoduli. The effect of a on elastic and strength prop-erties is negligible up to a ≤ 0.5, but the anisotropy incrack directions is considerably reduced.To complete the discussion on disorder, we need todescribe the distribution function that is mainly usedfor breaking thresholds, namely the WEIBULL distri-bution

P(x) = 1− e−( xτ )m

for x > 0. (7)

It uses the scale and shape parameter τ and m, and byvarying only one parameter m, failure probabilities ofdifferent materials can realistically be described. TheExtreme Value Theory suggests, that the WEIBULLdistribution will successfully represent failure formechanisms for which many competing similarfailure processes due to pre-existing voids or microcracks on the sub-model scale are independently”racing” to failure and the first to reach it (i.e., theminimum of a large collection of roughly comparablerandom failures) produces the relevant failure. This

model assumption is often named weakest linkapproximation (11) and is only valid under elonga-tion. Realistic values for engineering materials are2 ≤ m ≤ 10 with low values for highly disorderedand high values of m for brittle materials with lowdisorder.

Simulation Scheme

Similar to molecular dynamical simulations (26), thetime evolution of the system is followed by simulta-neously solving the NEWTON’s equations of motionfor all particles N for the translational and rotationaldegrees of freedom, namely

mixi = ∑Nj=1 Fi j (8)

IipΘi = ∑N

j=1 Mi jz , (9)

with the moment of inertia Iip of polygon i with

respect to its center of mass. The force Fi j con-tains all forces from particle contacts (Eqs.5), ele-ment elongation (Eq.1) and bending (Eq.2) as wellas possible damping or volume forces like gravity,while the torque Mi j

z contains the flexural torque fromthe beams (Eq.3) and particle contacts (Eqs.5) re-spectively. A fifth order GEAR Predictor-Correctorscheme (26) is used to numerically solve the differ-ential equations for the time increment ∆t (Eqs.8).Adaptive time steps can be realized, using the sum ofthe correction terms of the corrector step. Thereforewe use an implicit time stepping integration schemeoutlined in Tab.1 and have all the tools at hand andcan address to the question of representing and sim-ulating failure in composite materials. These simu-lations can be considered as numerical experiments,and like in real world experiments, one needs to aver-age over many realizations of disorder to obtain quan-titative statements.

DEM FOR FAILURE IN COMPOSITES

Discrete Element Models for composite materialsshould be structured in accordance with physical andgeometrical properties of the abstracted medium. Forunderstanding the relevance of this requirement, weneed to discuss how breaking is represented in themodel. Basically, the details of the microscopicphysics of rupture are contained in the breaking rule.A breaking rule that reflects the various rupture modes

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I model generation(1) particles (positions/orientations)(2) bondings (element properties)

II predictor step(1) relative velocities(2) relative positions(3) relative orientations(4) boundary conditions

III force calculation(1) from bondings(2) from contacts(3) gravitational forces(4) damping forces(5) sum of all forces and moments(6) forces on boundaries

IV corrector(1) sum of all corrections (adaptive ∆t)(2) corrected state variables(3) relative displacements(4) relative orientations

V fracture criterionVI updates

(1) element orientations(2) neighbors

Table 1. General DE simulation scheme.

as sum of corresponding terms, inspired by the VON-MISES yielding criterion is

pi j =

(

εi j

εmax

)2

+max(|θi|, |θ j|)

θmax≥ 1 (10)

with the longitudinal strain εi j ≥ 0 of the elementand if beam elements are used, the rotation angle ofthe beam ends θ j ,θ j with respective threshold valuesεmax,θmax that can be WEIBULL distributed (Eq.7).Discreteness implies that stresses in an element areequal to the mean of all local stresses in the vol-ume, represented by this element. Therefore it is notpossible to consider cases, where stresses are redis-tributed from failed to intact regions, smaller than thecharacteristic model length. Generally spoken, allthose small defects below the model scale can onlybe treated statistically, but when they join, grow andreach the model scale, there needs to be a mechanismthat stops them - determining the characteristic lengthscale. Such mechanisms are for example crack deflec-tions out of the preferred growth direction at compo-nent interfaces or regions with increased energy dis-sipation etc.. As result, the stiffness of one elementis reduced, leading to elastic stress redistribution toneighboring elements. If they now fulfill the breakingcondition on their part, we observe crack growth.Since models need to be adopted to the structural pe-

culiarities of specific composite materials and load-ing situations, diverse models with varying complex-ity can be employed. In this article we selected threeexemplary situations in fiber reinforced compositesfrom the first part of this book, but with increasingintricacy from the viewpoint of simulation, that canbe classified following Tab.2.

