CONSTITUTIVE MICROPLANE AND INTERFACE LAWS FOR MULTISCALE ANALYSIS OF STEEL FIBER CONCRETElaboratorios.fi.uba.ar/lmni/paper caggiano/Proceedings... · 2012. 9. 21. · 2.1 Composite

European Congress on Computational Methodsin Applied Sciences and Engineering (ECCOMAS 2012)

J. Eberhardsteiner et.al. (eds.)Vienna, Austria, September 10-14, 2012

CONSTITUTIVE MICROPLANE AND INTERFACE LAWS FORMULTISCALE ANALYSIS OF STEEL FIBER CONCRETE

Guillermo J. Etse1,3, Antonio Caggiano1,2,3, and Sonia M. Vrech3

1CONICET, University of Buenos AiresAv. Las Heras 2214, C1127AAR, Buenos Aires, Argentina

e-mail: [email protected]

2 DICiv, University of SalernoVia Ponte don Melillo, 84084, Fisciano, Salerno, Italy

e-mail: [email protected], [email protected]

3 CONICET, National University of TucumanAv. Independencia 1800, 4000, San Miguel de Tucuman, Argentina

e-mail: [email protected]

Keywords: SFRC, Fracture, Microplane, Mixture Theory, Plasticity, Zero-thickness Interface

Abstract. Non-linear failure behavior of Fiber Reinforced Concrete (FRC) is analyzed andmodeled by means of two different approaches. On the one hand a microplane theory based onthe continuum (or smeared-crack) approach is formulated while, on the other hand, an inter-face model based on the discrete crack approach is also considered. The predictions of thesetwo model approaches are comparatively evaluated when brittle and ductile failure modes ofFRC components are considered at the macro and mesoscopic levels of observation. To moreobjectively compare the predictive capabilities of the discontinuum and continuum model ap-proaches similar constitutive considerations are taken into account in both model formulationsto describe key mechanisms of FRC failure behavior. The well-known “Mixture Theory” isconsidered to describe the fiber-to-concrete interactions in terms of fiber debonding and doweleffects. As failure criterion, defining the onset of cracking, a second order hyperbola is con-sidered which simulates the interaction between maximum shear and normal stresses of bothmicroplane and interface interaction. After describing the features of both constitutive mod-els, this work focuses on numerical analysis of FRC failure behavior. The results in this workclearly demonstrate the advantages and shortcomings of both model approaches for FRC fail-ure behavior analyses while clearly show the relevant properties of the mechanical responsesof this composite material during ductile and brittle failure modes.

Guillermo J. Etse, Antonio Caggiano, and Sonia M. Vrech

1 INTRODUCTION

Fiber Reinforced Concretes (FRCs), obtained by mixing short fibers (made out of steel,plastic, natural and/or recycled fibers) and cement-based material, is becoming a largely usedmaterial for structural design applications [30, 21].

The main structural benefit of FRC is represented by the significant enhancement of residualstrengths in the cracked configuration of the material [2]. Actually, the stress-crack evolution ei-ther in pure mode I [33] than in mixed-fracture modes [19] deals with a more ductile compositein post-peak response.

Several experimental researches have recently been carried out by the scientific communityleading with the mechanical characterization of FRCs at both fresh and hardened state. Severalaspects are well documented: i.e., the workability related to the fiber distributions [29], thefiber orientations in function of the compaction procedures [32], fiber pull-out mechanisms[50], flexural post-crack behavior in three- [10] and four-point [53] bending tests, multiaxialcompressive state behavior [27], Brazilian disk tests [40], and so on.

A large amounts of theoretical models and numerical tools have been proposed aimed atrealistically predicting the pre- and post-cracking behavior of both concrete and FRC at materialand structural level. An extended literature review for modeling cracking phenomena in quasi-brittle materials has been proposed by [35].

