European Journal of Mechanics A/Solidsbingweb.binghamton.edu/~cjin/publications/eu.pdf · 2016. 4. 8. · (Cusatis et al., 2011b). LDPM is an improvement over the conﬁnement-shear

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European Journal of Mechanics A/Solids 57 (2016) 85e107

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European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

Lattice discrete particle modeling of fiber reinforced concrete:Experiments and simulations

Congrui Jin a, Nicola Buratti b, Marco Stacchini b, Marco Savoia b, Gianluca Cusatis a, *

a Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USAb DICAM-Structural Engineering, University of Bologna, Bologna, Italy

a r t i c l e i n f o

Article history:Received 31 August 2014Accepted 4 December 2015Available online 17 December 2015

Keywords:Fiber reinforced concreteExperimentsDiscrete models

* Corresponding author.E-mail address: [email protected] (G. C

http://dx.doi.org/10.1016/j.euromechsol.2015.12.0020997-7538/© 2015 Elsevier Masson SAS. All rights re

a b s t r a c t

Naturally accounting for material heterogeneity, the Lattice Discrete Particle Model (LDPM) is a meso-scale model developed recently to simulate the meso-structure of quasi-brittle materials by a three-dimensional (3D) assemblage of polyhedral particles. A meso-scale constitutive law governs the inter-action between adjacent particles and simulates various features of the meso-scale response, includingcohesive fracturing, strain softening in tension, strain hardening in compression and material compac-tion due to pore collapse. LDPM has been extensively calibrated/validated, showing superior capabilitiesin predicting qualitative and quantitative behavior of concrete. As a natural extension for this discretemodel to include the effect of dispersed fibers as discrete entities within the meso-structure, LDPM-Fincorporates this effect by modeling individual fibers, randomly placed within the volume according to agiven fiber volume fraction. In this investigation, the theoretical basis for LDPM-F is reviewed, and tocalibrate/validate the numerical model, an extensive experimental study has been conducted to inves-tigate the mechanical properties of various prismatic specimens containing different types (steel andsynthetic) and dosages of fibers. Excellent predictive capability of LDPM-F is demonstrated through arigorous calibration/validation procedure.

© 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Concrete is by nature a brittle material that performs well incompression, but is considerably less effective when in tension, andthus reinforcement is often used to absorb these tensile forces sothat cracking is either eliminated, e.g., in the case of prestressedmembers, or, at least, reduced. For many years, steel in the form ofbars or mesh has been used as reinforcement for a large variety ofstructural elements. In recent years, new forms of reinforcement toimprove concrete toughness have been developed, such as shortrandomly distributed fibers of small diameters, made out of steel,plastic, natural materials, recycled reinforcements, etc. (Li, 2003).Fiber reinforced concrete, characterized by a significant residualtensile strength in post-cracking regime and enhanced capacity toabsorb strain energy due to fiber bridging mechanisms across thecrack surfaces, shows a dramatic improvement in service proper-ties over plain concrete. ACI Committee 544 reported that theaddition of fibers in a concrete matrix improves some mechanical

usatis).

served.

properties of concrete, including impact strength and flexuraltoughness (ACI Committee 544, 1996). The most beneficial aspectsof the use of fibers in concrete structures include durability im-provements through reduction in crack width and permeability(Belletti et al., 2008; Lepech and Li, 2009); potential cost savingswhere fibers are substituted for labor-intensive steel-reinforcingbars (di Prisco et al., 2009); and the mitigation of deteriorationowing to shrinkage and the associated early age cracking. Fiberreinforced concrete has found many applications in tunnel linings,bridge decks, airport pavements, slabs on grounds, industrial floors,dams, pipes, fire protection coatings and marine structures. Inaddition, it can be also used for repair, rehabilitation, strengtheningand retrofitting of existing concrete structures (ACI Committee 544,1996).

Many numerical models have been developed to simulate thebehavior of fiber reinforced concrete at different scales. Most ap-proaches simulate the mechanical response of these composites bymeans of phenomenological macroscopic models combined withfracture mechanics techniques, such as damage, crack and micro-plane models (Fanella and Krajcinovic, 1985; Li and Li, 2000; Pengand Meyer, 2000; Caner et al., 2013; Ferreira, 2007; Han et al.,2003). Fanella1 and Krajcinovic developed a nonlinear analytical

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C. Jin et al. / European Journal of Mechanics A/Solids 57 (2016) 85e10786

model based on the principles of the continuum damagemechanicsfor the stress-strain behavior of fiber reinforced concrete subjectedto monotonic compressive and tensile loading (Fanella andKrajcinovic, 1985). This model was later modified by Li and Li tocharacterize the tensile stress-strain response of the fiber rein-forced concrete (Li and Li, 2000). Peng and Meyer proposed asimplified continuum damage mechanics model to describe theinelastic behavior of concrete reinforced with randomly distributedshort fibers, including anisotropic damage, unilateral effect,coupled volumetric and deviatoric inelastic deformation andcomplicated failure mechanisms (Peng and Meyer, 2000). Caneret al. proposed a M7f model for fiber reinforced concretes understatic and dynamic loads, which features many improvement overthe earlier versions of microplane model, including a more realisticdescription of the fiber pullout and breakage (Caner et al., 2013).Ferreira addresses the use of R-curves to study the fracturebehavior of high-strength concrete and steel-fiber-reinforced con-crete subjected to cracking in a three-point bending configuration,where the R-curves are modeled through an effective approachbased on the equations of linear-elastic fracture mechanics(Ferreira, 2007). Han et al. proposed a constitutive model based ontotal strain and applied it to simulate structural component tests,which captures fiber reinforced concrete's unique reversed cyclicloading behavior (Han et al., 2003).

However, experimental studies on fiber reinforced concretehave confirmed that the mechanisms responsible for the macro-scopic mechanical responses mainly involve phenomena that occurat the meso-structural level. The cement fracture is the mechanismthat triggers the failure of fiber reinforced concrete, but the sub-sequent mechanisms leading to complete failure is completelymodified by the relative contents of fibers in the composite, andmore importantly, by the bond characteristic at the fiber-matrixinterface and all the phenomena associated with this effect(Guerrero and Naaman, 2000). Consequently, many discreteframeworks, such as rigid particle and lattice models, have beendeveloped, taking explicitly into account the meso-scale phenom-ena (Bolander and Saito, 1997; Li et al., 2006; Schlangen and vanMier, 1992; Bolander and Sukumar, 2005; Bolander et al., 2008).A sequential multi-scale homogenization approach has been pro-posed by Kabele (2007), whose framework links analytical andcomputational models covering scales from micro-to macro-level.Recently, Oliver et al. (2012) proposed a two-scale approach inwhich the macroscopic model, at the structural level, takes intoaccount the meso-structural phenomenon associated with thefiber-matrix interface bond/slip process.

In the present work, as first proposed by Cusatis et al. (2010),Schauffert and Cusatis (2012) and Schauffert et al. (2012), fiberreinforced concrete is analyzed by a multi-scale model, calledLDPM-F, in which the fine-scale fiber-matrix interaction is solvedindependently and the overall response is analyzed in a 3D meso-scale framework based on the recently formulated lattice discreteparticle model (LDPM). LDPM is a realistic 3D model of concretemeso-structure developed by Cusatis et al. (2011a), which has beenextensively calibrated/validated under a wide range of quasi-staticand dynamic loading conditions, showing superior capabilities inpredicting qualitative and quantitative behavior of concrete(Cusatis et al., 2011b). LDPM is an improvement over theconfinement-shear lattice model (Cusatis et al., 2003a, 2003b),which, along with LDPM, has constitutive laws analogous to thoseof the microplane model (Ba�zant et al., 2000; Di Luzio and Cusatis,2013). As a natural extension for this discrete model to include theeffect of dispersed fibers as discrete entities within the meso-structure, LDPM-F incorporates this effect by modeling individualfibers, randomly placed within the framework according to a givenfiber volume fraction (Smith et al., 2014).