Problem Inter- Element Contact

action

Transverse ply l central HERTZ

cracking force (cf)

Anisotropic de- l bending/cf –

gradation of wood

HPFRCC l/n-l bending/cf Polygon

Table 2. Classification of simulation examples (l de-notes local, n-l non-local particle interaction).

Transverse Ply Cracking

The first macroscopically visible form of damagein long fiber reinforced composites is intra-laminarcracking of transverse plies. Due to the stronglyorthotropic properties of uni directional plies, trans-verse strength perpendicular to fiber direction is ratherlow. Due to cracking of transverse plies, effectiveYOUNG’s moduli, POISSON’s numbers and thermalexpansion coefficients change rather fast. The immi-nent danger arises from the mechanism of micro de-laminations and fiber failure that are activated oncethe transverse ply crack reaches the ply interface.Through the network of cracks, corrosive agents canpenetrate and further reduce stiffness and strength val-ues. Transverse ply cracking is studied rather welland numerous models, with diverse significance wereproposed, most of whom represent the ply in a ho-mogenized fashion (27; 28; 29). However, the dam-age process involves damage on several length scaleswith intrinsic disorder (see Fig.4).Disorder leads to randomly distributed micro cracks,

that are first experimentally observed to form in thevicinity of fiber concentrations, resin rich zones orvoids and develop along single fiber matrix interfaces,leading to debonding. Transverse cracks form, as mi-cro cracks and fiber-matrix debondings coalesce, ex-hibiting an extended fracture process zones in the or-der of the system size (see. Fig.5). Unfortunately themain cracks shield most small cracks, that close again

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Figure 4. Length scales involved in the damage pro-cess of uni directional [(0,90)2]s glass fiber rein-forced laminate.

and fall out out the resolution of the utilized LaserScanning Microscope (LSM). However, when cracksreach the ply interface, they are deflected, leadingto micro delaminations and isolated fiber fractures.Stresses are transfered to the transverse ply by shear

Figure 5. In-situ LSM growth observations of a trans-verse ply crack with micro cracks at the crack tipin a [02/902]s laminate (a1 − 4) and magnificationwith arrows pointing at micro delamination and crackbranching (b).

at the ply interface. Due to micro delaminations thestress transfer is intermitted, leading to a character-istic damage state where no more transverse crackscan form, since the transfered stress simply doesn’treach the strength value of the ply any more. Mod-els for transverse ply cracking therefore need to be

able to deal with multiple cracking of many interact-ing cracks.Discrete element models can naturally deal with thecomplicated crack-crack and mechanism interaction.Additionally dynamical aspects of the crack forma-tion are considered giving direct access to physicalinterpretations at any time. The possibility to studythe influence of characteristic properties like systemsize, strength and/or disorder points out the poten-tial of such studies. The composite behavior is deter-mined by the interaction of constituents, calling fora direct representation. The transverse ply is repre-sented by a two dimensional model which is justifiedby the fact, that transverse cracks grow instable infiber direction, leading to translational invariance ofthe system in this direction. Circular disks of identicaldiameter 2r f represent fibers, oriented perpendicularto the modeling plane (see Fig.6). They are placedon an equilateral triangular lattice and connected byspring elements (Eq.1). The fiber radius r f is relatedto the fiber volume fraction v f by

r f = s

√

v f√

3√

2πwith 0 ≤ v f ≤ v fmax = 0.906.

