Modeling of High-Strength FRC Structural Elements withSpatially Non-Uniform Fiber Volume FractionPetr Kabele , Tereza Sajdlová Milan Rydval, , Jiří Kolisko

Journal of Advanced Concrete Technology, volume ( ), pp.13 2015 311-324

Corrosion Durability and Structural Response of Functionally-Graded Concrete BeamsMohamed Maalej, Shaikh F.U. Ahmed P. Paramasivam,Journal of Advanced Concrete Technology, volume ( ), pp.1 2003 307-316

Nonlinear Analysis for Reactive Powder Concrete Beams under Rapid Flexural LoadingsKazunori Fujikake, Takanori Senga Nobuhito Ueda, , Tomonori Ohno and Makoto KatagiriJournal of Advanced Concrete Technology, volume ( ), pp.4 2006 85-97

Use of Slurry Infiltrated Fiber Concrete in RC Corner Connections Subjected to Opening MomentsHamed M. Salem, Mohammed Elnono Ahmed M. Farahat, , Ashraf H.ElzanatyJournal of Advanced Concrete Technology, volume ( ), pp.7 2009 51-59

Exploring the Potential of the Functionally Graded SCCC for Developing Sustainable Concrete Solutions.Olga Rio , Viet D.Nguyen Khan Nguyen,Journal of Advanced Concrete Technology, volume ( ), pp.13 2015 193-204

Journal of Advanced Concrete Technology Vol. 13, 311-324, June 2015 / Copyright © 2015 Japan Concrete Institute 311

Scientific paper

Modeling of High-Strength FRC Structural Elements with Spatially Non-Uniform Fiber Volume Fraction Petr Kabele1*, Tereza Sajdlová2 , Milan Rydval3 and Jiří Kolísko4

Received 7 January 2015, accepted 13 June 2015 doi:10.3151/jact.13.311

Abstract The research presented in this paper has been motivated by the need to numerically simulate performance of high strength fiber reinforced concrete (HSFRC) structural elements with given spatially variable fiber volume fraction. The intended applications include prediction of load and deformation capacity of HSFRC members with imperfect fiber dis-tribution or design and verification of functionally graded HSFRC members. In order to achieve a predictive capability, modeling is based on micromechanics of fiber debonding, pullout and crack-bridging. The concept of cohesive crack is employed for implementation in the finite element method (FEM). A strong emphasis is placed on the feasibility of the model identification. To this end, a procedure which uses data from conventional notched-beam fracture tests and in-verse analysis to determine the model parameters is proposed. The model and the identification method are verified and validated both on micro and macro scales by comparing predicted results with experimental data.

1. Introduction

Growing demands of the construction industry toward reducing the weight and volume of structures while in-creasing their load capacity and durability has been ad-dressed, among other means, by the development of modern materials, which consist of dense fine-grained cementitious matrix reinforced by short fibers (usually made of steel). A prominent position among these mate-rials belongs to the Ultra-High Performance Fiber Rein-forced Concretes (UHPFRC’s). To qualify as UHPFRC the material should exhibit characteristic compressive strength ranging up from 150 MPa, direct tensile strength over 7 MPa, and the fiber reinforcement should be sufficient to prevent it from brittle failure (AFGC/SETRA 2013; FHWA 2013; JSCE 2006). Nev-ertheless, fiber reinforced concretes (FRC), which do not strictly meet the criteria of UHPFRC but still exhibit compressive strength in the range of 100-150 MPa, ten-sile strength over 5 MPa, quasi-brittle behavior, and superior durability offer a good opportunity for applica-tion as well (e.g. Walraven 2009; Kolísko et al. 2012). Their popularity can be attributed mainly to their better affordability resulting from lower price and easier pro-duction (simpler mix design, possibility to use locally

available constituents and conventional mixing equip-ment, less strict requirements on production discipline and curing). Nomenclature for these FRC’s is not as rigorously established as in the case of UHPRFC. In the present paper, we will call them high-strength fiber rein-forced concretes (HSFRC’s).

Owing to their properties, HSFRC’s can be used to design structural elements in which conventional rein-forcement by steel bars is greatly reduced or even com-pletely eliminated. As thinner or no concrete cover is required to protect reinforcement from corrosion and due to the high material strengths, the structural ele-ments can be much slenderer than when ordinary con-crete is used. When such an element is exposed to bend-ing, its load bearing capacity is usually determined by the material’s tensile behavior. In terms of response un-der uniaxial tension, HSFRC’s can be classified as sof-tening materials, that is, the stress starts to decrease as soon as the concrete cracks. However, the stress de-crease is gradual, which is mainly attributed to the crack-bridging effect of fibers. These materials typically exhibit only one or a few localized cracks under tension. Consequently, the tensile behavior is characterized by a relation between the stress transmitted across a crack (cohesive traction or bridging traction) and the crack width. We will also use the established terms “traction-separation relation” or “cohesive relation/law” for this property throughout this paper.

It follows from the discussion above that any non-uniformity of the fibers’ spatial distribution and orienta-tion has a strong impact on the load capacity of struc-tural elements. Correlations among the workability of the FRC fresh mix, the mixing and casting procedure, the fiber distribution and the material strength or struc-tural member load capacity have been extensively stud-ied (e.g. Barnett et al. 2010; Benson et al. 2005; Benson