LDPM-F has many salient and unique features: (1) a realistic,three-dimensional modeling of concrete mesostructure, includinga discrete representation of individual fibers randomly distributedtherein; (2) a multiscale approach in which the effect of embeddedfibers on the structural response is based directly on the micro-mechanics of the fiber-matrix interaction; (3) the ability to simulatenot only tensile fracturing but also multiaxial compressive loading;and (4) true predictive capability demonstrated through a rigorouscalibration/validation procedure. In this investigation, the theo-retical basis for LDPM-F is reviewed, to calibrate/validate the nu-merical model, an extensive experimental study has beenconducted to investigate the mechanical properties of variousprismatic specimens containing different types (steel and syn-thetic) and dosages of fibers. The predictive capability of LDPM-F isdemonstrated through a rigorous calibration/validation procedure.

2. Review of LDPM-F

In this section, the theoretical basis for LDPM-F provided bySchauffert and Cusatis (2012) is reviewed, and details of the pre-viously established (Yang et al., 2008) micromechanical fiber pull-out model adopted for this study are presented.

2.1. Extension of LDPM to include fiber-reinforcing capability

LDPM simulates concrete mesostructure by considering onlycoarse aggregate. Particles with assumed spherical shape and inaccordance with typical mix designs and granulometric distribu-tions, are introduced randomly into the volume by a procedurewhich avoids particle overlapping and ensures that all particles arecontained within the volume of interest. A Delaunay tetrahedrali-zation of the particle centers, along with the nodes used to describethe external surface of the volume, is used to define the latticesystemwhich represents the mesostructure topology. A 3D domaintessellation, anchored to the Delaunay tetrahedralization, creates asystem of polyhedral cells. Each cell contains one aggregate particle(or a surface node), and adjacent cells interact through the trian-gular facets where they are in contact. Fig. 1 (a) shows two adjacentparticles along with their polyhedral cells and the associated tet-rahedron edge. In the LDPM formulation, the interface facets areinterpreted as potential crack surfaces. A detailed description of thetessellation procedure and LDPM idealization of concrete meso-structure can be found in Cusatis et al. (2011a, 2011b).

The extension to LDPM-F to include fiber-reinforcing capability isperformed by inserting individual fibers with randomly generatedpositions and orientations (Schauffert and Cusatis, 2012; Schauffertet al., 2012). The geometry of an individual fiber is characterized bydiameter df, length Lf, and curvature. In this study, fibers areassumed to be straight, and non-circular cross-sections are simu-lated through an equivalent diameter calculated as df¼ 2(Af/p)1/2,where Af is the fiber cross-sectional area. For a given fiber volumefraction Vf, the number of fibers Nf contained in a concrete volume Vis determined by Nf ¼

l4Vf V=ðpd2f Af Þ

m, where QxS indicates the

ceiling function. The fiber system is then overlapped with the cellsystem and all the facets intersected by fibers are detected. For eachintersection, the shorter and longer fiber lengths, denoted as Ls andLl, respectively, on each side of the facet, along with the orientationsof the fiber and the facet, characterized by unit vectors nf and n,respectively, are computed. In accordance with the current LDPMformulation, the facet depicted in Fig. 1(b) is the projection of theoriginal tessellation facet onto a plane orthogonal to the associatedtetrahedron edge. The projected facet is used in order to avoid non-symmetric behavior for the case of purely tangential movementbetween two particles and the consequent stress-locking effectwhich can occur (Cusatis et al., 2011a).

Fig. 1. LDPM simulation for concrete mesostructure: (a) Two adjacent particles; (b) Facet with intersecting fiber.

C. Jin et al. / European Journal of Mechanics A/Solids 57 (2016) 85e107 87

LDPM describes meso-structure deformation through theadoption of rigid body kinematics. For given displacements androtations of the particles associated with a given facet, the relativedisplacement EuCF at the centroid is used to define measures of

strain as εN ¼ nTEuCF=[, εM ¼ mTEuCF=[, and εL ¼ lTEuCF=[, where [is the tetrahedron edge length associated with the facet, andm andl are twomutually orthogonal unit vectors tangential to the facet. Itcan be demonstrated analytically that this measure of strainscorrespond to the projections onto the facets of the classical straintensor of continuum mechanics (Cusatis and Zhou, 2013). Thestrains are then used to compute facet stresses assc¼sNnþ sMmþ sLl according to the constitutive equationreviewed in the next section. When a facet strains beyond thetensile elastic limit, the meso-scale crack opening can be calculatedas w¼wNnþwMmþwLl, where wN ¼ [ðεN � sN=ENÞ,wM ¼ [ðεM � sM=ET Þ, and wL ¼ [ðεL � sL=ET Þ with EN, ET being theelastic normal and tangential LDPM stiffnesses. At this point fibersare engaged and equilibrium considerations at the facet level allowfor the reasonable assumption of a parallel coupling between thefibers and the surrounding concrete matrix. Total stresses on eachLDPM facet are then computed as s ¼ sc þ ð P

f2Ac

Pf Þ=Ac, where Ac is

the facet area, Pf is the crack-bridging force for each fiber crossingthe given LDPM facet. The following assumptions are made: (1)Pfz 0 for both εN< 0 andw¼ 0; and (2) the bridging force dependson the micromechanical crack-bridging mechanisms associatedwith an increasing meso-scale crack opening, Pf¼ Pf(w). The crack-bridging force Pf is calculated by analyzing separately the crack-bridging effect of one individual fiber on one single meso-scalecrack.

2.2. LDPM constitutive law

A vectorial constitutive law governing the behavior of the ma-terial is imposed at the centroid of each facet. The elastic behavior isdescribed by assuming that the normal and shear stresses are pro-portional to the corresponding strains: sN¼ ENεN; sM¼ ETεM;sL¼ ETεL; where EN¼ E0, ET¼ aE0, E0¼ effective normal modulus,and a¼ shear-normal coupling parameter. E0 and a are assumed tobe material properties that can be identified from results of exper-imental tests in the elastic regime. At themacroscopic level, concreteelastic behavior is statistically homogeneous and isotropic. As such,it can be modeled effectively through the classical theory of elas-ticity characterized by Young's modulus, E, and Poisson's ratio, n. Therelationship between the meso-scale LDPM parameters, a and E0,and the macroscopic elastic parameters, E and n, can be obtained byconsidering a limiting case in which an infinite number of facetssurrounds one aggregate piece. In this case, the LDPM formulationcorresponds to the kinematically constrained formulation of the

microplanemodel without deviatoric/volumetric split of the normalstrain component, and it can be written as:

E0 ¼ E1� 2n

⇔E ¼ 2þ 3a4þ a

E0 (1)

and

a ¼ 1� 4n1þ n

⇔n ¼ 1� a

4þ a(2)

For stresses and strains beyond the elastic limit, LDPM meso-scale failure is characterized by three mechanisms as describedbelow.

The first one is fracture and cohesion due to tension andtension-shear. For tensile loading (εN> 0), the fracturing behavior isformulated through an effective strain and stress:

ε ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2N þ a

�ε2M þ ε

2L

�q; s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2N þ ðsM þ sLÞ2

.a

r(3)

which define the normal and shear stresses as

sN ¼ εNðs=εÞ; sM ¼ aεMðs=εÞ; sL ¼ aεLðs=εÞ (4)

An internal variable u characterizes the coupling between

normal strain εN and total shear strain εT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2M þ ε

2L

q, or the normal

stress sN and total shear stress sT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2M þ s2L

q, as:

tanðuÞ ¼ εN� ffiffiffi

ap

εT ¼ sNffiffiffia

p �sT (5)

The effective stress s is incrementally elastic ( _s ¼ E0 _ε), and mustbe less than or equal to the strain-dependent boundary:

sbt ¼ s0ðuÞexp ½ � H0ðuÞhε� ε0ðuÞi=s0ðuÞ� (6)

where ⟨x⟩ ¼ maxfx;0g. For pure tensile stress (u ¼ p/2), the tensileboundary represents strain-softening with exponential decay,characterized by: (1) the mesoscale tensile strength st; and (2) aninitial slope of the exponential decay, or softening modulus Ht. Inorder to preserve the correct energy dissipation during mesoscaledamage localization, the softening modulus in pure tension isexpressed as:

Ht ¼ 2E0=ðlt=le � 1Þ (7)

where lt ¼ 2E0Gt=st2; le is the length of the tetrahedron edge under

consideration; and Gt is the mesoscale fracture energy. For pureshear stress (u ¼ 0), the effective stress boundary representsperfectly plastic behavior characterized by the mesoscale shear

Fig. 2. Fibers used in the experiments. MS2 represents synthetic fibers, and SF rep-resents steel fibers. Scales are in cm.