(11)Using a regular lattice is a clear neglect of the topo-

Figure 6. Micro structure of the transverse ply model.

logical disorder, characteristic for real fiber packings.However, it is assumed that all the relevant disorder inthe material is cumulated in the distribution of break-ing thresholds controlled by one disorder parameterm (Eq.7). Fibers can contact after fracture of the co-hesive elements using a two dimensional form of theHERTZ contact law (Eq.4) allowing only for local in-teraction.Load is introduced in the system by carefully stretch-ing rails, representing the sub-laminates, attached tothe upper and lower system border by interface ele-ments, whose strength is ai times the strength of thebulk springs. Therfore internal stresses can be re-leased either by the formation of cracks inside the

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system or by the fracture of interface bonds. Notethat segmentation cracks are identified as clusters ofbroken elements, spanning the total width of the spec-imen. A detailed model description can be found in(15) and for the special case of thermal loading anddegradation in (16). With the model, we can simulatethe degradation process of the transverse ply startingfrom microscopic fiber-matrix debonding to segmentcrack formation, micro delaminations up to the satu-rated crack state.In the simulations we first observe randomly dis-tributed cracking, depending on the disorder of thesystem from very early stages on (see Fig.9). Withfurther straining ε we observe a fast increase in frac-ture activity and the first segmentation cracks formin uncorrelated fashion, meaning that the average dis-tance of segmentation cracks < L > is larger that thestress recovery length, so all cracks form basicallyindependently under similar stress conditions. Thescaling of the stress field in between segmentationcracks can clearly be seen in Fig.9 (c). From a cer-

Figure 7. Scaling of systems with model thicknessny = 10, ny = 20 and WEIBULL modulus m = 4 com-pared to predictions with Eqs.12.

tain crack density on, this is not possible any more,and cracks form somewhere in between existing seg-mentation cracks. The two regimes of correlated anduncorrelated segment formation can be described bythe scaling laws

< L >∝ ε−α and < L >∝ ε−α/(α+2) = ε−β, (12)

with the meso scopic shape parameter α of theWEIBULL distribution of the ply. This approach canalso be used for an inverse determination of the micro-scopic WEIBULL parameter, giving realistic values of

3 ≤ m ≤ 5 for glass fiber reinforced composites.Simulations give us the possibility to easily vary in-herent system properties like ai, m or the systemthickness ny for a fixed value of v f , to study the ef-fect on the crack evolution, micro structure of dam-age and degradation of elastic ply properties. First we

Figure 8. (a) Segment definition with micro delam-inations d1,d2 on a ply with half thickness t1, evo-lution of segment geometry (b) depending on the in-terface strength multiplier ai, (c) inherent disorder mand (d) system thickness ny showing size effects. Notethat all results are averaged over several realizationsand presented in dimensionless form with Nseg as to-tal number of segmentation cracks, and L meaning thetotal system size.

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fix all system properties like size or disorder and varyonly the strength multiplier, increasing the breakingthresholds to ai times the ones of the bulk material.As we can observe, the strain for first ply crackingis not effected as the ply interface is weakened (see.Fig.8(b)). Since the simulations were stopped whenthe system reaches a relaxed state, cases with smallai stop early due to complete delamination of thetransverse ply. When we get to values of ai around2−4, we observe, that segments remain larger, whichis mainly due to more pronounced micro delamina-tion zones. It is interesting to note that in the caseof ai → ∞, delaminations can not be suppressed, butform in the bulk material just below the interface el-ements. We now fix ai and modify the inherent dis-order m. It is not surprising that we observe a largeamount of randomly distributed micro cracks fromthe very beginning of the loading for low disordervalues. The first segmentation cracks also form atlower strains. The qualitative shape of the segmen-tation curves with the fast increase and saturation isthe same for all cases, but higher disorder (meaninglower values of m) lead to slightly larger segments(see. Fig.8(c)). Increasing the system thickness nyfor a realistic value of m = 4 leads to a rather pro-nounced size effect, not for the segmentation crackinitiation strain, but for the segment geometry aspectratios (see. Fig.8(d)). For thick plies, we find effectsof dynamic crack growth like crack branching, thatcan also be observed experimentally (comp. Fig.5 andFig.9(e)).