Concrete cracks have traditionally been treated by means of classical continuum approachesin which the fracture zone is considered as a smeared localized deformation in a certain region ofthe solid [20]. Classical models for concrete, based on the so-called Smeared Crack Approach(SCA), are characterized by strong finite element size dependence of the localization band widthon the element size [45][49]. Regularization procedures to avoid this kind of problems areavailable in the scientific literature, such us non-local plasticity [48], strain-gradient approaches[56, 60, 58] or by means of simple techniques regarding the discretization mesh dimensions andfinite element types [4, 28].

Discrete Crack Approaches (DCAs), aimed at incorporating strain or displacement discon-tinuities into standard Finite Element (FE) procedures have been more and more developed inthe last years. Several proposals are currently available to introduce crack discontinuities withinFE domains. It can be distinguished between the Embedded strong discontinuities (E-FEM)by [25, 1], the eXtended Finite Element Method (X-FEM) [59, 41], lattice model approaches[55, 61], particle models [3, 34], the hybrid-Trefftz stress-based formulation [36], the Element-free Galerkin method [8, 51] and zero-thickness interface models [18, 22].

This paper presents a micro-macro modeling approach aimed at reproducing and predictingthe failure behavior of FRC. A continuum model based on the microplane theory, which orig-inal contributions are outlined in [5, 17, 38, 47], is derived by means the macroscopic stressintegration starting from microplane stress-strain relationships. The stress-strain laws at mi-croplane lever are now treated by means of an interface cracking model, originally outlined in[18] for reproducing failure behavior on plain concrete in mesoscale analyses [42, 43] and suc-cessively reformulated by the authors [57, 14, 13] to include the interaction between steel fibersand concrete.

After this Introduction, the proposed fracture-based interface model, following the DCA, hasbeen briefly reported in Section 2. The basic assumptions of the composite theory adopted, inwhich the well-known “Mixture Theory” by [54] is considered to account the fiber-to-concreteinteractions, are also outlined. The smeared crack model, based on microplane theory combinedwith the interface approach, is presented in Section 3.

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The numerical results of both proposed numerical approaches are given in Section 4. Par-ticularly, the analyses at both material and mesoscale levels, based on the interface crackingproposal, are firstly shown and compared against experimental data. The interface formulationis used in this paper for simulating the fracture behavior of FRC specimens at both material andmesoscale level. Then, the numerical predictions corresponding to the continuum microplanemodel are presented with the aim to evaluate the predictive capability of the proposed formula-tion to reproduce the mechanical behavior of the FRC material.

2 STRESS-CRACK OPENINING INTERFACE MODEL FOR FRC

The cracking behavior of FRC has been formulated at the interface level in terms of nor-mal/shear stress components, tt = [σ, τ ], related to the corresponding relative displacements,ut = [u, v], being “t” in these notations the transpose vectorial operation. The detailed descrip-tion of the model, its key features and capabilities, are well documented in [14].

The well-known “Mixture Theory” by [54] is employed for reproducing the actions of fibersin the cracking process of FRC. On the one hand, the bridging effect of fibers under axial stressesis explicitly considered by taking into account the bond between the fiber to the surroundingconcrete matrix. On the other hand, their dowel action is simulated as a possible restrain ofrelative displacements at the two sides of the crack in the transverse fiber direction. The lattercontribution is relevant for steel reinforcements, but can be mainly neglected for plastic ones.

2.1 Composite material model and mixture theory

FRC is assumed to be a composite material made out by a plain concrete matrix with ran-domly distributed fibers. According to the basis of the mixture theory, the rate of the compositeinterface stress t can be obtained as the weighted sum of stresses referred to each constituent[46]. Based on this hypothesis, the following expression of the stress vector takes place

t = w[ρi]ti +

nf∑f=0

w[ρf ] (σf [εN ]nN + τf [γT ]nT) (1)

beingw[ρ#] the weighting functions defined by means of the volume fraction of each constituentρ#, while i and f are indices referring to interface and fibers, respectively; σf and τf mean thebond-slip action and dowel effect of the single considered fiber which are related to the axialand tangential fiber strains, εN and γT , respectively; nf represents the number of fibers crossingthe interface and finally, nN and nT are the fiber direction and its orthogonal, respectively.