1Professor, Faculty of Civil Engineering, Czech Technical University in Prague, Prague, Czech Republic*Corresponding author, E-mail: petr.kabele@fsv.cvut.cz 2Ph.D. student, Faculty of Civil Engineering, Czech Technical University in Prague, Prague, Czech Republic3Ph.D. student, Klokner Institute, Czech Technical University in Prague, Prague, Czech Republic. 4Director, Klokner Institute, Czech Technical University in Prague, Prague, Czech Republic.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 312

and Karihaloo 2005; Ferrara et al. 2010; Kang et al. 2011; Kang and Kim 2012; Martinie et al. 2013; Wille and Parra-Montesinos 2012). Some of these findings have been implemented in recommendations for mate-rial design and production of UHPFRC (AFGC/SETRA 2013; JSCE 2006). The provision by AFGC/SETRA (2013), for example, introduces experimentally obtained K-factors to account for the effects of fiber alignment. Nevertheless, it still has been of an interest to research-ers and designers to predict, analytically or numerically, the effect of imperfect fiber distribution on the structural performance, especially if HSFRC or UHPFRC is used without conventional reinforcement. This motivation has been further elevated by the recent research of func-tionally graded fiber reinforced cementitious composites (e.g. Dias et al. 2010; Shen et al. 2008), where non-uniform fiber distribution is introduced in a controlled manner in order to optimize the properties of structural members. Park et al. (2010), for example, used a cohe-sive fracture model adapted for non-uniform material to simulate the performance of functionally graded FRC beams by the finite element method (FEM). They as-sumed a multi-linear cohesive law, which was defined using the fracture energy associated with matrix and fiber bridging. The model, however lacked any direct link to the fiber volume fraction and other microme-chanical characteristics of the composite. Kang and Kim (2012) employed the micromechanical model proposed by Li et al. (1991) to predict the cohesive law of UHPFRC using an experimentally measured non-uniform distribution of fibers’ spatial orientation. The cohesive law was then approximated by polynomial and linear functions and used in a finite element model to reproduce the results of bending tests on UHPFRC beams. It is noted that vast literature exists on models relating the tensile behavior of fiber-reinforced brittle-matrix composites to the fiber volume fraction and other micromechanical properties of fibers, matrix and their interface. Some will be cited later in the text. For a broader overview, a reader can refer, e.g., to Bentur and Mindess (2006).

The research presented in this paper has been moti-vated by the desire to numerically simulate performance of HSFRC structural members with given spatial varia-tion of the fiber volume fraction. The material non-uniformity may be induced inadvertently or intention-ally. In the former case, the proposed methodology can be used, for example, to predict the bounds of load and deformation capacity of HSFRC members if imperfect fiber distribution occurs due to flaws in the production process. In the latter case the methodology can be used to assist design and verification of functionally graded HSFRC members. Given the anticipated practical use, the modeling approach should meet the following re-quirements. On one hand, it should take into account the dominant micromechanical phenomena that govern the tensile response of HSFRC materials, such as, matrix cracking, fiber debonding and pullout, and crack bridg-

ing effects. On the other hand, it should be possible to identify the model parameters based on a reasonable number of easy-to-perform tests. To this end, the pro-posed approach employs a combination of microme-chanical modeling, inverse analysis and the FEM.

2. Modeling

2.1 Modeling concept Since our objective is to simulate the mechanical behav-ior of structural elements of arbitrary shapes and under general loading conditions, modeling is based on the finite element method. We constrain our attention to the class of tension-softening or low tension-hardening HSFRC’s and UHPFRC’s. Thus the post-cracking ten-sile response can be modeled using the concept of a cohesive crack (Hillerborg et al. 1976). In this approach, a crack is perceived as a displacement discontinuity, which, however, is capable of transferring stress be-tween its faces. This cohesive stress is related to the crack opening displacement through a traction-separation relationship. The mathematical form and the dependence of this relation on the fiber volume fraction and other micromechanical parameters will be discussed in detail in subsections 2.2 to 2.4. Particular aspects of employing the derived cohesive law in the FEM frame-work will be then outlined in subsection 2.4 2.2 Traction-separation relationship of cementi-tious matrix When formulating the composite traction-separation relationship we consider that the cohesive effects on cracks in FRC materials result from the combination of matrix bridging and fiber bridging. The matrix bridging model accounts for the growth and coalescence of mi-crocracks and subsequent aggregate interlock on the macrocrack faces. As fine-gained matrices are usually used in HSFRC and UHPFRC, the matrix exhibits a steep tension-softening behavior. We approximate the matrix traction-separation relation by the following function, which was proposed by Hordijk (1991):

( ) ( )3

3

, 1 2 1 2

max, max, max,

,( ) 1 exp 1 expb m

m m m

t m c c c cδ δ δ

σ δ σδ δ δ

= + − − + −⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎪

⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

(1)

where ,b mσ is the matrix bridging traction, δ is the crack opening displacement (COD), σt,m is the matrix cracking strength, δmax,m is the crack width at which the matrix cohesive stress is completely released and c1, c2 are fitting parameters. Although the function in Eq. (1) has been originally proposed for ordinary concrete, we will show later in subsection 4.1 that it can adequately represent the data experimentally obtained for high-strength cementitious matrices. In general, the matrix cracking strength of a composite may depend on the fiber content. However, since in the present study we constrain our attention to cases when the fiber volume fraction varies in a narrow range between 0% and 3% we neglect this effect.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 313

2.3 Traction-separation relationship due to fiber bridging To describe fiber bridging we adapt the theoretical deri-vation set forth by Li (1992) and Li and Leung (1992). We consider that the fiber bridging effect can be ex-pressed by cohesive traction, which corresponds to the sum of forces carried by fibers spanning faces of a crack of a unit area. As the crack opens, each individual fiber undergoes debonding and pullout from the matrix. These processes have been studied and modeled by many researchers in the past (e.g. Wang et al. 1988; Naaman et al. 1991; Lee et al. 2010 and references cited there).

During the debonding phase, part of the fiber remains bonded to the matrix, while the pulled side gradually detaches from the matrix. The debonded part of the fiber undergoes elastic stretching, which results in relative slip at the fiber-matrix interface while frictional bond acts on the interface. As the relative slip in this phase is very small (on the order of 1-10 μm), the frictional bond strength may be assumed to remain constant with re-spect to the slip displacement. Naaman (1972), however, has found that the average frictional bond strength de-creases with increasing fiber density (expressed as the number of fibers per unit crack area Nb/Ac). This phe-nomenon is sometimes referred to as fiber group reduc-tion effect. Following Naaman (1972), we assume that this relation is linear:

0 1 bs

c

Na

Aτ τ

⎛ ⎞= − ⋅⎜ ⎟

⎝ ⎠ (2)

where τs is the initial average bond strength of a single fiber in a large body of matrix, such that Nb/Ac → 0 and a is a fitting parameter.