Material scientists probably find more interest inthese studies than applied engineers, since the modelis valid for a broad spectrum of materials and loadingsituations by modifying inherent parameters, that aremostly not really accessible from a practical point ofview. Nevertheless the model can also generate evo-lution laws for the damage mechanism of transversecracking, by reducing the damage state to one scalardamage parameter via homogenization. The damageparameter can be directly be used for example to en-rich phenomenological failure predictions (see Chap-ter ??). Using the elastic energy Epot stored in thesystem, the effective YOUNG’s modulus Ee f f can becalculated as Ee f f = 2Epotε−2 and leads to a damageparameter d = Ee f f /E0 by using the initial modulusE0. For low disorder we observe an abrupt loss ofstiffness at the critical strain, while increased disorderleads to a smooth transition of the damage parameter(see. Fig.10).To subsume, DEM give a realistic representation of

cracks in heterogeneous materials, accounting for theheterogeneity of the medium, dynamic crack growthand multiple cracking, leading for example to sizeeffects. To describe transverse ply cracking central

Figure 9. (a) LSM picture and (b)− ( f ) snap shots ofthe micro structure of damage in models of size nx =800 in length and (b) ny = 50 (c)− ( f ) ny = 10 ele-ments in thickness direction. ( f ) disperse distributesmicro cracks with the first segmentation crack; (e)quasi periodic crack patter with (d) correspondingstress distribution and (c) saturation state. (b) sat-uration state for a five times thicker model.

force networks proved to be sufficient due to the factthat it is mainly tension failure with negligible fric-tional effects. Nevertheless more accurate resultscould be obtained by beam lattices like the one usedin the next study of failure in cellular solids.

Anisotropic Degradation of Wood

Many natural and artificial materials exhibit a cellu-lar micro structure, determining their macroscopic be-

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Figure 10. Stiffness loss of transverse ply due to fail-ure.

havior. Nature is limited in its material choice to bio-logical processes, which is compensated by an adap-tation of structure on all length scales (see Fig.11).The damage evolution in cellular materials is closelyrelated to the material-structure composition of ele-ments like ligaments, cell walls, fibers that bear allelastic interactions inside the material. DE modelsgive the possibility to directly represent the cellu-lar material structure and therefore to get around theproblem of formulation complicated fabric tensors.We limit ourselves to the study of failure in sprucewood in the radial-tangential (RT) plane. The damageevolution and basic failure mechanisms were charac-terized using in-situ observations for various speci-men orientations and we refer to Chapter ?? for a de-tailed description.The model micro structure is determined by the den-

sity distribution of a single year ring and by elasticand geometric properties of cell walls, which are crosssections of wood fibers (17). To represent macro-scopic properties of wood realistically, a hexagonalmicro structure with variable cell geometry and wallthickness close to the observed micro structure ofwood is necessary. First, the relative density ρ∗/ρc ismeasured by evaluating the distribution of white pixelalong a representative year ring divided by the numberof pixels which can be approximated with the analyticfunction

ρ∗ = ρmin(1+2 ·ax′′br · ecx

′′r ) (13)

with a = 1,85, b = 0,2, c =−6 and ρmin = 370kg/m3

(see Fig.12). Density variations in wood mainly aredue to variable fiber geometry and cell wall thickness.Cells show variable geometry lR only in the radial di-rection, hence lT can be considered as constant suffi-

Figure 11. Length scales involved in the damage pro-cess of spruce wood.

ciently far form the tree axis. The fiber geometry ra-tio lR/lT was measured using a LSM along one yearring (see Fig.12) and finally the beam width wi j of thehexagonal cell can be estimated as

wi j(ρ∗) = A · (h+2l)−

1A·√

(h+2l)2− cosθ(2h+ l sinθ)2lρ∗

Aρc(14)

with A = 1+ sinθ+ cosθ (see. Fig.12.After assigning the micro structure to spring-beam el-ements of variable width, we need to modify the frac-ture criterion to meet the details of failure in wood,namely the two generic failure mechanisms peelingand cell wall fracture. Kinking of cell walls is ex-cluded, since specimens were loaded mainly undertension. Peeling is activated, when the force differ-ence in opposite directions in a node exceeds a thresh-old, calculated from the critical stress p1 and the af-fected area A (see Fig.13). This mechanism only ef-fects the beams oriented in tangential direction andp1 does not depend on the position inside the year

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Figure 12. Density distribution and micro structuremodel parameters. Insets show fiber geometries forlate-, transmission- and early-wood, a microscopicwood section and the corresponding model represen-tation for one year ring.