strength ss. LDPM provides a smooth transition between puretension and pure shear. With the ratio of shear strength to tensilestrength given as rst ¼ ss/st, the parabolic variation for the strengthis expressed as:

s0ðuÞ ¼ str2st

�� sinðuÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2ðuÞ þ 4a cos2ðuÞ�r2st

q ��

�h2acos2ðuÞ

i(8)

The second physical phenomenon simulated is compaction andpore collapse from compression. For compressive loading (εN< 0),the normal stress is computed by imposing theinequality �sbc(εD,εV)�sN� 0, where sbc is a strain-dependentboundary depending on the volumetric strain εV and the devia-toric strain εD. The volumetric strain is computed at the tetrahedronlevel as εV¼ (V� V0)/3V0, where V and V0 are the current and initialvolumes of a tetrahedron, respectively. The upper limit of elasticbehavior is defined by a volumetric strain value given as: εc0¼ sc0/E0, where sc0 is a material parameter termed the mesoscalecompressive yield stress. Beyond this limit, the compressiveboundary is assumed to have an initial linear evolution, modelingpore collapse, up to a volumetric strain value of εc1 ¼ kc0εc0. Dis-regarding the deviatoric-to-volumetric strain ratio, rDV ¼ εD/εV, thestiffness of this linear segment is some fraction of the elastic stiff-ness and is expressed as Hc0. For axial compression with lateralconfinement, rDV can significantly affect the response. To simulatethe observed horizontal plateau featured by typical experimentaldata, the stiffness of the pore collapse segment must tend to zerofor increasing values of rDV. This can be achieved by setting:

HcðrDV Þ ¼ Hc0=ð1þ kc2hrDV � kc1iÞ (9)

where Hc0, kc1, and kc2 are material parameters. The boundary forthe pore collapse phase (εc0 � �εV � εc1) can now be expressed as:

sbc ¼ sc0 þ h � εV � εc0iHcðrDV Þ (10)

Compaction and rehardening occur beyond pore collapse(�εV � εc1). In this case one has the following:

sbc ¼ sc1ðrDV Þexp ½ð � εV � εc1ÞHcðrDV Þ=sc1ðrDV Þ� (11)

sc1ðrDV Þ ¼ sc0 þ ðεc1 � εc0ÞHcðrDV Þ (12)

Finally, the LDPM constitutive equation simulates friction due tocompression-shear. The incremental shear stresses are computedas:

_sM ¼ ET�_εM � _ε

pM

�; _sL ¼ ET

�_εL � _ε

pL

�(13)

where _εpM ¼ _lv4=vsM , _εpL ¼ _lv4=vsL, and l is the plastic multiplierwith loading-unloading conditions 4 _l � 0 and _l � 0. The plasticpotential is defined as:

4 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2M þ s2L

q� sbsðsNÞ (14)

where the nonlinear frictional law for the shear strength isassumed to be as follows:

sbs ¼ ss þ ðm0 � m∞ÞsN0½1� expðsN=sN0Þ� � m∞sN (15)

where sN0 is the transitional normal stress, and m0 and m∞ are theinitial and final internal friction coefficients.

2.3. Constitutive law for fiberematrix interaction

In the present study, a semi-empirical formulation based on anumber of assumptions, including that the fiber is initially straight,elastic, and has negligible bending stiffness, is adopted (Yang et al.,2008). Prior to a purely frictional pull-out stage, an embedded fibersegment must completely debond from the surrounding matrix.The debonding stage described as a “tunnel-type” cracking processis characterized by two parameters: the bond fracture energy Gd,and a constant value of frictional stress t0. The slippage has acritical value vd representing full debonding. For a genericembedment length Le, it can be expressed as (Lin et al., 1999):

vd ¼ 2t0L2eEf df

þ 8GdL

2e

Ef df

!1=2

(16)

where Ef is the modulus of elasticity of the fiber. For the debondingstage (v< vd), fiber load is given as (Lin et al., 1999):

PðvÞ ¼"p2Ef d

3f

2ðt0vþ GdÞ

#1=2(17)

For the pull-out phase (v> vd), the resistance is entirely fric-tional and one has (Lin et al., 1999):

PðvÞ ¼ P0

�1� v� vd

Le

�"1þ bðv� vdÞ

df

#(18)

where P0 ¼ pLedft0, and b is a dimensionless coefficient.In a general situation, the orientation of the embedded segment

and the free segment, with associated crack-bridging force Pf, aredifferent. The original deflection angle between the bridgingsegment and the embedded end is denoted as 4f. At the pointwhere the fiber exits the matrix and changes orientation, bearingstresses are created in the underlying matrix. When this localizedstress field reaches a sufficient intensity, spalling occurs, and theembedment length of the fiber is reduced by a correspondinglength sf. The force in the bridging segment experiences a suddendrop, and the deflection angle between the two fiber segments isreduced to 40

f . In the current study, the spalling length is obtainedby Yang et al. (2008):

Table 1Mechanical properties of fibers used in the experiments.

Type Fiber code Lf [mm] df [mm] Lf/df [-] Ef [GPa] suf [MPa] Shape

Steel SF 50 1 50 210 1100 HookedPolystyrene MS2 40 0.83 48 11.3 600 Hooked

Table 2Fiber dosage used in the experiments.

Mix code Number of specimens Dosage [kg/m3] Volume fraction [%]

MS2-5 7 5.0 0.37MS2-10 7 10.0 0.74SF-20 7 20.0 0.26SF-35 7 35.0 0.45


sf ¼PfNsinðq=2Þ

kspstdf cos2ðq=2Þ(19)

where PfN is the normal component of the total force Pf, st is thematrix tensile strength, ksp is a material parameter, andq ¼ arccosðnf

TnÞ.The snubbing effect is taken into account by assuming that at

the point where it exits the tunnel crack, the fiber wraps around theintact matrix in a perfectly flexible manner. The summation of all

Fig. 3. LDPM simulation for plain concrete: (a) Specimen geometry and discretization in num(d) Meso-scale crack patterns at CMOD¼ 0.8 mm. The experimental curve (red line) reprepresents the average response from two distinct meso-structural models. (For interpretatversion of this article.)

slip-friction and debonding forces acting parallel to the embeddedlength is denoted as P. The additional friction and bearing forces atthe exit point imply that Pf> P. The following analytical formulationis adopted:

Pf ¼ expksn40

f

PðvÞ (20)

where P has now been properly expressed as a function of relativefiber slippage v, and ksn is a dimensionless material parameter.

The formulations above describe the fiber-matrix interactiononly when the load does not cause the rupture of the fiber, it is thennecessary to check for each fiber that:

sf ¼ 4Pf.pd2f � suf e

�krup40f (21)

where krup is a dimensionless material parameter, and suf is theultimate tensile strength of the fiber.

erical simulation; (b) Three divisions of the cross section; (c) Load vs CMOD curve; andresents the average of two experimental tests, and the numerical curve (green line)ion of the references to colour in this figure legend, the reader is referred to the web

Table 3LDPM concrete parameter values.

E0 [GPa] a [-] st[MPa] lt [mm] Gt [N/m] ss/st [-] nt [-]

51 0.25 2.21 900 43 2.5 0.2

sc0 [MPa] Hc0/E0 [-] kc0 [-] kc1 [-] kc2 [-] m0 [-] m∞ [-]

195 0.2 2.7 1 5 0.25 0.0

sN0 [MPa] Ed/E0 [-] d0 [mm] c [kg/m3] w/c [-] nF [-] r [kg/m3]

600 1 10 357 0.49 0.5 2400

a/c [-] da [mm]

5.02 22

Table 4LDPM-F parameter values used in numerical simulations.

SF-20 SF-35 MS2-5 MS2-10

Gd [N/m] 5.0 5.0 5.0 5.0t0 [MPa] 6.0 6.0 1.5 1.5b [-] 0.0 0.0 0.0 0.0ksp [-]a 6.2 6.2 6.2 6.2ksn [-] 0.5 0.5 0.5 0.5suf [MPa] 1100 1100 600 600krup [-] 0.0 0.0 0.3 0.3Ef [GPa] 210 210 11.3 11.3

a In numerical simulation, a simplified procedure has been adopted. Most studiesprovide the macroscopic compressive strength s0c of the concrete or mortar beinginvestigated. Using a mid-range estimate, it is assumed that s0t ¼ s0c=10, which al-lows for the approximate calculation of kspst as 6:2s0c=10z150 Mpa.