Li and Leung (1992) used the shear-lag theory to ex-press the fiber pullout force Pd during the fiber debond-ing phase as:

( ) 301

2f

d f fP E d e ϕπ η τ δ= + ⋅ (3)

where Ef and df are fiber Young modulus and diameter, respectively, ( )1f f m fE V E Vη ⎡ ⎤= −⎣ ⎦ , Vf is the fiber volume fraction, Em is matrix Young modulus. The ex-ponential term accounts for fiber snubbing at the point where it exits matrix; f is the snubbing friction coeffi-cient, and ϕ is the angle between fiber embedment di-rection and pullout direction. Note that Eq. (3) applies to each individual fiber from a group of fibers bridging a crack, thus we use Eq. (2) for τ0. Variable δ stands for the crack opening displacement; it is considered that fiber debonding takes place on both sides of the crack and thus the fiber slip against matrix at the crack faces is δ/2.

Debonding is completed once the interface becomes separated from the matrix along the entire embedded length Le of the fiber. The corresponding crack opening displacement is according to Li and Leung (1992):

( )24

1e 0

0f f

LE dτ

δη

=+

(4)

Subsequently, the process of fiber pullout starts. In this phase, the fiber slides from the matrix, as a result of which their contact area decreases. The main resisting mechanism is the friction at the fiber-matrix interface. Experimental studies (e.g. Wille and Naaman 2010) show that the frictional stress significantly varies with increasing fiber slip. We will show later that this behav-ior can be well described by a polynomial function:

2 3

0 1 2 3( ) 1f f f

b b bd d d

τ τ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ Δ Δ⎢ ⎥Δ = + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

(5)

where Δ is the relative slip of a fiber against matrix and b1 thru b3 are fitting parameters. The fiber group effect is again taken into account by using Eq. (2) for τ0. Throughout the forthcoming derivations we will con-sider that only the shorter embedded part of a crack-bridging fiber undergoes pullout while the other part remains partly bonded to the matrix and that the fiber displacement due to its elastic deformation is negligible compared to its total slip. Thus the slip Δ is assumed to be equal to the crack width δ. The fiber force Pp for the pullout phase can be expressed following (Li 1992). Note that while (Li 1992) assumed the interfacial fric-tion τ to be constant, we use Eqs. (5) and (2) to express its dependence on the fiber-matrix slip and group reduc-tion effect, respectively:

( ) ( ) 1 fp e f

e

P L d eL

ϕδδ πτ δ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(6)

To obtain the cohesive traction acting on crack faces due to fiber bridging, the forces carried by individual fibers spanning a unit area of a crack must be integrated while taking into account the fibers’ random spatial dis-tribution and orientation. To this end, we follow the pro-cedure described by Li (1992). Using Eqs. (2) through (6), we obtain the result as:

12* *

, * *( ) 2 forb fδ δσ δ σ δ δδ δ

⎡ ⎤⎛ ⎞⎢ ⎥= − <⎜ ⎟⎢ ⎥⎝ ⎠

⎣ ⎦

(7)

2 3 2

*

, 1 2 3

*

2( ) 1 1

2

b f

f f f f

f

b b bd d d L

for L

δ δ δ δσ δ σ

δ δ

= + + + −

≤ ≤

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (8)

where

( )

22

*

21

1

ff s

f

f f

VL a

d

E d

τπ

δη

⎛ ⎞− ⋅⎜ ⎟⎜ ⎟

⎝ ⎠=+

(9)

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 314

*2

21

2f f

f sf f

L VgV a

d dσ τ

π⎛ ⎞

= − ⋅⎜ ⎟⎜ ⎟⎝ ⎠

(10)

Parameter Lf stands for the fiber length and ( ) ( )/ 2 22 1 4fg e fπ= + + is the snubbing factor de-

fined by Li (1992). In the above derivation we consid-ered that for randomly distributed and oriented fibers with circular cross section, the number of bridging fi-bers per crack area can be expressed according to Naaman (1972) as:

2

212

f fb

c f f

V VNA A dπ

= = (11)

2.4 Composite traction-separation relationship The composite traction-separation relationship is ob-tained by summing up the contributions due to bridging effects of matrix (Eq. 1) and fibers (Eq. 7 or 8):

, ,( ) ( ) ( )b b m b fσ δ σ δ σ δ= + (12)

By reviewing the above equations it is obvious that the traction-separation relationship is now defined in terms of several micromechanical parameters, which are all independent of the fiber content (σt,m, δmax,m, c1, c2, τs, a, Ef, df , Lf, Em, g, b1 thru b3) and the fiber volume frac-tion Vf. If all the former parameters are properly deter-mined, and the spatial distribution of Vf is given, Eq. (12) can be used to model the spatially variable cohe-sive law in structural elements with non-uniform fiber content.

2.5 Finite element implementation Numerical simulations of HSFRC structural elements discussed hereafter are carried out with the finite ele-ment program Atena v. 5 (Červenka et al. 2008). The package includes implementation of several variants of the fracture-plastic material model described by Čer-venka and Papanikolaou (2008). In this class of material models, compressive behavior is treated by the theory of plasticity with Menétrey-Willam yield/failure surface. Failure in tension is represented by a fracture model with Rankine failure criterion. The fracture model is based on the smeared crack formulation and crack band approach. This means that the cohesive traction vs.

crack opening displacement relation is transformed into stress vs. inelastic strain relationship; however, by using an element-dependent characteristic length in the proc-ess, the result remains objective with respect to the size of finite elements used. We use a version of the fracture-plastic model called CC3DNonLinCementitious2User, which accepts a user-defined traction-separation law as an input. We utilize the σb-δ data obtained by Eq. (12) for this purpose. 3. Materials

Calibration and application of the models described in the preceding section will be demonstrated using data from experiments on specimens of a HSFRC material with fiber content varying between 0% and 3% by vol-ume (Rydval et al. 2014). The matrix consisted of ce-ment, microsilica, slag, silica fume, quartz sand, super-plasticizer, and water. Brass-coated high-strength steel fibers were used as reinforcement. Details of the mix composition and fiber parameters are listed in Tables 1 and 2, respectively. The matrix composition was kept constant for all composites, except for superplasticizer content, which was slightly increased for mixes with higher Vf to improve workability. In addition to fracture and bending tests, which will be described in the follow-ing sections, compressive strength was measured on cast 100 mm cubes and fragments of notched beams (160×40×40 mm) according to the CSN EN 196-1 stan-dard (CSI 2005). Prior to destructive testing, the beams were also used to measure the dynamic Young modulus according to the CSN EN 12504-4 (CSI 2002). The re-sults are summarized in Table 3. All test were con-ducted 34 days after casting.