The proposed interface model can be employed in mesoscopic simulations with the aim toanalyze failure processes possibly developed at the FRC mortar-mortar or mortar-aggregateinterfaces as proposed in Fig. 1.

The number of crossing fibers per interface is evaluated by means of the expression proposedby [52]

nf = α ~N

ρfAf

Ai (2)

being ρf the fiber content, Af and Ai are the cross-sectional area of a single fiber and theinterface area, respectively. The values assumed for the orientation factors α ~N are based on thetheoretical proposal given by [24].

3


(b)

(a)

(c)

pull-out

dowel

Zero-thickness interface

Figure 1: Concrete mesoscale specimen (a), interfaces between mortar-mortar and mortar-aggregate (b) and FRCinterface modeling (c).

2.2 Cracking modeling of plain concrete interface

The proposed formulation is founded on the idea outlined in [18], originally aimed at sim-ulating fracture phenomena in plain concrete. For the sake of brevity, the relationships whichdescribe the cracking process on plain concrete matrix are shown in Table 1. They refer to astrain-softening elasto-plasticity approach and formally consider the crack-opening, ucr, as arelative plastic displacement. The constitutive equation, in the framework of rate-independentplasticity theory, can be expressed in rate format as proposed in Table 1, where C defines a fullyuncoupled normal/tangential elastic stiffness operator connecting the incremental stress vector,ti, and the rate of the relative joint displacement u = [u, v]t. The three-parameter hyperboladefined in [18] has been considered as failure criterion, f (ti, κ) which dimension in the stress-space (σN − σT ) is function of the tensile strength χ, the cohesion c and the internal frictionangle φ.

The plastic flow law is represented by a general non-associated rule which defines the direc-tion of fracture displacements, m, by means of the transformation operator, A, applied to theassociated direction, n. The evolution of the yield criterion in the post-cracking regime is basedon the ratio between the work spent in fracture processes (in mode I and/or mixed ones), wcrand the corresponding fracture energy parameters, GI

f and GIIaf , respectively. The evolution

laws, proposed in Table 1, are considered for each internal parameter pi of the yield condition,which alternatively equals to χ, c or tan(φ). This equation defines the evolution of the inter-nal parameter from its initial value, pi = p0i, to the residual one, pi = rp · p0i, in terms ofthe scaling function S[ξpi ]. Finally, the constitutive integration of the model can be achievedby means of the classical Kuhn-Tucker loading/unloading conditions. Further details of thematerial formulation are given in [14].

2.3 Bond-slip action of fibers on concrete cracks

The fiber axial action, derived from a fracture opening process, results in the bridging effectat the two sides of a general crack. The bond between fibers and the surrounding concrete

4


Table 1: Interface constitutive model for plain concrete.

Fracture− based energy formulation

Constitutive equationti = C · (u− ucr)u = uel + ucr

Yield condition f (ti, κ) = σ2T − (c− σN tanφ)2 + (c− χ tanφ)2

Flow ruleucr = λmm = A · n

Cracking work evolution

κ = wcrwcr = σN · ucr + σT · vcr if σN ≥ 0

wcr = σT · vcr(1− |σN |tan(φ)

σT

)if σN < 0

Evolution laws pi = [1− (1− rp)S(ξpi)] p0i

Kuhn− Tucker conditions λ ≥ 0, f (ti, κ) ≤ 0, λ f (ti, κ) = 0

matrix controls these bridging mechanisms. The axial (tensile) stresses at crack level, derive byequilibrium in shear stresses throughout the lateral contact surfaces of fibers embedded withinthe concrete matrix. Under these simplified assumptions and hypothesizing that each genericfiber crosses the fracture line at its mid-point, i.e. lemb = lf/2, the following equilibriumequation can be founded

dσf [x]

dx= −4τa[x]

df(3)

where σf is the axial tensile stress and τa the local bond stress between fiber and surroundingconcrete; df represents the fiber diameter.