For steel fibers, the experimental studies usually provide suf, andsometimes the elastic modulus, usually close to Ef¼ 200 GPa.

Consider a fiber, with initial orientation nf, subject to a crackopening of vector w. The tangential component of w is defined as

Fig. 4. LDPM simulation for steel fiber reinforced concrete SF-20 with Vf¼ 0.26%: (a) Specim(c) Meso-scale crack patterns at CMOD¼ 0.8 mm. The experimental curve (red line) reprerepresents the average response from seven distinct meso-structural models. (For interpretversion of this article.)

wT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2

M þw2L

q. Assuming that the spalling length sf is the same

on both sides, the vector for the crack-bridging segment can becomputed as w′¼wþ 2sfnf. The bridging segment force vector isassumed to be coaxial with the fiber Pf ¼ Pfn0

f , where

n0f ¼ w0=kw0k. The frictional pulley model applies equally to both

the shorter and longer ends, with initial lengths Ls and Ll, relativeslippages vs and vl, and pullout resisting forces Ps and Pl, respec-tively. Equilibrium of the fiber bridging segment leads toPf ¼ expðksn40

f ÞPsðvsÞ ¼ expðksn40f ÞPlðvlÞ, where 40

f ¼ arccosðnTf n

0f Þ,

which implies that Ps(vs) ¼ Pl(vl), and compatibility has to beenforced, which is jjw′jj¼2sfþ vsþ vl.

As discussed in the sections above, the LDPM-F response de-pends on two sets of parameters: (a) the LDPMmaterial parametersgoverning the behavior of plain concrete; and (b) the parametersgoverning the fiber-matrix interaction model, which needs to beidentified by fitting experimental data.

3. Experimental data and test setup

The bending test of prismatic specimens is themost popular andwidely used experimental technique for assessing the post-cracking behavior of fiber reinforced concrete. It is relatively sim-ple to perform and can be representative of many practical situa-tions. To calibrate/validate the proposed numerical model, anextensive experimental study was conducted to investigate thethree-point bending behavior of various prismatic specimenscontaining different types (steel and synthetic) and dosages offibers.

The fracture properties of the fiber reinforced concrete weredetermined according to the European Standard EN14651 (UNI EN14651, 2005) and EN14845 (UNI EN 14845, 2007) which requirebending test (3PBT) on small beam specimens(150� 150� 550 mm). The width and depth of test specimens are

en geometry and discretization in numerical simulation; (b) Load vs CMOD curve; andsents the average of seven experimental tests, and the numerical curve (green line)ation of the references to colour in this figure legend, the reader is referred to the web

Fig. 5. LDPM simulation for steel fiber reinforced concrete SF-35 with Vf¼ 0.45%: (a) Specimen geometry and discretization in numerical simulation; (b) Load vs CMOD curve; and(c) Meso-scale crack patterns at CMOD¼ 0.8 mm. The experimental curve (red line) represents the average of seven experimental tests, and the numerical curve (green line)represents the average response from seven distinct meso-structural models. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)

Table 5Numbers of the fibers bridging the cracks in three divisions of the cross section.

Experiments on SF-20

Specimen #1 #2 #3 #4 #5 #6 #7Bottom 8 10 22 18 12 15 18Middle 9 15 17 9 11 13 14Top 15 16 23 4 13 24 25

Experiments on SF-35


Experiments on MS2-5


Experiments on MS2-10


Table 6Numbers of the fibers bridging the cracks in three divisions of the cross section.

Numerical Simulation on SF-20

Specimen #1 #2 #3 #4 #5 #6 #7Bottom 17 12 10 16 15 8 17Middle 17 11 21 14 15 14 16Top 11 13 12 9 5 13 10Specimen #8 #9 #10 #11 #12 #13 #14Bottom 10 11 28 21 21 19 12Middle 18 15 9 18 15 20 16Top 9 12 14 9 4 12 19

Numerical simulation on SF-35

Specimen #1 #2 #3 #4 #5 #6 #7Bottom 25 31 20 27 29 19 33Middle 22 23 31 26 32 31 37Top 22 16 23 27 19 10 16Specimen #8 #9 #10 #11 #12 #13 #14Bottom 27 28 24 35 25 21 16Middle 26 25 22 27 28 29 29Top 15 17 19 19 21 19 20

Numerical simulation on MS2-5


Numerical simulation on MS2-10



greater than three times the length of the fibers used in this study,so the preferential fiber alignment effect is minimized (ASTMC1609/C 1609M-07, 2007). Fracture tests were carried out with aclosed-loop hydraulic testing machine by using displacement ascontrol parameter. The experimental results were reported in theform of load versus Crack Mouth Opening Displacement (CMOD)curves, and the latter was measured by means of a resistivedisplacement transducer positioned astride a notch (having a depthof 25 mm and a width of 4 mm) at mid-span.

In the present work, only a subset of the obtained experimentaldata was used. In particular, the experimental data on the behaviorof the specimens made of plain concrete and concrete containingthe fibers were used, as shown in Fig. 2. Fibers SF are hooked steel

Fig. 6. (a) Load increment versus number of active fibers in BottomþMiddle of thesteel fiber reinforced concrete; (b) Load increment versus number of active fibers inBottom of the synthetic fiber reinforced concrete; (c) Load increment versus number ofactive fibers in BottomþMiddle of the synthetic fiber reinforced concrete.


fibers, and Fibers MS2 are polystyrene fibers. The most importantproperties of these fibers are listed in Table 1. For each type of fiber,two different dosages were tested, corresponding to the minimumand the maximum dosages suggested by the fiber manufacturers,as shown in Table 2. For each dosage, 7 specimens were tested.Although the experimental setup and test procedure have beendescribed in Buratti et al. (2011), the experimental data used in thecurrent study have not been published before.

Existing literature has shown that to improve the compressiveconcrete strength, flexural strength, and shrinkage, deformed steelfibers are more effective than straight fibers due to mechanical

anchorage created by the deformed shape (Banthia and Trottier,1994; Ezeldin and Lowe, 1991; Ramakrishnan et al., 1989), andthus it is not surprising that almost all commercially available steelfibers at present are mechanically deformed, such as hemi-circular,hooked, and corrugated fibers (Banthia and Trottier, 1994; Cunhaet al., 2010; Robins et al., 2002). Steel fibers with hooked endswere used in this research. These fibers represent the most widelyused steel fibers' geometries used in concrete reinforcement(Laranjeira et al., 2010).

4. Model calibration and validation

The proposed numerical model is calibrated by simulating threepoint bending tests on prismatic notched specimens and bycomparing the numerical results with the experimental data. TheLDPM concrete parameters are determined by fitting a simulationto the plain concrete response. The fiber model calibration is thenperformed by fitting a simulation to one of the non-zero Vf

response curves. For validation purposes, the numerical responsesfor other fiber contents, determined by altering only the fiber vol-ume fraction Vf, are then generated, and the results are comparedwith the experimental curves.

4.1. Plain concrete

In three-point bending experiments, damage typically occursonly in a small region of the beam centered about the mid-spannotch. Therefore, in order to reduce the computational cost of thesimulations, the mid-span portion was modeled with LDPM-F, andthe ends of the beams were modeled with standard elastic finiteelements connected to the LDPM-F region through a master-slavealgorithm, as shown in Fig. 3(a). The simulation results obtainedfrom the discrete model with LDPM-F defined over the wholespecimen are exactly the same as those obtained from the modelmixed with LDPM-F in the middle and finite elements at two ends.In Fig. 3(c) the load vs Crack Mouth Opening Displacement (CMOD)curves are shown for plain concrete. In this case the experimentalcurve (red line) represents the average of two experimental tests,and the numerical curve (green line) represents the averageresponse from two distinct meso-structural models (The figuresappear in color in the electronic version of this article). The LDPMconcrete parameter values determined by matching the numericalresults with experimental data are shown in Table 3.

4.2. Steel fiber reinforced concrete

For steel fiber reinforced concrete, LDPM-F was calibrated byfitting numerical simulation results to the SF-20 (Vf¼ 0.26%) curve,and the fiber parameters determined from that fit are summarizedin Table 4. Fig. 4 shows the specimen geometry and discretization inthe numerical simulations, the load vs CMOD curve, and the meso-scale crack patterns at CMOD¼ 0.8 mm. In this case the experi-mental curve (red line) represents the average of seven experi-mental tests, and the numerical curve (green line) represents theaverage response from seven distinct meso-structural models.