Table 1 Matrix composition of tested HSFRC’s (in weight ratio per unit of cement).

Sand, fraction [mm] Cement 1) Microsilica 2) Slag 3) Silica fume4) Superplasticizer 5) 0.3 – 1.66) 0.1 – 0.57) Water

1 0.143 0.114 0.071 0.072-0.075 1.121 0.564 0.243 1) CEM II A-S/ 42.5R; 2) Elkem 940-U; 3) Dětmarovice; 4) Minorit MT 300; 5) Glenium G300, 6) Provodínské písky PR30,7) Provodínské písky PR33

Table 2 Parameters of fibers (as provided by the supplier).

Fiber type Material Lf [mm] df [mm] Ef [GPa] Tens. strength [MPa] BASF MASTERFIBER® 482 brass coated steel 13 0.2 200 ≥ 2200

Table 3 Measured compressive strengths and Young moduli.

Material Vf [%]

Cube100 mm,strength [MPa]

Fragment ~80×40×40 mm, strength [MPa]

Beam 160×40×40 mm,

Young mod. [GPa]

Vf-00 0.0 89.9 108.1 49.2 Vf-10 1.0 N/A 126.9 51.4 Vf-15 1.5 111.4 129.0 49.6 Vf-20 2.0 N/A 145.0 52.2 Vf-30 3.0 117.8 148.6 51.4

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 315

4. Identification of model parameters

In this section we propose a pragmatic methodology for identification of the parameters defining the traction-separation relation developed in section 2. We recall that our principal goal is a simulation of fracture in HSFRC members with non-uniform fiber content but random fiber orientation, which can be encountered, for exam-ple, in functionally graded structures. Therefore, the fiber volume fraction Vf will be considered as a parame-ter whose spatial distribution is known for a given prob-lem.

Then, there is a group of parameters, whose values are usually provided by a fiber manufacturer or can be easily measured directly: the fiber length Lf , diameter df, and elastic modulus Ef. Values of these parameters for fibers used in the present study are listed in Table 2. Determination of the remaining parameters, which de-scribe the fracture properties of the matrix and the fiber-matrix interaction will be discussed in more detail in the next subsections.

4.1 Parameters of matrix bridging Matrix bridging effect is described by Eq. (1), which involves the matrix cracking strength σt,m, crack width at which the matrix cohesive stress is completely re-leased δmax,m and parameters c1 and c2, which affect the shape of the cohesive curve. These parameters can be calibrated by carrying out a series of direct tension tests on plain matrix without fibers and fitting Eq. (1) to the measured stress vs. crack width data. However HSFRC’s are usually produced with high-strength fine-grained matrices, which are very brittle. Performing a direct tension test on such materials is extremely diffi-cult, as it requires very stiff testing frame and precise displacement control and measurement. Therefore, we opt for the indirect method described hereafter.

First, fracture tests are performed on notched beams under three-point bending. We used beams with dimen-sions of 400×100×100 mm, span 300 mm and notch depth 20 mm. Loading was controlled by the crack mouth opening displacement (CMOD). The crosshead displacement rate was 0.05 mm/min up to CMOD of 0.1 mm; subsequently, it was increased to 0.2 mm/min. The applied force, deflection, crack mouth opening dis-placement and crosshead displacement were monitored and recorded. The test series included three specimens and the results are plotted in Fig. 2.

Subsequently, inverse analysis is applied to each measured load-deflection (F-u) curve to obtain data of the cohesive relation. The inverse analyses can be con-veniently performed using program Consoft, which is based on an evolutionary algorithm with local neighbor-hood attraction (Slowik et al. 2006). The program, which is available at the author’s website (Slowik 2014), allows a user to select from several softening functions and to specify limits in which the function’s parameters are optimized so as to minimize the error between the

measured and reproduced load-deflection curve. We selected the tri-linear function, which allowed us enough flexibility to capture the cohesive fracture be-havior of the matrix. Figure 3 shows the cohesive curves obtained for the three tested specimens.

As the next step, the traction-separation data from a test series involving several (three in our case) experi-ments on the same material should be combined to ob-tain an “average” cohesive law. Plain averaging of the tri-linear curves would result in a multi-linear relation-ship. However, for the forthcoming combination with fiber effects (Eq. 12) we prefer to approximate the ma-trix cohesive law by the smooth function specified in Eq. (1). This is achieved by least-square fitting Eq. (1) to the data obtained from the inverse analyses. To this end, the tri-linear curves must be first resampled at con-

Fig. 1 The finite element mesh used for the simulations of the three-point bending tests.

0

2

4

6

8

10

12

0.00 0.05 0.10 0.15 0.20 0.25 0.30

F [k

N]

u [mm]

specimen A

specimen B

specimen C

numerical reproduction

Fig. 2 Load-deflection curves of plain matrix notched beam specimens (400×100×100 mm): experiments and numerical reproduction.

0

2

4

6

8

10

12

14

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

σ b,m

[MP

a]

δ [mm]

spec. A, inverse analysis

spec. B, inverse analysis

spec. C, inverse analysis

model

Fig. 3 Traction-separation relations of plain matrix ob-tained by inverse analysis and approximation by Eq. (1).

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 316

stant intervals of COD so as to assign weight uniformly to all the branches - from our experience 50 samples proved to be sufficient. The fitting can be easily per-formed using an implementation of the nonlinear least-squares Marquardt-Levenberg algorithm, which is available in program gnuplot (Anonymous 2014). Since parameters c1 and c2 in Eq. (1) are strongly correlated with δmax,m, the value of δmax,m is calculated first as the average of COD of the last points of the tri-linear curves obtained by the inverse analyses (i.e., COD at which the cohesive traction becomes equal to zero). Then the re-maining parameters σt,m, c1, and c2 are identified by the least-square method. The values are listed in Table 4. The traction-separation curve defined by Eq. (1) and these parameters is plotted in Fig. 3. Note that the value of c1 = 3.05 obtained in the present study is very close to c1 = 3, which was proposed by Hordijk (1991) for ordinary concrete. Our value of parameter c2 = 8.86 is slightly higher than Hordijk’s c2 = 6.93. The higher value of c2 results in a steeper slope of the initial part of the softening curve, which is plausible for the fine-grained matrix of HSFRC.