A bi-linear shear-slip law, similarly approached by [16] for FRP laminates under pull-outtesting, is proposed with the aim to model the fiber-concrete debonding. These relationships aregiven as follows

τa[x] =

−kEs[x] s[x] ≤ se

−τy,a + kS (s[x]− se) se < s[x] ≤ su0 s[x] > su

(4)

where s[x] is defined as the debonded displacement between the fiber and concrete, at the pointof the abscissa x. The positive constants kE and kS represent the elastic and softening slopes ofsuch bond-slip relationships, respectively; τy,a is the shear bond strength while se and su are theelastic and ultimate slips, respectively. The complete derivation of this numerical model, andits validation against bond-slip experimental tests, were proposed in a previous work publishedby the authors, see [15].

2.4 Dowel behavior of fibers on concrete cracks

The dowel action resulting in a shear transfer mechanism across cracks represents an im-portant component on the overall bridging effect of steel fibers in fracture processes of FRC.A simple analytical model for reproducing the dowel action of fibers crossing cracks has beendeveloped. It is based upon the definition of both stiffness and strength of a generic fiber em-bedded in a concrete matrix and subjected to a transverse force/displacement behavior at thefracture level.

5


Firstly, the well-known Winkler beam theory is used to describe the dowel force, Vd, cor-responding to the transversal displacement, ∆, in the elastic regime. Its analytical solution isgiven by

Vd = EsJsλ3f∆ (5)

where Es is the steel elastic modulus and Js the moment of inertia of the considered fiber. TheWinkler parameter, λf , is analytically derived as

λf = 4

√kcdf

4EsJs= 4

√√√√ 16kcEsπd3

f

(6)

where kc is the foundation stiffness (herein, the surrounding mortar).Finally, the empirical expression proposed by [23] for RC-structures is taken as maximum

dowel strength Vd,uVd,u = kdowd

2f

√|fc||σy,s| (7)

being kdow a non-dimensional coefficient whose typical value 1.27 could be assumed as refer-ence for RC-structures [9], df is the diameter of the fiber, while fc and σy,s are the strengths ofconcrete and the steel, respectively.

3 MICROPLANE CRACKING MATERIAL MODEL FOR FRC

A fracture-based energy microplane model is developed for simulating FRC failure behaviorat macroscopic point of view. The formulation is based on the thermodynamically consistentapproach of the microplane plasticity formulation proposed by [17] and [38].

3.1 Basic assumptions

(b) (c)

3 4

12

3 4

12

(a) (d)

n

Figure 2: Concrete specimen (a), continuum discretization scale (b), 4-node continuum finite element (c) andspherical microplane region at Gauss-point (d).

The microplane kinematic constraint assumes that the normal and shear strains (εN and εT ,respectively) on the microplane direction n (see Fig. 2) are calculated as the projections of themacroscopic strain tensor ε

εN = n · ε · nεT = ε · n− εNn

(8)

6


or in index notationεN = ninjεijεTk = εkjnj − εNnk.

(9)

Then, stress equilibrium between the micro and macro-level can be achieved by means theapplication of the virtual work principle applied in the spherical microplane region

4π

3σ : δε = 2

∫Ω

(σNδεN + σT · δεT )dΩ (10)

where σ is the macroscopic stress tensor while Ω is the boundary surface of one hemisphere.Combining Eqs. (9) and (10), the following relation for the macroscopic stress tensor can be

derivedσij =

3

2π

∫Ω

(σNninj +

σTk2

[niδkj + njδki])

dΩ. (11)

The microplane components of stress, σN and σTk , are obtained from the microscopic free-energy potential, ψmic0 ,

σN =∂ [ρ0ψ

mic0 ]

∂εN, σTk =

∂ [ρ0ψmic0 ]

∂εTk(12)

being ρ0 the material density.The macroscopic free-energy potential per unit mass of material in isothermal conditions,

ψmac0 (ε,κ), with κ a set of thermodynamically consistent internal variables, results

ψmac0 =3

2π

∫Ω

ψmic0

(σmic,κ

)dΩ (13)

being σmic the vector collecting the normal and shear strain components for the microplane.