Numerical prediction of the higher Vf response was obtained bychanging only Vf's value to 0.45%. The numerical load vs CMODcurves were compared with the experimental curves, as shown inFig. 5. In this case the numerical results show higher values than theexperimental data. The difference between the experimentalaverage and the numerical average, calculated at CMOD¼ 0.8 mm,is about 45%. As shown in Figs. 3(d), 4(c) and 5(c), for each set ofspecimen analyzed, the maximum crack opening is about 80% ofthe CMOD value.

Fig. 7. Crack opening isosurfaces at CMOD¼ 0.8 mm for steel fiber reinforced concrete: (a) Vf¼ 0.26%; (b) Vf¼ 0.45%.

Fig. 8. LDPM simulation for synthetic fiber reinforced concrete MS2-5 with Vf¼ 0.37%: (a) Specimen geometry and discretization in numerical simulation; (b) Load vs CMOD curve;and (c) Meso-scale crack patterns at CMOD¼ 0.8 mm. The experimental curve (red line) represents the average of seven experimental tests, and the numerical curve (green line)represents the average response from seven distinct meso-structural models. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)


Fig. 9. Single fiber two way pull-out test: (a) Relationship between fiber slippage andcrack opening for steel and synthetic fibers; (b) Relationship between load and crackopening for steel and synthetic fibers.

Fig. 10. LDPM simulation for synthetic fiber reinforced concrete MS2-10 with Vf¼ 0.74%: (curve; and (c) Meso-scale crack patterns at CMOD¼ 0.8 mm. The experimental curve (red liline) represents the average response from seven distinct meso-structural models. (For inteweb version of this article.)


In order to clarify the discrepancy in the simulation withVf¼ 0.45%, the fiber arrangement within each specimen wasanalyzed. It was hoped that such an approach would develop abetter understanding of the observed discrepancy, as shown inFig. 5(b). In the experiments, each specimen was analyzed at theend of the test, and the number of fibers bridging the crack in thethree divisions of the cross section, as shown in Fig. 3(b), wascounted and the relevant values are reported in Table 5. To improvereliability of the comparison, seven more numerical simulationswere conducted for each case, and the number of fibers bridgingthe cracks was extracted from the output files of the numericalsimulation, as shown in Table 6.

In Fig. 6(a), at CMOD¼ 0.8 mm, the load increment due to theeffect of fibers, calculated as 100(P*/P0�1), where P* is the load forthe fiber reinforced concrete specimens and P0 is the average loadfor the plain concrete specimens, is plotted against the number ofactive fibers in the lower two-thirds of the ligament length, i.e.,BottomþMiddle. It can be observed that there is a clear linearrelationship between the load increment and the number of activefibers, and that both the experimental data and the numerical re-sults demonstrate a similar trend. However, for the numericalspecimens, the range of active fibers increase from [22e39] to[47e70] as the fiber volume fraction nearly doubles from Vf¼ 0.26%to Vf¼ 0.45%, while for the experimental specimens, the increase ofthe number of active fibers, from [17e39] to [17e50], is notconsistent with the doubling of the fiber volume fraction. It can beconcluded that, for the higher value of Vf, the experimental speci-mens contain less active fibers than expected, probably due to theinherent difficulty of obtaining a uniform fiber distribution duringcasting. This could be one of the major reasons for the discrepancyobserved in the curves shown in Fig. 5(b), and at the same time ithighlights the fact that the actual fiber distribution within thespecimens is a crucial information needed for the numerical

a) Specimen geometry and discretization in numerical simulation; (b) Load vs CMODne) represents the average of seven experimental tests, and the numerical curve (greenrpretation of the references to colour in this figure legend, the reader is referred to the

Fig. 11. Crack opening isosurfaces at CMOD¼ 0.8 mm for synthetic fiber reinforced concrete: (a) Vf¼ 0.37%; (b) Vf¼ 0.74%.


simulation to accurately capture the mechanical behavior of fiberreinforced concrete.

It is worth observing here that the reason that the number ofactive fibers in the BottomþMiddle part of the specimen wasplotted was motivated by analyzing the crack pattern in the nu-merical models. Fig. 7 shows the color-scale isosurfaces from twotypical meso-scale crack openings. Each isosurface corresponds to adifferent crack opening value. The minimum value that was plottedcorresponds to a crack opening magnitude of 2vd. This is the overallcrack opening associated with a full debonded straight fiber,orthogonal to the crack surface. At this level of crack opening, fiberscan be considered fully active. As depicted in Fig. 7, the zonecharacterized by fully active fibers extends to approximately two-thirds of the ligament length above the notch. The choice of plot-ting isosurfeces at CMOD¼ 0.8 mm for the comparison is due to thefact that at this value the macroscopic load becomes almostconstant.

4.3. Synthetic fiber reinforced concrete

For synthetic fiber reinforced concrete, LDPM-F was calibratedby fitting numerical simulation results to the MS2-5 (Vf¼ 0.37%)curve, and the fiber parameters determined from that fit aresummarized in Table 4. Fig. 8 shows the specimen geometry anddiscretization in numerical simulation, the load vs CMOD curve,and the meso-scale crack patterns at CMOD¼ 0.8 mm. Table 4shows that, besides Ef and suf, only krup and t0 are modifiedcompared to the previous case of steel fibers.

Fig. 9 shows the comparison between the synthetic fiberbehavior and the steel fiber behavior during a twoway pull-out test.Fig. 9(a) shows that the synthetic fiber undergoes a greater

deformation before the complete debonding of the short embed-ment end since its elastic modulus is much lower than that of thesteel fiber. It can be seen that, for the synthetic fiber, the completedebonding of the short embedment end is achieved atv¼ 0.268mm,while for the steelfiber it is achieved at v¼ 0.086mm.Fig. 9(b) shows that, due to the differences in elastic modulus andtensile strength, the synthetic fiber undergoes a much lower loadlevel (0.101 kN), before the load drop, than the steel fiber does(0.474 kN). Besides, the synthetic fiber load drop is about 0.014 kN,whereas the steel fibers load drop is close to 0.071 kN.

Numerical prediction of the higher Vf response was obtained bychanging only the value of Vf to 0.74%. The numerical load vs CMODcurves were compared with the experimental curves, as shown inFig. 10. In this case, LDPM-F is able to predict very accurately theresponse for the higher fiber content. The error, compared to theexperimental results, is only a small percentage at the peak andinitial post-peak, and remains less than 20% even in the far post-peak.

The numerical simulations are further investigated in Figs. 3(d),8(c) and 10(c), where contours of the meso-scale crack openings atCMOD¼ 0.8 mm, are reported for plain concrete and synthetic fiberreinforced concrete. Due to the presence of the notch and theassociated stress concentration, one main macroscopic crack de-velops in all three cases. This crack, however, is the result of a bandofmeso-scale cracking characterized by a certain width. This result isin general agreement with the theoretical interpretation providedby Ba�zant and Oh (1983). For plain concrete, the crack is verylocalized, and in the vicinity of the notch tip, the crack bandwidth isabout 20 mm. This is commensurate with the maximum aggregatesize, suggesting very little crack-bridging effect, and is in accordancewith the nearly complete loss of load-carrying capacity indicated in

Fig. 12. Fracture process zone for plain concrete.


Fig. 13. Fracture process zone for steel fiber reinforced concrete with Vf¼ 0.26%.


Fig. 3(b) for CMOD¼ 0.8mm. For syntheticfiber reinforced concrete,the width of the crack band increases significantly with increasingfiber volume fraction. For the case with Vf¼ 0.74%, the crack bandwidth is approximately 75 mm, nearly 3.5 times the maximumaggregate size and nearly twice the fiber length.

Notwithstanding the good agreement between the experi-mental data and the numerical results for the model validation, thesame analysis on the fiber arrangement within each specimen wasconducted. Experimental data for the number of fibers bridging thecrack in the three divisions of the cross section have been shown inTable 5. Seven numerical models were analyzed for each case, andthe number of fibers bridging the cracks was extracted and shownin Table 6.