ring, since the properties of the middle lamella, thatglues fibers together, is the same throughout the sam-ple. In contrast, cell wall fracture depends stronglyon the wall thickness. A fracture criterion for tensionfailure and for bending failure with parameter p2 canbe used, where either equation needs to be fulfilledto delete the element stiffness. Note that the mass isconcentrated in the nodes without physical extension,so no contact forces need to be considered.First, a rectangular section is loaded by moving the

vertical boundaries in opposite directions with sup-pressed failure, to measure the constitutive systembehavior. The moduli ratio ET /ER are measured asET /ER = 0.65 which is in good agreement to exper-imental values (30). In the next numerical experi-ment, single loading pins located at a distance fromthe specimen are connected via rigid elements to thevertical boundaries. The system is only loaded bymoving the loading pins in opposite direction, allow-ing for rotations of the specimen, as cracks grow. Thissetup is especially chosen for comparison with ex-perimental observations (compare Chapter ??). Asdemonstrated in Fig.14, already small systems gener-ate the mesoscopically observed damage characteris-tics like peeling (TR) with smooth crack surfaces, cellwall fracture (RT) with stepped surfaces and the com-plex crack path of the intermitted specimen (RT45). A

Figure 13. Generic failure mechanisms of sprucewood, their respective model and breaking law.

detailed description and more simulation results canbe found in (17).To subsume, DEM is a top candidate for calculatingthe constitutive behavior of cellular solids, includingactivation of dynamically interacting failure mecha-nisms. Due to the open access to micro structureand rheological elements, extensions to many kindsof problems (with short range interaction) are possi-ble in a straightforward way giving direct insight intodamage mechanisms for material design.

Fracture in High Performance Fiber ReinforcedCement Composites (HPFRCC)

The simulation of damage and failure in short fiberreinforced composites (comp. Chapter ??) likehigh performance fiber reinforced cement compos-ites (HPFRCC) is still a challenging task, due to therichness of failure mechanisms introduced by fiberson the meso-scale of the material (comp. Chap-ter ??. Randomly oriented, hooked steel fibers, likethe BAEKAERT Dramix fibers sustainable modify themacroscopic failure behavior of the composite likeincreased compressive, tensile, flexural and shearstrength. Smaller crack opening displacements withless water permeability and wear, an increased frac-ture toughness and higher impact resistance due toadditional, multiple energy dissipations are observed.On the meso-sale fibers introduce a multitudes of ad-ditional mechanisms like fiber fracture, fiber bridging,fiber bending and matrix spalling for inclined fibersand finally fiber pull-out (see Fig.16), just to name afew (31).

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Figure 14. Simulation snapshots (a) RT, (b) T R and(c) RT 45◦ compared to damage situations from in-situ LSM pictures.

The damage simulation for these materials is facedwith two serious problems:

Figure 15. Scales in HPFRCC ranging from macro tomicro.

P1 The representation of concrete as cohesive, fric-tional material with micro structure, and

P2 the representation of different phases of fiberpull-out (comp. Chapter??) for 3D randomlyoriented fibers.

Problem P1 has a long tradition in DE-Modelingstarting in the 70s with the pioneering works ofCUNDALL(9; 10) up to the advanced models of VAN-MIER (32) and KUN& HERRMANN and D’ADDETTA(19; 33). This work is based on the two dimen-sional model of KUN&HERRMANN that proved tosuccessfully describe fracture and fragmentation forconcrete (19). The model construction is describedabove (Fig.3), only originating from a square lattice

13

Figure 16. Fiber bridging of crack surfaces withoff-axis pull-out leading to fiber bending and matrixspalling.

with lattice spacing Lc and the disorder parametera, leading to polygonal particles with the above de-scribed contact formulations (Eqs.5). Spring-beamelements (Eqs.1,2,3) are used to connect particle cen-ters of neighboring particles with the possibility tofail (Eq.10). For a detailed study on the physical be-havior of the model, we refer to the (33). Neverthe-less it should be mentioned, that particles do not di-rectly correlate to grains of concrete. For good con-crete grains of all sizes are required to achieve a densepacking of strong aggregate bonded by as few weakcement paste as possible, which is not geometricallyrepresented by the model. To represent fiber rein-forced concrete, the model is extended by randomlythrowing additional fibers (Eq.1) with the projectedfiber length L2D