3.2 Microplane constitutive formulation

In a considered microplane, characterized by its normal direction n, the inelastic responsestarts once the stress state reaches certain cracking surface in the σN -σT space. The yieldcondition, the strain-softening evolution and the non-associated flow rule are defined for eachmicroplane by means of the constitutive model previously proposed in Section 2.

A macro-meso procedure has been implemented into a FE framework for FRC failure anal-yses. Once the non-linear FE solution of the structural problem is achieved, in terms of nodaldisplacement parameters, the strain field can be evaluated at the typical Gauss-points of anyFE. Then, the strain computed in each integration point, is projected on the microplanes. Eachmicro-strain, εmic = [εN , εT1 , εT2 ]

t, is transformed in terms of crack-opening displacementthrough the following expressions

εN =u

lIcsεT1 =

v1

lIIacs

εT2 =v2

lIIacs

(14)

being u, v1 and v2 the relative separation displacements of a general 3-D interface formulation(see i.e. [11, 12]), while lIcs and lIIacs are characteristic lengths for failure modes I and II, similarlyto the proposal given in [31]. Each of them mainly represents the crack spacing in direct tensionand under shear modes with very high compression (without dilatancy effect), respectively.

7


Table 2: Microplane constitutive model for plain concrete/mortar.

Fracture− based energy microplane

Constitutive equationσmic = Cmic ·

(εmic − εp,mic

)εmic = εel,mic + εp,mic

Yield conditionf(σmic, κ

)= ‖σT‖2 − (cmic − σN tanφmic)2+

(cmic − χmic tanφmic)2

Flow ruleεp,mic = λmmic

mmic = Amic · nmic

Cracking evolution

κ = wcrwcr = σN · εp,micN · lIcs + σT,k · εp,micT,k · lIIcs if σN ≥ 0

wcr = σT,k · εp,micT,k · lIIcs(1− |σN |tan(φmic)

‖σT ‖

)if σN < 0

Evolution laws pmici =[1−

(1− rmicp

)S[ξpmic

i]]pmic0i

Kuhn− Tucker conditions λ ≥ 0, f(σmic, κ

)≤ 0, λ f

(σmic, κ

)= 0

Finally, the microplane formulation can be derived in terms of stress (normal/shear)-crackopening constitutive law representing the non-linear fracture of FRC. Table 2 outlines the mi-croplane proposed formulation where a strain-softening elasto-plasticity approach is proposed,considering the strains at microplane level as the sum of an elastic and a plastic component,εel,mic and εp,mic, respectively.

The Cmic tensor defines a fully uncoupled normal/tangential elastic stiffness operator at mi-croplane level

Cmic =

EN 0 00 ET 00 0 ET

(15)

which connects the incremental rate stress vector, σmic, and the rate of the microplane elasticstrains εel,mic. The microscopic elastic moduli are related to the macroscopic ones [6, 7] asfollows

EN = 3K + 2G ET = 2G (16)

where K and G are the bulk and the shear modulus, respectively.The three-parameter hyperbola defined in Section 2 has been considered as failure surface

f(σmic, κ

). Its expression at the microplane stress-space is function of three material param-

eters: i.e., the tensile strength χmic, the cohesion cmic and the internal friction angle φmic. Theplastic flow is represented by a general non-associated law defining the direction of inelasticstrains, mmic, by means of the transformation operator, Amic, applied to the normal direction,nmic. The ratio between the work spent in fracture, wcr, and the corresponding fracture energyparameters, GI

f and GIIaf , defines a scaling parameter, ξpmic

i= wcr/G

#f , used to calculate the

evolution of the yield criterion in the post-cracking regime. The adopted evolution laws areformally the same to the expressions used for the interface model, as outlined in Table 1.