To test the quality of the inverse analysis and ap-proximation, we numerically reproduced the fracture tests using Eq. (1) with parameters from Table 4 as the cohesive law in the model. The calculations were per-formed under plane stress conditions and nearly uniform mesh of bilinear quadrilateral finite elements with 16 elements in the ligament above notch was used (Fig. 1). It is obvious in Fig. 2 that a very close match was ob-tained, verifying of the applied method.

4.2 Parameters of fiber bridging The cohesive law describing the effect of fiber bridging is defined by Eqs. (7) and (8), which involve several parameters that yet remain to be identified. Reviewing Eq. (5) it is obvious that parameters τs, b1 b2 and b3 were introduced to characterize the frictional bond strength of the fiber-matrix interface. Most often, these parameters are determined by fitting Eq. (6) to results of single-fiber pullout tests. This method, however, poses several difficulties, such as, fabrication of special specimens and testing of tiny samples under very small forces. Fur-thermore, the results typically exhibit a significant scat-ter, which leads to the requirement of testing large sets of specimens. Therefore, we adopt a top-down approach, in which we extract these parameters from tests of the composite fracture behavior on a larger scale. The pro-posed method, besides being easier to execute and in-volving common testing procedure, also alleviates the problem of scatter among responses of individual fibers, as a large group of fibers is tested at once.

First, notched HSFRC beams are tested in three-point

bending. In order to calibrate parameter a, which cap-tures the reduction of the apparent fiber bond with in-creasing fiber density (group effect - Eq. (2)), at least two series of beams with different fiber volume frac-tions need to be tested. We tested two sets with Vf = 1.5% and Vf = 3%. Each series comprised of three specimens (however, only two valid datasets were ob-tained for Vf = 3%). Specimen dimensions, loading con-ditions and monitoring were the same as in the case of tests on plain matrix (see subsection 4.1). The results are shown in Figs. 4 and 5.

Next, an inverse analysis with Consoft program is applied on each of the obtained sets of load-deflection data. Multi-linear cohesive relation with six segments was used to allow sufficient flexibility while capturing the composite fracture behavior. Thus, for each data set it was necessary to determine twelve parameters - COD’s and bridging tractions corresponding to the six points defining the piecewise linear function. Note that the first point corresponds to zero COD and the last point to zero cohesive traction. As is obvious from Figs. 4 and 5, all beams exhibited a mild softening behavior, which implies that the “tail” of the traction-separation relationship, which is associated with fiber bridging, had a major effect on the response. Thus, it can be an-ticipated that the inverse analysis would not capture

Table 4 Parameters of matrix traction-separation rela-tionship calibrated by the proposed method.

σt,m [MPa] δmax,m [mm] c1 [-] c2 [-]11.8 2.9×10-2 3.05 8.86

0

5

10

15

20

25

30

0 2 4 6 8 10

F [k

N]

u [mm]

specimen A

specimen B

specimen C

numerical reproduction

Fig. 4 Load-deflection curves of notched beams (400×100×100 mm) made of HSFRC with Vf = 1.5%: experiments and numerical reproduction.

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

F [k

N]

u [mm]

specimen A

specimen C

numerical reproduction

Fig. 5 Load-deflection curves of notched beams (400×100×100 mm) made of HSFRC with Vf = 3%: ex-periments and numerical reproduction.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 317

accurately the initial part of the cohesive relation at small crack widths, which is dominated by matrix bridg-ing. For this reason the value of the traction at the first point of the sought cohesive curve (i.e., the composite cracking strength) was set to the average matrix strength σt,m which was obtained in section 4.1 (see Table 4). Furthermore, we considered that according to the theo-retical model of fiber bridging (Eq. 8), the cohesive traction should vanish when COD reaches the value of Lf/2. Therefore the value of COD at the last point of the sought curve was set equal to Lf/2. Consequently, only ten parameters had to be identified by the inverse analy-sis for each test. The obtained multilinear softening curves are shown in Figs. 6 and 7.

The cohesive relations obtained by the above proce-dure represent the combined matrix and fiber bridging effect. To isolate the cohesive traction σb,f (δ) due to fibers only, we have to subtract the cohesive traction due to matrix σb,m (δ) from the total composite traction σb (δ), as evident from Eq. (12). To this end, we use the “average” matrix cohesive law, which was identified in subsection 4.1 (Eq. (1) with parameters from Table 4), and the composite multilinear traction-separation rela-tions determined in the previous paragraph. Subse-quently, we obtain the fiber bridging traction vs. COD data for each of the tested composite beams. The proce-dure is illustrated for one of the specimens in Fig. 8. The figure shows that due to the rather coarse represen-tation of the composite cohesive curve, the fiber bridg-ing traction at very small crack widths in the pre-peak region cannot be relied upon. Therefore, we use only the post-peak data and, accordingly, Eq. (8) to identify the parameters of the fiber bridging model.

Turning our attention to Eq. (8) we can see that the general shape of the softening curve, which is deter-mined by the second and third term, is independent of fiber volume fraction, while only the first term, σ*, in-volves Vf (Eq. 10). Thus, as the next step, we perform multi-branch least-square fitting of Eq. (8) to the σb,f -δ data obtained in the preceding paragraph. To this end we use the data of all specimens and all tested volume frac-tions. The fitting parameters are b1, b2, and b3 (common for all data) and σ*(Vf) (different values of σ* are con-sidered for each tested volume fraction). In our case, valid experimental data were obtained for three beams with Vf = 1.5% and two beams with Vf = 3%. Thus, fit-ting was executed on two sets of data (branches) yield-ing the values of σ*(Vf = 1.5%), σ* (Vf = 3%), b1, b2, and b3 (Table 5).