f = cos(Θ) ·L f > Lc (for fibers inclinedwith respect to the modeling plane) on the disorderedtwo-dimensional system. It is important to note, thatthis does not represent a superimposed, central forcedelaunay network, since fibers are basically indepen-dent and interact through the elastic spring-beam-particle model. However if the fiber volume fractionv f increases strongly, an interconnected central forcenetwork forms out. This effect corresponds to zoneswith nested fibers, that still remain a technologicalproblem for the broad application of HPFRCC, lead-ing to a prescribed limitation to v f > 1.5% or 80kg/m3

in terms of fiber mass fraction.(34). Depending onthe extensions of fibers and aggregate, fibers can beconcentrated in regions with smaller aggregate size.We focus on the situation, where fibers are larger thangrains and a homogeneous distribution of fibers canbe obtained.To approach problem P2, the complicated fiber-matrix interaction is reduced to a force vector acting

in opposite direction at the sites of the initial fiberends, that first are moved and bonded to the nearestparticle centers before their real initial length is calcu-lated. Hence, problem P2 calls for additional rheolog-ical elements to adopt the spring behavior to representthe experimentally determined single fiber pull-outbehavior (comp. Chapter??). In experiments, threepull-out phases could be identified, that are shortlyrecalled. Without cracks in the cement matrix, theinfluence of fibers is negligible, however their grandis marked by macroscopic cracks, bridged by fibers.For simplification purposes, we define a linear elasticphase A for fiber/matrix debonding, phase B for defor-mation and pull-out of the hook and the purely fric-tional phase C, when the straightened fiber is pulledout, following simple shear-lag assumptions with theinterfacial shear strength τi (see Fig.17). For simpli-fication purposes we assume, that cracks always formin the middle of a fiber.

Figure 17. Pull-out of one fiber measured with themodel with thee phases and force components for in-clined pull-out.

A This phase is characterized by the fiber stiffness,the maximum pull-out load F1.

B Fibers that bridge a crack will always be pulledout of the weaker portion of the material (seeFig.16 right side). The phase is determined bythe straightening of the hook and therefore bythe corresponding pull-out length d f (see Fig.16.Also the stiffness of the fiber is decreased lin-early with the pull-out length.

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C In the purely frictional phase, the fiber stiffnessis kept constant. As we assume simple shear lagover the whole fiber length due to weak inter-faces, the reaction force is reduced linearly withthe pull-out length.

Off-axis pull-out under the inclination angle Θ isconsidered by transforming the forces in the initialcoordinate system of the fiber and applying the samepull-out law as for aligned fibers, but only with theforce component Fx‘ acting in fiber direction (seeFig.17inset). This rather crude estimate is in goodagreement with experimental values for the maximumpull-out load (35). Therefore, the same pull-out curve(see Fig.17), defined by the strain where debondingstarts (d1/Lc), the force reduction factor F2/F1 andthe pull-out length defined by the portion of the hookd f /Lc, can be utilized for all inclinations. Note that

Figure 18. Micro structure of the model for short fiberreinforced composites with various fiber volume frac-tion.

the pull-out law can be randomizes with probabilitydistributions like Eq.7. Additionally a fiber failurecriterion or a cut off value for sudden pull-out caneasily be considered. This introduces an additionalsource for energy dissipation, representing frictionfrom fiber pull-out. Now that the model is set, welook at two typical loading situations for structuralelements made of HPFRCC, namely the failureof a truss in tension and a notched beam in threepoint bending, that was also discussed in Chapter ??.

Typical model properties:Ep Eb E f γN γT

10MPa 40MPa 210MPa 105 0

ν ρ ∆t εbmax θb

max0 5g/cm3 10−6s 0.03 3◦

a Lc L f0.8 1 8

Tension failure Under uni-axial tension, we canobserve different failure situations, depending for ex-ample on the fiber content or fiber matrix interfaceproperties. For an extremely week fiber matrix inter-face, we observe a macroscopic crack, spanning thesystem, followed by pullout of bridging fibers, with-out additional failure activity. The other extreme ofa very stiff fiber-matrix interface would lead to com-plete decohesion of elements and a system of particlesinterconnected by fibers and fiber failure. We choseintermediate properties for the simulations shown inFig.19. For low fiber contents, we observe sequen-tial crack branching. Already small fiber contentsprohibit branching and the larger v f gets, the widerthe crack zone becomes. For high values of v f weobserve stress recovery and the formation of parallelcracks. For increasing fiber content, we observe stiff-ening and strengthening and additional energy dissi-pation. It is interesting to note, that the failure straindecreases with increasing fiber content, but for highvalues of v f , the macroscopic failure occurs in sev-eral steps.