Finally, the constitutive integration can be reached by means of the classical Kuhn-Tuckerloading/unloading conditions. The interaction between concrete and steel fibers, namely fiberdebonding (see Section 2.3) and dowel effects (see Section 2.4), are treated by means of thecomposite model given in Section 2.1 and applied at microplane level.

8


4 NUMERICAL RESULTS

This section presents the main features and capabilities of the proposed formulations com-paring some numerical results against experimental data at both material and structural level ofanalysis.

4.1 Meso-scale analyses

The key assumptions of the non-linear analyses performed with the discontinuous interfacemodel for FRC aimed at predicting mesoscale tests are shown in this section.

Figure 3: Meso-structural composite geometry: (a) initial regular 2D distribution, (b) randomly perturbed posi-tions, (c) superposition of the points and (d) Voronoi/Delaunay tessellation.

A convex polygonal representation is adopted for representing both large aggregates andmortar matrix. The latter phase simulates effect of cement mortar, small aggregates and possiblyfibers. The polygonal geometry is numerically generated though standard Voronoi/Delaunaytessellation [37] from a regular array of points which is slightly perturbed as shown in Fig. 3.Both the polygonal particles and the space between them (surrounding matrix) are meshed withfinite elements. The continuum elements obtained by means of the above procedure are assumedas linear elastic, whereas all non-linearities are concentrated within zero-thickness interfaceelements defined throughout the adjacent edges of the meshed elements. Non-linear fracture-based laws and fiber actions (in terms of both bridging and dowel effects) are introduced in thoseinterface elements according to the formulation outlined in Section 2. In particular, aggregate-matrix interfaces do not consider the effect of fibers, while matrix-matrix ones take into accountthe contribution of passing through fibers. As a matter of fact, interface elements represent thepotential crack patterns which can develop during the fracture process.

Figure 4 gives the 2-D composite geometry and the subsequent structural FE discretization

9


Figure 4: Detail of the FE discretization: (a) Pre-cracked beam tested under 3PB, (b) zone with an explicit meso-mechanical discretization and (c) coarse-aggregate and FR Mortar-FR Mortar interfaces.

Figure 5: Mixture components of the continuum material.

employed in the present study. Three continuum materials are considered for the proposednumerical analyses, as can be seen in Fig. 5: (i) coarse aggregate, (ii) fiber reinforced mortar and(iii) fiber reinforced concrete. The mechanical properties of the three continuous can be definedwithin the framework of the classical theory of elasticity. However, the “Mixture Theory”,already mentioned in Section 2, is taken into account for defining the generic elastic parameter

pc = ρApA + ρMpM + ρFpF (17)

being pc the elastic modulus or the Poisson’s ratio; ρA, ρM and ρF are the volume fractions ofthe coarse aggregates, matrix and fibers, respectively; while pA, pM and pF are the consideredelastic parameter of each component of the composite component.

The simulations on 5×10×40 cm3 pre-cracked concrete specimens tested under three-pointbending by [19] are reported in this section. Plane stress hypothesis and displacement-basedcontrol are assumed. Test specimens present a vertical notch (2.0mm wide) at the bottom ofthe beam characterized by a depth of about 33 cm. The distance between the mid-length of thebeam and the notch position varies from zero (pre-cracked beam with centroidal notch) to 0.25l

10


Plain concrete ρf = 0.25 % ρf = 0.50 %

(a)

(b)

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.50 1.00 1.50 2.00

load

[k

N]

deflection [mm]

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.50 1.00 1.50 2.00

load

[kN

]

deflection [mm]

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.50 1.00 1.50 2.00

load [

kN

]deflection [mm]

Figure 6: Test results on both plain concrete (PC) and SFRC beams under three-point bending with centroidalnotch by [19]: (a) cracked configuration at ultimate state and (b) load-deflection comparisons (experimental datagiven with dotted lines).