Since σ* is now known for, at least, two different fi-ber volume fractions, Eq. (10) can be used to determine the remaining parameters τs and a, which characterize the initial fiber-matrix frictional bond strength and its

dependence on Vf (Eq. 2). If σ* has been determined for two volume fractions, then the parameters are solved directly; otherwise, they are obtained by least square fitting Eq. (10) to the σ*-Vf data. The values of τs and a determined from our test data are listed in Table 5. Note, however, that the described procedure relies on the con-dition that all remaining parameters in Eq. (10) except Vf are known. This is true for Ld and df (see the first paragraph of section 3), but not for the snubbing factor g. This parameter is related to the mechanism of snub-bing (or belt friction), which is assumed to occur during pullout of an inclined fiber at the point, where the fiber exits matrix (Li 1992). If we want to keep the concept of the present top-down methodology and refrain from micromechanical testing, there is no way to determine this parameter unless, for example, composites with aligned and randomly oriented fibers are tested. As such testing was not feasible within the scope of our study,

σb

[MP

a]

δ [mm]

spec. A, inverse analysisspec. B, inverse analysisspec. C, inverse analysismodel

0

2

4

6

8

10

12

0 1 2 3 4 5 6

Fig. 6 Traction-separation relations of HSFRC with Vf =1.5% obtained by inverse analysis and approximation by Eq. (12).

Table 5 Parameters of fiber bridging calibrated by the proposed method.

σ*(Vf = 1.5%) [MPa] σ*(Vf = 3%) [MPa] b1 [-] b2 [-] b3 [-] τs [MPa] a [mm2] g1) [-] 7.05 8.72 1.49×10-2 -5.92×10-3 2.72×10-4 10.73 1.16 1.86

1) Value adopted from (Li et al. 1993)

σb

[MP

a]

δ [mm]

spec. A, inverse analysisspec. C, inverse analysismodel

0

2

4

6

8

10

12

0 1 2 3 4 5 6

Fig. 7 Traction-separation relations of HSFRC with Vf =3% obtained by inverse analysis and approximation by Eq. (12).

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 318

we adopted the value of g = 1.86 (which corresponds to f = 0.75) from the literature (Li et al. 1993).

4.3 Verification of the cohesive law identifica-tion method By completing the previous steps, all parameters of the matrix (Eq. 1) and fiber (Eqs. 7 and 8) bridging models are calibrated. The combined cohesive effect is then expressed by Eq. (12). Results produced by this equa-tion with parameters from Tables 2, 4, and 5 are plotted together with the σb -δ data obtained from the inverse analyses in Figs. 6 and 7. It is obvious that the proposed model captures well the tendency of the data, especially in the “tail” part, which is controlled by the fiber bridg-ing.

To validate the proposed methodology of model iden-tification, we numerically reproduced with FEM the fracture tests on notched beams for Vf = 1.5% and Vf = 3%. To this end, Eq. (12) and parameters from Ta-bles 2, 4, and 5 were used to define the traction-separation relation of the material model (subsection 2.5). The calculations were carried out under plane stress conditions using the FE mesh shown in Fig. 1. Figures 4 and 5 show that the numerical results match very closely the experimental data, which verifies soundness of the proposed identification method.

5. Discussion of the obtained micro- and meso-mechanical characteristics

Since we have identified various micromechanical pa-rameters (Tables 2, 4, and 5) of the HSFRC material from macro-scale tests, it is of some interest to discuss the findings and compare them with those found by other authors.

Wille and Naaman (2010) experimentally investi-gated the pullout behavior of high strength straight brass-coated steel fibers from ultra high performance cementitious matrices, that is, on similar constituents as those used in the present study. They performed pullout tests on individual fibers, from which they directly ob-tained the force-slip (P-Δ) data. From these data they calculated the slip-dependent frictional bond stress τ(Δ)

using Eq. (6), and the equivalent bond strength, which is defined using the pullout work as:

02

2 ( )eL

eqf e

P d

d Lτ

π

Δ=

Δ=Δ Δ

= ∫ (13)

All of these characteristics can be readily retrieved for the HSFRC tested in the present study using the model described in section 2 with parameters calibrated in section 4. To this end, we consider straight pullout (ϕ = 0) of a single fiber from plain matrix (Vf = 0%) with embedment length Le = Lf/2 = 6.5 mm (the same as Wille and Naaman (2010)). The resulting force-slip curve according to Eqs. (3) and (6) is plotted in Fig. 9 and the corresponding relation between the frictional bond stress and slip is shown in Fig. 10. The curves fairly well correspond with the range obtained by the direct pullout tests by Wille and Naaman (2010). In par-ticular, the bond-slip relation exhibits the strong harden-ing character, which was pointed out by Wille and Naaman (2010) and was attributed to scratching of the fiber surface and clogging of the interface with debris. The obtained maximum pullout force Pmax = 43.8 N and the equivalent bond strength τeq = 11.3 MPa fall within the bounds reported in the cited publication: 22.0 - 57.8 N and 6.1 - 22.1 MPa, respectively.

Fig. 8 Isolation of the fiber bridging traction from the result of inverse analysis of a fracture test on a composite beam(example for Vf = 1.5%, specimen B). Left: detail for small crack width. Right: complete curves.

0

10

20

30

40

50

60

0 1 2 3 4 5 6

P [N

]

Δ [mm]

range of experimental resultsmodel

Fig. 9 Result of a single fiber pullout test predicted by theproposed model compared with the range of experimen-tal results reported by Wille and Naaman (2010).

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 319

The model also allows us to investigate the effect of the fiber volume fraction on the crack-bridging cohesive traction. Figure 11 shows the cohesive curves predicted by the model for Vf varying between 0% and 4%. We can see in the graph on the left side that the curves for Vf > 0 exhibit a peak (denoted σb,max), whose value ex-ceeds the matrix strength (σt,m). This peak is attributed to the combined effect of the matrix and fiber bridging tractions, where the former initially decreases at lower rate than the latter increases (see Eqs. (1) and (7)). However, note that the peak occurs at very small COD < 1 μm and thus it may be interpreted as the actual cracking strength of the composite. Figure 12 shows, that σb,max initially slightly increases with increasing fiber content, but the tendency changes at higher Vf (3.5% in our case). In this way, the model may represent the slight variation of composite tensile strength with changing fiber content, which has been reported in some experimental studies (e.g. Wille et al. 2014). The right-side graph in Fig. 11 illustrates the complete cohesive curves including the tail portions, which are controlled solely by fiber bridging. As dictated by Eq. (8), the tails of the curves scale with σ* but otherwise their shape is independent of the fiber content. Equation (10) indicates that, due to the inclusion of the linear term that accounts for the fiber group effect, σ* is a quadratic function of Vf.