The crack pattern in the numerical models was analyzed inFig.11,which shows the color-scale isosurfaces for twotypicalmeso-scale crack openings. Theminimumcrack opening (2vd) is 0.300mmin this case. As depicted in Fig. 11, the zone characterized by fullyactive fibers extends to approximately one half of the ligamentlength above the notch. For this reason the comparison between the

experimental data and the numerical results of the load incrementversus the number of active fibers was now performed by consid-ering two cases: (a) only the numbers of active fibers in the Bottom,as shown in Fig. 6(b); and (b) the numbers of active fibers in theBottomþMiddle, as shown in Fig. 6(c). Again, it can be observed thatthere is a linear relationship between the load increment and thenumber of activefibers, and that both the experimental data and thenumerical results demonstrate a similar trend.

4.4. Comparison and discussion

The LDPM-F numerical simulations capture many salient fea-tures of the experimental results remarkablywell: (a) by comparingthe load vs CMOD curves for plain concrete and fiber reinforcedconcrete, the increased load-carrying capacity of fiber reinforcedconcrete proportional to fiber volume fraction can be observedduring the post-peak period; (b) due to the difference in fibers'mechanical properties, despite the increased fiber volume fractionused in the case of synthetic fibers, the load vs CMOD curves of the

Fig. 14. Fracture process zone for synthetic fiber reinforced concrete with Vf¼ 0.74%.


synthetic fiber reinforced concrete show much lower load levelsduring the post-peak compared with steel fiber reinforced con-crete; (c) at the peak load, it can be observed that there are nocorrelations between the load level and the fiber volume fractionsince, at that time, the crack opening is so small that the fibers arenot active yet and the material response is mainly driven by theconcrete matrix, whose maximum strength depends on its randommeso-scale structure; and (d) for fiber reinforced concrete, becausethe fibers' bridging effect leads to the load transmission to thecracked concrete matrix, the size of the crack propagation zone ismuch greater than that in the case of plain concrete.

However, it can be seen that the numbers of fibers in the nu-merical models tend to be less scattered than those obtained fromthe experimental results. This incongruence is caused by the dif-ference in the approaches by which the fibers are arranged withinthe specimen. During the experiments, the fibers are introducedinto the cement matrix non yet aged and then distributed bymixing, whereas during the numerical simulation, LDPM-F

uniformly distributed the fibers by a random iterative procedure. Tomake an efficient and reliable comparison between the experi-mental and the numerical data, it must be highlighted the impor-tance of recording the exact number of fibers arranged in thevolume fraction in which the cracking leads to the specimen failureduring the test. It is interesting to note that to predict the evolutionof distribution and orientation of fibers in the fiber reinforced self-compacting concrete during the casting process, a two-waycoupled model for flow of suspension of rigid solid particles in anon-Newtonian fluid with a free surface has been developedrecently (Svec et al., 2012). This model could be adopted by LDPM-Ffor the extension to include non-uniform fiber distribution.

5. Analysis of the fracture process zone

For cementitious materials, the Fracture Process Zone (FPZ) isdefined as the inelastic zone around the crack tip inwhich energy isdissipated at a certain stage of analysis. The behavior of the FPZ is


dominated by complicated mechanisms, such as micro-cracking,crack deflection, bridging, crack face friction, crack tip blunting byvoids, and crack branching. Because its length is related to thecharacteristic length of the cementitious materials, the size, extent,and location of the FPZ have been the subject of countless researchefforts for several decades (Cedolin et al., 1983; Castro-Monteroet al., 1990; Maji and Shah, 1988; Hu and Wittmann, 1990). Theunderstanding of FPZ has been significantly hindered by the factthat, inmost experiments, only global quantities, such as the overallstrength of the structure, are determined, and local quantities, suchas the spatial distribution of dissipated energy and fracture pat-terns, could not be obtained. Experimentally, local quantities so farhave been evaluated by acoustic emission measurements (Landis,1999; Otsuka and Date, 2000; Haidar et al., 2004; Muralidharaet al., 2010), X-ray tomography (Landis et al., 2003), moir�e inter-ferometry (Guo et al., 1993), and optical microscopy (Hadjab et al.,2007). Digital image correlation techniques have also been appliedto study fracture processes in concrete (Choi and Shah, 1997; Corret al., 2007; Huon et al., 2009) and other materials (Gregoireet al., 2011; Espinosa et al., 2011). However, more detailed resultsare required to fully understand the mechanics of the evolution offracture processes of concrete, especially fiber reinforced concrete.

The main purpose of this section is to create a 3D numericalmethodology to identify the change in fracture characteristics ofconcrete reinforced with different types of fibers. A somewhatsimilar analysis, but in 2D, was proposed recently by Grassl et al.(2012). For this purpose, the numerical models analyzed in previ-ous sections for plain concrete and fiber-reinforced concrete werepost-processed to understand how the fibers affect the shape andsize of the FPZ, as shown in Figs. 12e14, where contours of themeso-scale combined stresses are reported. The combined stress ofeach facet is defined as the square root of the sum of the squares ofthe normal stress sN, and the two shear stresses sM and sL. It shouldbe noted that only the facets that undergo tensile stress, i.e., sN> 0,are considered, since their progressive failure is responsible for thecrack propagation.

It is well-known that the fracture process zone is subdivided intoan inner softening zone and an outer micro-fracture zone. To studythe inner zone where the energy dissipation is concentrated, thefacets located around it, weakly contributing to the dissipation, arenot plotted. Figs. 12e14 are obtained by setting a minimum dissi-pated power value of 0.001 J/s and a minimum combined stressvalue of 0.2MPa, i.e, the facetswith power or stress lower than thesevalues are not shown.During all the tests, theminimumvalue set forpower is less than 1.5% of themaximum, and theminimumvalue setfor combined stress is less than 5.5% of the maximum.

5.1. Plain concrete

The evolution of the FPZ for plain concrete is shown in Fig. 12.The minimum power value of 0.001 J/s equals to 1.24% of the

Table 7Fracture process zone features at CMOD¼ 0.1 mm and CMOD¼ 1.0 mm.

CMOD [mm] FPZ width [mm

Plain concrete 0.1 75.5Plain concrete 1.0 36.1SF-20 0.1 75.6SF-20 1.0 76.2SF-35 0.1 80.8SF-35 1.0 73.8MS2-5 0.1 84.3MS2-5 1.0 38.4MS2-10 0.1 82.2MS2-10 1.0 65.8

maximum at the peak load, which is 0.0803 J/s, and 0.33% of themaximum value at CMOD¼ 0.8 mm (when the load becomesalmost constant), which is 0.3023 J/s. The minimum stress of0.2 MPa is 5.21% of the maximum value at the peak load, which is3.84 MPa, and 4.07% of the maximum value at CMOD¼ 0.8 mm,which is 4.92 MPa. In the beginning, localized presence of highenergy dissipation can be observed just above the notch where thestress is concentrated. With the load increasing, the FPZ size be-comes larger and it reaches its maximum width immediately afterthe peak load. The FPZ width starts to decrease after that becausethe specimen fails by cracking, and correspondingly, the high stressvalues move from the notch to the top surface where the concretematrix has not been damaged yet. In the meantime, the facetslocalized just above the notch undergo a significant stress drop. Atthe end of the test, only the facets near to the top surface stillcontribute to the energy dissipation by enduring combined stressvalues greater than zero. However, it can be observed that somefacets close to the notch, where the concrete matrix has been fullydamaged, still have non-zero combined stress values. This apparentinconsistency is caused by the frictional effect between twoopposite crack faces while they undergo relative displacement dueto not perfectly symmetric specimen deflection.

5.2. Steel fiber reinforced concrete

The evolution of the FPZ for steel fiber reinforced concrete withVf¼ 0.26% is shown in Fig. 13. The minimum power value of 0.001 J/s equals to 1.27% of the maximum at the peak load, which is0.0789 J/s, and 0.13% of the maximum value at CMOD¼ 0.8 mm,which is 0.7672 J/s. The minimum stress of 0.2 MPa is 4.74% of themaximum value at the peak load, which is 4.22 MPa, and 3.74% ofthe maximum value at CMOD¼ 0.8 mm, which is 5.35 MPa. Fig. 13shows that the FPZ size keeps increasing until the peak load isreached, but different from the previous case of plain concrete, dueto the bridging effect between fibers and matrix by which theapplied load can be transmitted to the facets even with large crackopening displacements, the facets close to the notch still undergo acertain amount of stress at the end of the test.