(pre-cracked beam with eccentrical notch), being l = 36 cm the distance between the beam endsupports.

To better simulating the crack evolution during the load-tests, a finer meso-mechanical meshis considered in the notched zone. Outside of that zone a coarser mesh discretization is assumed,as lower stress levels are expected. The explicit meso-mechanical discretization, achieved bymeans of the Voronoi/Delaunay theories [44], contains 36 coarse aggregates embedded in amatrix representing both fiber reinforced mortar and small aggregates. As previously declared,two families of fracture lines are considered (see Fig. 4):

• fractures developing throughout the interfaces between coarse aggregate and FR matrix;

• cracks which cross two adjacent parts of fiber-reinforced cementitious mortars.

Three material types have been considered in the numerical analyses: (1) plain concrete (PC),(2) steel fiber reinforced concrete (SFRC) specimens with ρf = 0.25% and (3) ρf = 0.50%.The numerical analyses have been performed by referring to the main geometrical and materialproperties deduced by the experimental evidences reported in [19]. The proposed applicationconsiders the model parameters previously calibrated by the authors in [26].

Figures 6(a) and 7(a) point out the cracked configurations at ultimate states on both exper-imental tests by [19] and numerical simulations. In both cases, failure process leads to onlyone macro-crack, starting at the top of the notch. Regarding the specimens with pre-fixed notchat mid-length of the beams, it can be noted that the crack evolves in an almost vertical direc-tion (mode I of fracture), while an inclined crack path (mixed fracture modes) can be observed

11


Plain concrete ρf = 0.25 % ρf = 0.50 %

(a)

(b)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.50 1.00 1.50 2.00

load

[k

N]

deflection [mm]

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.50 1.00 1.50 2.00

load

[k

N]

deflection [mm]

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.50 1.00 1.50 2.00

load [kN

]deflection [mm]

Figure 7: Test results on both plain concrete (PC) and SFRC beams under three-point bending with eccentricalnotch by [19]: (a) cracked configuration at ultimate state and (b) load-deflection comparisons (experimental datagiven with dotted lines).

in case of eccentrical notch. Furthermore, Figs. 6(a) and 7(a) outline that the crack evolvesespecially at the aggregate-matrix interfaces, which in fact represent the weakest interface inconcrete composites. The numerical results confirm the predictive capability of this formula-tion in terms of fracture behavior and failure modes.

Moreover, Figs. 6(b) and 7(b) show the force-deflection curves against the correspondingexperimental results in both cases. The load-displacement response of SFRC beam tests em-phasize the significant influence of the fiber-related effects which mainly influence the peakstrength and the post-peak behavior of the FRC. Fiber bridging effect on cracked concrete iswell captured by the proposed meso-mechanical approach. Thus, the laws introduced for sim-ulating both bridging effect and dowel action are mechanically consistent, though they requirea proper calibration whose values clearly depends on the nature of fiber and their bond to theconcrete matrix.

Finally, the peak load mainly depends on test configuration, obtaining lower strengths inthose cases characterized by failure mode I respect to the higher resistance obtained in mixedmodes of loading.

4.2 Microplane predictions

Numerical predictions carried out in both plain and SFRC by adopting the microplane for-mulation, given in Section 3, aimed at analyzing tensile test cases, are presented in this section.The comparisons between model predictions and the experimental data by [39] are reportedbetween Fig. 8 and 9. The same set of experimental tests have been already analyzed by the

12


authors in [13], where the stress-strain responses were mainly realized by using the interfacemodel presented above.

Table 3: Fiber types employed in the experimental tests by [39].