This dependence is illustrated in Fig. 12. A similar ten-dency was also observed by Wille et al. (2014). Figure 12 shows that the peak of σ* occurs at Vf = 2,7%. Thus, this value can be considered as the optimal fiber content, at which the fiber bridging is most efficient, for the ma-trix and fibers used in the present study.

6. Testing and numerical simulation of layered beams

As stated in the introduction, the main objective of the present study is to develop methodology for modeling of HSFRC structural elements with non-uniform spatial distribution of fibers. To validate the presented models and the calibration method, we carried out numerical simulations of layered HSFRC beams. The beams were physically tested, but the information from these tests was used only for comparison with the analytical results and not for the model calibration or adjustments. 6.1 Experiments The beams with dimensions of 400×100×100 mm con-sisted of four equally thick layers of HSFRC with fiber volume fractions of 3%, 2%, 1%, and 0% (from bottom to top). The material composition was the same as de-scribed in section 3. The specimens were produced by subsequent casting of the layers into a steel mold. The interval between emplacements of the individual layers

σb

[MP

a]

δ [mm]

σb,max

0

2

4

6

8

10

12

14

0 0.01 0.02 0.03 0.04 0.05

σb

[MP

a]

δ [mm]

Vf = 0%Vf = 1%Vf = 2%Vf = 3%Vf = 4%

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6

Fig. 11 Traction-separation relations predicted by the proposed model for different fiber contents. Left: detail for small crack widths. Right: complete curves.

Coh

esiv

e tra

ctio

n [M

Pa]

Vf [%]

σb,max

σ* 0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3 3.5 4

Fig. 12 Effect of fiber content on the maximum cohesive combined matrix and fiber bridging (σb,max) and fiber bridging only (σ* ... Eq .(10)).

0

10

20

30

40

50

60

0 1 2 3 4 5 6

τ [M

Pa]

Δ [mm]

range of experimental resultsmodel

Fig. 10 Frictional bond-slip relation retrieved by the pro-posed model compared with the range of experimentalresults reported by Wille and Naaman (2010).

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 320

was about 20 minutes, which implies that each new layer was cast while the material of the previous layer was still fresh. The experimental series consisted of three specimens.

The beams were tested in symmetric four-point bend-ing. The span was 300 mm and the distance between loading lines was 100 mm (see Fig. 14). The experiment was controlled by the crosshead displacement, which was applied at the rate of 0.25 mm/min. The acting force, deflection and crosshead displacement were monitored and recorded. In addition, the front surface (which had been sprayed with a random pattern before the test) was photographed by a high-resolution digital camera at regular intervals of 5 s. The acquired image data were subsequently processed by digital image cor-relation (DIC) software Ncorr (Balber et al. 2014), by which we retrieved the evolution of the displacement and strain fields during the tests. The obtained load vs. mid-span deflection curves are presented in Fig. 13. Figure 14 shows the distribution of maximum principal strain obtained by DIC at the loading stages indicated by points (a), (b), and (c) in Fig. 13. Note that the strain concentrations can be attributed to formation and open-ing of cracks. The figures show that, just prior to attain-ing the peak load (a), several cracks were propagating from the bottom side of the beam in the space between the load-lines. Interestingly, cracking was most intense away from the symmetry plane, with the crack on the right side being dominant. Figure 14 (b) shows that subsequent softening was associated with localization of fracture to the dominant crack, which, however, still exhibited some branching. As seen in Figure 14 (c), in the final stage of the experiment, the crack extended all the way to the upmost layer, where it showed a tendency to propagate toward the symmetry plane. A similar be-havior was observed for all the tested beams.

6.2 Numerical simulation The experiment on the layered beam was simulated by FEM program Atena (see subsection 2.5). A uniform mesh of four-node quadrilateral plane stress elements with size of 5 mm was used in the central part of the beam, while coarser mesh was generated toward the ends (Fig. 15). The model from section 2 with parame-ters identified in section 4 was used to predict the trac-tion-separation relationship for the HSFRC with differ-ent fiber volume fractions (Fig. 11). These relations were assigned through the program input to element groups corresponding to the individual layers. A perfect bond between layers was assumed. The analyses were carried out incrementally, typically in about 80 steps, using load control with the arc-length method, which allowed us to capture the softening response.

At first, unperturbed properties within each layer were assumed (model NP). Figure 16 shows that the analysis predicted higher peak load and post-peak load capacity than the experiment. Figure 17 depicts the crack patterns at loading levels denoted by points (a)

and (b) in Fig. 16. It is obvious that the calculation pre-dicted formation of multiple cracks in the central part, which exhibited nearly uniform opening all the way up to the peak load (a). In the post-peak phase (b), fracture localized into two major cracks, but still there were many open finer cracks. Comparing with the experimen-tal results (Fig. 14), it must be admitted that the analysis did not capture well the fracture localization. However, this is quite a common problem, which can be attributed to the fact that, after bifurcation of the solution due to material softening occurs, the nonlinear algorithm may not find the equilibrium path, which corresponds to the global minimum of the total potential energy of the

(a)

(b)

(c)

Fig. 14 Evolution of cracks in the layered beam inter-preted form DIC measurement. Contour bands indicate the maximum principal strain.

(a)

(b)

(c)

0

10

20

30

40

50

60

70

0 1 2 3 4

F [k

N]

u [mm]

specimen A

specimen B

specimen C

Fig. 13 Experimentally obtained load (total) vs. mid-span deflection curves of layered HSFRC beams.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 321

structure and traces another, less stable, path. This diffi-culty arises especially often, when both material proper-ties and stress distribution are nearly uniform, as it is the case in the horizontal direction in the central part of the beam. In such a situation, the localization can be trig-gered by imposing a small perturbation to the model.