Failure under bending Many studies on failure inHPFRCC are made on three point bending setups, al-lowing for a rather moderate crack propagation (36).Normally the load-deflection of the upper support ismeasured and correlated to the observed damage. Onecan distinguish between an elastic stage, the crack-ing and the failure stage. It is visible in Fig.22 thatthe stiffness of the material can severely be increasedby fibers. Due to fibers the strength also increasesstrongly and for high fiber contents, the system stillcan take strong loads and dissipate additionally en-ergy, even though large crack systems are present inthe system (see Fig.21).

To subsume, failure in short fiber composites likeHPFRCC can be modeled using models like the de-scribed beam lattice / particle model with superim-posed elements to represent the behavior of pulled outfibers. In this article, the effect of the fiber contenton the constitutive behavior and damage morphologywas demonstrated for an arbitrary material combina-

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Figure 19. Tensile system failure. Left the com-plete system for v f =0.001 is shown with severe crackbranching. On the right, the crack zones for differentfiber contents (v f = 0.2, 0.4, 0.81 from top to bottom)are shown.

tion. Due to the possibility to use force curves fromsingle fiber pullout, the model can be used for manydifferent material combinations to study the effect offiber properties their amount and embedment on thefailure behavior of the system.

CONCLUSION AND OUTLOOK

We discussed models for various composite materialsand demonstrated their capabilites to describe damage

Figure 20. Constitutive behavior of systems with var-ious fiber contents.

and multiple failure. The representation of a mate-rial region by a set of particles and cohesive elementswith short range interactions allows for the simulationof many simultaneously initiated, growing and inter-acting crack systems. This was for example demon-strated for transverse ply cracking in long fiber rein-forced composites using a central force network. Dueto the natural representation of material micro struc-ture inside the models, effects that originate in theinterplay of geometry and physics are captured in astraight forward way. This is of course true for all ex-amples, but wood with its complicated cellular microstructure is an excellent material for demonstratingthis potential of DEM using a spring-beam network.Since the time evolution of the system in followed,additionally cracking phenomena that originate in thedynamic nature of every fracture process can be ob-served and represented easily. Nevertheless DE mod-els are not limited to short range interactions and forthe example of HPFRCC non-local interactions wereimplemented via a superimposed central force ele-ment grid on a spring-beam model with polygonalparticles. This last example points out the easy ac-cessibility of DE models for enrichment, modificationand specialization towards many different problemswhile they maintain the possibility to realistically de-scribe the complex failure behavior of composite ma-terials.Today particle models and Discrete Element Modelsin special are on the verge of becoming widely ap-plicable engineering tools for the material design andfirst commercializations are in progress (37). Espe-

16

Figure 21. HPFRCC under tree point bending bend-ing (left). The dependence of the fiber content onthe damage morphology is demonstrated for differentfiber contents (v f =0.001, 0.2, 0.4, 0.81 from top tobottom) for identical displacement.

cially when advances in structural optimization is sat-urated, adaptation of materials to meet special localmaterial requirements is an elegant solution. On onehand this leads to mass reductions but material op-timization requires on the other a good understand-ing of the failure evolutions and tools that enable en-gineers to gain access to failure processes inside thematerial. As computers power increases, system sizesincrease too, leading to refined results, longer simula-tion times, better discretization, statistics etc. in otherwords more meaning full results for the description

Figure 22. Force-displacement in arbitrary units,showing stiffening and strengthening effects of theadded fibers.

of failure mechanisms in composites. In this book wedare to make a small step in this direction and includeone DE model, namely the transverse ply crackingmodel in a Finite Element framework as part of a pro-totypical implementation of a so-called Micro Mech-anism Toolbox, described in detail in Chapter ??.

Acknowledgments The authors are indebtedfor additional financial support of NATO grantPST.CLG.977311. F. Kun is grateful for financialsupport of the Humboldt Foundation and OTKAT049209,M041537 and by the Gy. Bekesi Founda-tion of HAS.

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