5.3. Synthetic fiber reinforced concrete

The evolution of the FPZ for synthetic fiber reinforced concretewith Vf¼ 0.74% is shown in Fig. 14. The minimum power valueequals to 1.28% of the maximum at the peak load, which is 0.0781 J/s, and 0.23% of the maximum value at CMOD¼ 0.8 mm, which is0.4376 J/s. The minimum stress is 3.87% of the maximum value atthe peak load, which is 5.15 MPa, and 3.85% of the maximum valueat CMOD¼ 0.8mm,which is 5.20MPa. Similar to the previous casesof plain concrete and steel fiber reinforced concrete, Fig. 14 showsthat the FPZ size keeps increasing until the peak load is reached,and then the high stress values move from the notch to the top

] Number of facets Total power [J/s]

6050 93.31934 36.95566 85.14876 90.16866 79.76170 126.46664 99.42648 56.76180 89.73108 43.4

Fig. 15. Fracture process zone for (a) plain concrete at CMOD¼ 0.1 mm, and (b) at CMOD¼ 1.0 mm; (c) steel fiber reinforced concrete with Vf¼ 0.26% at CMOD¼ 0.1 mm, and (d) atCMOD¼ 1.0 mm; (e) steel fiber reinforced concrete with Vf¼ 0.45% at CMOD¼ 0.1 mm, and (f) at CMOD¼ 1.0 mm; (g) synthetic fiber reinforced concrete with Vf¼ 0.37% atCMOD¼ 0.1 mm, and (h) at CMOD¼ 1.0 mm; (i) synthetic fiber reinforced concrete with Vf¼ 0.74% at CMOD¼ 0.1 mm, and (j) at CMOD¼ 1.0 mm.


surface where the concrete matrix has not been damaged yet.Comparison with the cases of steel fiber reinforced concrete showsthat the FPZ is much smaller at the end of the test in synthetic fiberreinforced concrete andmuch less facets experience non-negligiblestress levels although higher fiber volume fractions are used. Thishas been explained by the significantly lower capacity of the syn-thetic fibers in supporting high stress concentrations, as shown inFig. 9. For this reason, after an initial phase during which the fibersbridge the cracks close to the notch and transmit the load to theconcrete matrix, the fibers' pull-out occurs, leading to a significantdrop in the facets' stress levels.

Fig. 16. The trend of the total power during the test for the five different analyzedspecimens.

5.4. Comparison and discussion

To compare the FPZ shape and size during the tests for the abovediscussed cases, Table 7 reported the FPZ width, i.e., the width ofthe area composed of facets that undergo non-negligible power andstress values, the number of facets in the FPZ, and the value of thetotal power, both immediately after the peak load atCMOD¼ 0.1 mm when, for each analyzed specimen, the FPZ rea-ches its maximum width, and at the end of the tests whenCMOD¼ 1.0 mm and fibers are fully active. Fig. 15 shows the FPZwidths at CMOD¼ 0.1 mm and CMOD¼ 1.0 mm for the fivedifferent specimens. Fig. 16 shows the trend of the total powerduring the tests.

For plain concrete, Table 7 and Fig. 15 show that the FPZwidth atCMOD¼ 1.0 mm is much smaller than that at CMOD¼ 0.1 mm. Asshown in Fig. 16, the total power at the end of the test is about 60%lower than that at CMOD¼ 0.1 mm. This is caused by the concretematrix failure, which leads to a significant decrease of the numberof facets (68% less) in which energy is dissipated. The describedbehavior is representative of the inability of cracked concrete totransfer tensile stress across the crack. The LDPM-F simulationcaptures this feature very well.

By contrast, for the steel fiber reinforced concrete withVf¼ 0.26%, the FPZwidth at the end of the test is almost the same asimmediately after the peak load. The number of facets that make upthe FPZ at the end of the test is much larger than that in the case ofplain concrete and only 12% lower than the value atCMOD¼ 0.1 mm. As shown in Fig. 16, the trend of the total power ofsteel fiber reinforced concrete appears very different from that ofthe plain concrete. The energy dissipation for plain concrete at theend of the test is about one third of the value immediately after thepeak load, while for steel fiber reinforced concrete with Vf¼ 0.26%,the total power at the end is still 6% greater than the value atCMOD¼ 0.1 mm. The difference is caused by the fact that in steelfiber reinforced concrete, facets with large crack openings can stillsupport tensile loading due to the fibers' bridging effect.

A similar trend has been observed in steel fiber reinforced con-crete with Vf¼ 0.45%. But in this case, because of the higher fibervolume fraction, a larger number of facets in the FPZ and a highervalue of the total power are obtained at the end of the test. Thenumber of facets in the FPZ at CMOD¼ 1.0mm is 10% lower than thevalue at CMOD¼ 0.1 mmwhile the total power is about 60% higher.

For synthetic fiber reinforced concrete with Vf¼ 0.45%, the FPZwidth at the end of the test (38.4 mm) is much smaller than that atCMOD¼ 0.1 mm (84.3 mm). Correspondingly, the number of facetsthat make up the FPZ and the value of the total power is 62% smallerand 40% lower than the values immediately after the peak load,respectively. The specimen with synthetic fiber volume fraction of0.74% shows a better behavior due to the higher fiber content, butstill, the number of facets at the end of the test is 53% smaller thanthat immediately after the peak load, and the value of the totalpower is 52% lower. The poor fibers' contribution has been

explained by the significantly lower capacity of the synthetic fibersin supporting high levels of stress.

From Table 7, it can be seen that the reported values for the fivedifferent specimens at CMOD¼ 0.1 mm do not differ significantlyfrom each other. This is because at CMOD¼ 0.1 mm, the maximumcrack openings for fiber reinforced concrete specimens are:0.079 mm for SF-20, 0.064 mm for SF-35, 0.082 mm for MS2-5, and0.078 mm for MS2-10, respectively, and so the fibers are not activeyet. Recall that the crack openings at complete debonding are0.094 mm and 0.300 mm for steel fibers and synthetic fibers,respectively.

From Fig. 16, it is interesting to observe that the total power ofMS2-10 becomes lower than that of MS2-5 after CMOD¼ 0.7 mm,although MS2-10 has a much higher fiber volume fraction. Thiscould possibly be explained by the randomly distributed fibers andcracks, resulting in fewer active fibers in MS2-10 than in MS2-5.

6. Characterization through round panel test

As reported in previous sections, characteristic values deter-mined from three-point bending tests have a large scatter on theresults. It has been shown that this large scatter is due to anonuniform fiber distribution. Such a scatter becomes particularlylarge when low fiber volume fractions are used (Sorelli et al., 2005).One of the possible solutions to avoid the large scatter is to increasethe cracked area. For this reason, a round panel test, first developedby Bernard (2000, 2002), was proposed by ASTM (ASTM C1550-04,2004) in 2004. This type of test has a statically determinateconfiguration (a round slab having a diameter of 800 mm and athickness of 75 mm with three supports at 120�) where the crackpattern is predictable and the post-cracking material properties canbe adequately determined. It appears to be particularly suitable forevaluating the performance of fiber reinforced concrete because thetest method involves relatively large deflections. However, handlingand placing such a specimen is quite complicated due to the largesize and, consequently, high weight (91 kg). In addition, standardservo-controlled loading machines may not fit with the geometry ofthe panel, which is too large for most of them. To meet the need ofhaving a specimen easier to handle, Minelli and Plizzari (2010) andMinelli and Plizzari, 2011 have proposed the use of a smaller roundpanel having a diameter of 600 mm and a depth of 60 mm, with aweight of only 40 kg. Recently, Ciancio et al. (2014) have proposed touse even smaller panels having a diameter of 500 mm. It has been

Fig. 17. LDPM simulation for round panel test on small specimens. This figures shows the specimen geometry and discretization for plain concrete and steel fiber reinforced concretewith Vf¼ 0.26% and Vf¼ 0.39%.


shown that results from smaller round panels are consistent, reli-able and provide a repeatable and predictable crack pattern with aconsistently lower dispersion than classical beam tests.

In this section, LDPM-F is calibrated/validated by simulatinground panel tests on both large and small specimens and bycomparing the numerical results with the experimental data pro-vided by Minelli and Plizzari (2010) and Minelli and Plizzari (2011).