Considering that our goal was a predictive analysis, in which case the location of the localized crack was not a-priori known, we proceeded in the following system-

atic way. We constructed several models, in which a perturbation in the form of an element with thickness reduced by 15% was placed at different locations along the line of elements at the bottom side of the beam (see Fig. 15). The percentage of thickness reduction was determined by trial and error as the least value that initi-ated localization of fracture. The numbers in the models’ names correspond to the distance of the weaker element from the plane of symmetry in millimeters. By running

(c)

(d)

(e)

(a)

(b)

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4

F [k

N]

u [mm]

spec. A spec. B

spec. C model NP

model 9 model 18

model 30 model 36

50

55

60

65

70

75

80

0 0.2 0.4 0.6 0.8 1

F [k

N]

u [mm]

spec. A spec. Bspec. C model NPmodel 0 model 3model 6 model 9model 12 model 15model 18 model 24model 30 model 36model 42

Fig. 16 Experimentally and numerically obtained load-deflection curves of layered HSFRC beams. Left: representative results. Right: detail of peaks.

Fig. 15 Finite element model of layered HPFRC beam. Colors indicate position of weaker element in different models.

(a)

(b) Fig. 17 Deformed shape and crack pattern of layered HSFRC beam obtained with the numerical model without imperfec-tion (model NP). Contour bands indicate COD [m], (a) and (b) refer to points marked in Fig. 16.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 322

FE analyses we obtained load-displacement response for each model, as shown in Fig. 16. It is obvious that all the models with imperfection predicted much lower peak than the unperturbed model NP, and that all the numerical results are now very close to those of speci-men B. Nevertheless, even the perturbed models showed some slight differences in the values of peak loads. The lowest load capacity was predicted with model 15, which was thus deemed to represent the most likely failure mode. Figure 18 shows the cracks evolu-tion in this model at load levels indicated by points (c) - (e) in Fig. 16. It is evident, that the perturbation trig-gered formation and propagation of one major crack even before the peak (c). However, some fine cracking along the entire bottom side of the central part was also observed. After the peak, the major crack continued to open and propagate vertically toward the upper surface of the beam (d). In the final stage the crack slightly bent in the direction to the center (e). Some very fine branch-ing of the crack was seen in the uppermost layer, which lacked any fiber reinforcement. By comparison with Fig. 14 it is evident, that the position of the major crack, as well as its propagation path were very well predicted by the model.

7. Concluding remarks

In this article we present a methodology for numerical simulation of HSFRC structural elements with spatially variable fiber volume fraction. The envisioned applica-

tions for such analyses include prediction of load and deformation capacity of HSFRC members with imper-fect fiber distribution or design and verification of func-tionally graded HSFRC members. Thus, the emphasis is placed on predictive capability of the model with re-spect to variable fiber volume fraction, as well as feasi-bility of the model identification. The first goal is achieved by employing micromechanical models that capture the dominant phenomena governing the HSFRC’s tensile response: matrix cracking, fiber debonding and pullout and crack-bridging effects. The second requirement is met by utilizing conventional notched-beam fracture tests and inverse analysis to de-termine the model parameters. The calibrated material model is eventually used in the framework of FEM.

The proposed approach has been verified or validated on several levels: (1) Each step, which involved inverse analysis (calcu-

lation of cohesive relationship from results of bending tests and its approximation) was followed by successful reproduction of the original test using the obtained relation as input.

(2) The model, which was calibrated from macroscale experiments, was used to simulate a microscale test of a fiber pullout. The predictions both qualita-tively and quantitatively matched results reported in the literature, which were obtained directly by single-fiber pullout experiments.

(3) The model was used to predict behavior of a lay-ered beam, where each layer had a different fiber

(c)

(d)

(e) Fig. 18 Deformed shape and crack pattern of layered HSFRC beam obtained with the numerical model with imperfection (model 15). Contour bands indicate COD [m], (c) thru (e) refer to points marked in Fig. 16.

P. Kabele, T. Sajdlová , M. Rydval and J. Kolísko / Journal of Advanced Concrete Technology Vol. 13, 311-324, 2015 323

volume fraction. The simulation results matched very well with the experimental ones. Note that with exception of two layers (with Vf = 0% and Vf = 3%), the fiber volume fractions in the simulated beam were different than those on which the model was calibrated.

Although the model has been validated on a layered beam, where fiber volume fraction was varied in one dimension only (vertically), the presented modeling concept is applicable even to such cases when the ana-lyzed domain exhibits any known spatially non-uniform distribution of fiber content.

The material model and its identification method have been validated for tension-softening composites consisting of fine-grained high strength cementitious matrix and thin short high-strength steel fibers with Vf not exceeding 3%. As such, it is applicable to HSFRC and UHPFRC materials, which do not exhibit tensile hardening and multiple-cracking behavior. Use of the model for other similar materials should be precondi-tioned by consideration, whether some additional mi-cromechanical phenomena (e.g. fiber rupture) need not be included.

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Nomenclature Ac crack area a   fitting parameter for fiber group reduction effect b1, b2, b3 fitting parameters of fiber matrix frictional

bond c1, c2 fitting parameters of matrix cohesive law df fiber diameter Ef  fiber Young modulus Em   matrix Young modulus F load applied to beam specimens f snubbing friction coefficient g snubbing factor Le fiber embedment length Lf fiber length Nb number of fibers Pp   fiber force for pullout phase Pd   fiber force for debonding phase u deflection of beam specimens Vf   fiber volume fraction Δ relative slip of fiber against matrix δ crack opening displacement (COD) δmax,m crack width at which the matrix cohesive stress is

completely released δ0 crack opening displacement when a single fiber

completes debonding δ∗ crack opening displacement when all fibers

complete debonding σb total bridging traction σb,f fiber bridging traction σb,m matrix bridging traction σt,m matrix cracking strength σ∗ maximum fiber bridging traction τ0 frictional bond stress at the fiber-matrix interface τeq  equivalent frictional bond strength τs  initial frictional bond stress φ angle between fiber embedment direction and

pullout direction