Density [g/cm3] df [mm] lf [mm] σy,s[GPa] Es[GPa]Dramix type I 7.8 0.5 30 1.20 200Dramix type II 7.8 0.5 50 1.20 200

The considered SFRC specimens contain two fiber types, namely “Dramix type I” and“type II” whose fundamental characteristics are given in Table 3. The model parameters ofthe proposed numerical analyses, adjusted according to the experimental data given in [39],result: Ec = 39.5GPa and ν = 0.20, tanφmic0 = tan βmic = tanφmicr = 0.6, χmic0 =4.0MPa, cmic0 = 7.0MPa, GI

f = 0.12N/mm, GIIaf = 1.2N/mm . While, the param-

eters of the fiber-to-concrete mechanisms are: τy,a = 1.95MPa, kE = 52.5MPa/mm andkS = 1.70MPa/mm for the bond-slip strength; κ1 = 6.5, fc = 10 · χ0 and kdow = 0.23 for thedowel effect.

The stress-strain response for SFRCs with steel “Dramix type II”, characterized by fibercontents ranging between 3.0% to 4.0%, are given in Fig. 8. While Fig. 9 depict the numericaland experimental comparisons of SFRC tests characterized by “Dramix type I”, whose fibercontents are 6.0%, 7.0% and 8.0%, respectively.

The numerical predictions compared against experimental results demonstrate a very goodagreement. In fact, the model predicts in a very realistic mode the mechanical response of theanalyzed SFRC specimens.

It is worth noting that all the other numerical curves have been obtained by just changing thefiber contents (ρf ) and/or fiber types (e.g., lf ) according to the experimental properties. Thisaspect is the key advantage when the fiber effects are explicitly modeled, giving the possibilityof modeling variation in the macroscopic stress-strain response by just changing the fiber typeand geometry.

5 CONCLUSIONS

In this work two different approaches were developed for failure analyses of Fiber Rein-forced Comcrete (FRC). On the one hand, a discrete crack approach aimed at simulating thefracture behavior of FRC at mesoscopic level is proposed. Then, a macroscopic smeared-crackapproach based on the microplane theory at material level was developed. Both constitutivemodels consider the well-known “Mixture Theory” to simulate the combined bridging inter-actions of fibers in concrete/mortar cracks. The interactions between steel fibers and concretematrix associated with debonding and dowel mechanisms are explicitly considered in both con-stitutive formulations.

The numerical simulations at different scales of observation demonstrate that the constitutiveproposals mainly capture the fundamental behaviors of FRC. Very good agreement betweennumerical against experimental data, available in scientific literature, are achieved in terms ofpeak-strength and post-crack predictions.

ACKNOWLEDGMENT

The authors acknowledge the financial support for this work by CONICET (Argentine Na-tional Council for Science and Technology) through the Grant No. PIP 112-200801-00707,

13


°

°

°°°

°°

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ρ f 3.0

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

0

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4

6

8

10

12

ε

σM

Pa

°

°

°°°

°°

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ρ f 3.5

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

0

2

4

6

8

10

12

ε

σM

Pa

°

°

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°°

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ρ f 4.0

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030

0

2

4

6

8

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ε

σM

Pa

Figure 8: Comparison between the numerical predictions and the experimental results (dotted curves) by [39]considering the SFRC with “Dramix type II”.

by CIUNT (Research Council University of Tucuman) through the Grant No. E/26 479, byUniversity of Buenos Aires through the Grant No. 20020090100 139 and by the “EnCoRe”Project (FP7-PEOPLE-2011-IRSES n 295283; www.encore-fp7.unisa.it) funded by the Euro-pean Union within the Seventh Framework Programme.

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CONSTITUTIVE MICROPLANE AND INTERFACE LAWS FOR MULTISCALE ANALYSIS OF STEEL FIBER CONCRETElaboratorios.fi.uba.ar/lmni/paper caggiano/Proceedings... · 2012. 9. 21. · 2.1 Composite