6.1. Small round panel tests

The numerical simulation for small round panel tests was per-formed on round panels with a diameter of 600 mm (effectivediameter of 550 mm) and a thickness of 60 mm, with the loadapplied to the panels using a rigid cylinder (50 mm of diameter)above the center of the top surface. The panels were supported by

three pivots (40 mm of diameter), 120� apart radially and posi-tioned 25 mm from the edge of the panel. The geometry of thespecimen is shown in Fig. 17. The translation of the nodes on thebottom surface of the panel in contact with the three pivots is set tobe fixed in z direction only, and their rotation is fixed in both x and ydirections. The test is displacement controlled by means of thecentral displacement. The nodes on the top surface of the panel incontact with the loading cylinder are selected, and the loadinghistory is assigned to these nodes.

Since only part of the material parameters are provided inMinelli and Plizzari (2010) and Minelli and Plizzari (2011) and theyare the same as those used in previous sections, it is assumed herethat the material parameters for concrete and steel fiber used in theexperiments (Minelli and Plizzari, 2010; Minelli and Plizzari, 2011)are the same as those used in previous sections, and so the

Fig. 18. Experimental data and numerical simulation results for the relationship be-tween load and deflection (obtained from three numerical specimens in each case) fordifferent kinds of fiber reinforcement, together with the curves for plain concrete: (a)Small round panel test; (b) Large round panel test.


parameters shown in Table 3 are used in this section for plainconcrete, and the parameters for SF shown in Tables 1 and 4 areused for steel fibers.

The relationship between load and deflection (obtained fromthree numerical specimens in each case) for different kinds of fiberreinforcement, together with the curves for plain concrete, areshown in Fig. 18. The simulation results have shown that as thepanel was loaded, cracks first appeared at the bottom of the panelat its central point, radiated to the edges, and then opened up-wards. The typical crack pattern of each round panel was threecracks at angles about 120�, with the panel breaking into 3 pie-shaped wedges at large deflections. Fig. 19(a) shows the crackpropagation through the round panel obtained from one of thenumerical models on plain concrete at d ¼ 0.31 mm, d ¼ 0.41 mm,d ¼ 0.48 mm, and d ¼ 0.60 mm. Fig. 19(b) shows the crack propa-gation obtained from one of the numerical models on steel fiberreinforced concrete with Vf¼ 0.39 at d ¼ 0.32 mm, d ¼ 0.45 mm,d ¼ 0.54 mm, and d ¼ 0.63 mm.

Typically, the Ped curves of the panels are almost linear up tothe peak load. The central deflections at first crack load for bothsteel fiber reinforced concrete panels are similar, much larger thanfor the plain concrete. The increase in peak load due to steel fiberreinforcement was not significant. However, it can be seen that the

toughness of the concrete panel has been increased dramaticallydue to steel fiber reinforcement. Fig. 18 shows the comparisonbetween experimental data and numerical results. Note that theuse of experimental data is complicated by the fact that thedisplacement measurement is highly susceptible to inaccuraciescaused by several factors, such as the deformation of the supportsand the loading platen. To make it simple, the inaccurate part of thedisplacement is assumed to be linear with the loading, and thus allthe experimental data of displacement, d, aremodified as d*¼ (d�P/k)þ P/k* with k¼ 50 kN/mm and k* ¼ 90 kN/mm, intentionallyadjusting the slope of the elastic part of the curves to create amatchwith that of the numerical results. With this adjustment, while notan exact match, the agreement between experimental and nu-merical curves is quite reasonable.

For plain concrete, Fig. 18 shows three distinct load drops afterthe peak load, associated with the subsequent formation of thethree main cracks, and then a softening behavior by which the loadtends to become zero. For steel fiber reinforced concrete, load dropscan be observed during the test for all the cases, but the trend ismuch smoother than in the case of plain concrete, due to the factthat the fibers bridging effect leads to smaller load jumps. In fact, itis difficult to distinguish the three distinct load drops, becausewhen a crack starts to propagate from the panel center, and the loadstarts to drop, fibers can be intercepted, and the bridging effectleads to load transmission. Nevertheless, in all the cases of steelfiber reinforced concrete, notwithstanding this difference fromplain concrete, at the end of the test, three macro-cracks can againbe distinguished, but now around them a larger crack propagationzone caused by the fibers bridging effect can be observed.

6.2. Large round panel tests

The simulation process and material parameters for large roundpanel test are exactly the same as those used in small round paneltest except for the geometry of the specimen. The numericalsimulation for large round panel tests was performed on roundpanels with a diameter of 800 mm (effective diameter of 750 mm)and a thickness of 75 mm, using the same setup as the small roundpanel test. The panels were supported by three pivots (40 mm ofdiameter), 120� apart radially and positioned 25 mm from the edgeof the panel.

The loadedeflection curves for different fiber reinforcements,together with the curves for plain concrete, are shown in Fig. 18.Once again, the loadedeflection curves computed by LDPM-F areconsidered to exhibit a reasonable agreement with those deter-mined experimentally. Fig. 20(a) shows the crack propagationthrough the round panel bottom surface obtained from one of thenumerical models on plain concrete at d ¼ 0.31 mm, d ¼ 0.41 mm,d ¼ 0.48 mm, and d ¼ 0.60 mm. Fig. 20(b) shows the crack propa-gation obtained from one of the numerical models on steel fiberreinforced concrete with Vf¼ 0.39 at d ¼ 0.48 mm, d ¼ 0.57 mm,d ¼ 1.43 mm, and d ¼ 1.90 mm.

Since for each numerical specimen, three cracks occur, theoverall number of fibers involved in the load bearing is more similaramong the panels than in case of three point bending test, in whichonly a single crack occurs in the middle where the notch is located.That is the reason why the loadedeflection curve for each roundpanel shows approximately the same load values in the post-peaksoftening regime where all the three main cracks have been fullyestablished.

7. Conclusions

In this investigation, the fundamental concepts governing theformulation of LDPM-F were reviewed and the mathematical

Fig. 19. LDPM simulation for round panel test on small specimens. This figures shows the crack propagation through the panel: (a) Plain concrete at d¼ 0.30 mm, d¼ 0.38 mm,d¼0.45 mm, and d¼0.53 mm; (b) Steel fiber reinforced concrete with Vf¼ 0.39% at d¼0.32 mm, d¼0.45 mm, d¼0.54 mm, and d¼0.63 mm.

Fig. 20. LDPM simulation for round panel test on large specimens. This figures shows the crack propagation through the bottom surface: (a) Plain concrete at d¼ 0.33 mm,d¼ 0.49 mm, d¼ 0.50 mm, and d¼ 1.10 mm; (b) Steel fiber reinforced concrete with Vf¼ 0.39% at d¼ 0.48 mm, d¼ 0.57 mm, d¼ 1.43 mm, and d¼ 1.90 mm.



formulation was highlighted. Fiber-matrix interaction is modeledby adapting an established micro-mechanical theory providing thefiber crack-bridging force as a function of the crack opening. For allfibers that intersect an LDPM facet, the crack-bridging forces areadded together and assumed to act in parallel with the LDPMconstitutive behavior. For calibration/validation purpose, anextensive experimental study was conducted on various prismaticspecimens containing different types (steel and synthetic) anddosages of fibers. The numerical simulations conducted on three-point bending tests and round panel tests demonstrated thatLDPM-F is able to accurately reproduce the fiber tougheningmechanisms, and the associated increases in macroscopic strengthand ductility that can be seen in the results of experimental testingwhile keeping the computational cost manageable. In conclusion,the work presented in this paper demonstrates that LDPM-F is anaccurate and efficient simulation tool for studying themechanics ofthe evolution of fracture processes of fiber reinforced concrete.

Acknowledgment

This material is partly based upon work supported by the Na-tional Science Foundation under grant no. CMMI-1201087. M. Savoiaand G. Cusatis conceived and initiated the project. N. Buratti and M.Savoia designed and performed the experimental work. M. Stacchiniperformed the computation in Section 4 and Section 5. C. Jin per-formed the computation in Section 6, analyzed data and wrote themanuscript. G. Cusatis supervised all the computational tasks.

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Documents

European Journal of Mechanics A/Solidsbingweb.binghamton.edu/~cjin/publications/eu.pdf · 2016. 4. 8. · (Cusatis et al., 2011b). LDPM is an improvement over the conﬁnement-shear