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Procedia Structural Integrity 5 (2017) 848-855

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2nd International Conference on Structural Integrity, ICSI 2017, 4-7 September 2017, Funchal,

Madeira, Portugal

Fracture toughness of fibre-reinfocced concrete determined by

means of numerical analysin

Patrizia Bernardi*, Elena Michelim, Alice Sirico, Sabrin a Vantadori, Andrea Zanichelli

As is well-loiown, the addition of fibres to concrete mix (Fibre Reinforced Concrete, FRC) produces a positiv«; effect on caacking behaviour. nn this work, the results of an experimental campaign on FRC ^inec;imens with lannotnly distributed micro-synthetic polypropylene fibrillated fibres are examined. The tests concern single-notched beams under three-point bending, where the fibre content vanee. Such an experiпlentfl testiug is numerically onalysed through a nonlinear fieile element model, named 2D-PARC, wlrr;rr; a proper conftitutive law Vor fibre-reinforced concrete is impletnenteil. rr--e loanrcrack mouth opening nisplacement eCMOD) curves numerically obtained are employed to ileterпline the critical strest-mtensity factor (fracture toughnrst) for different values of fibre content, according to the two-parameter model. The comparison between such numerical results and thosr obtainedby applying the two-pacamerer model fo the rxperimental load-CMOD cueves is performed.

© e017 The Autoons. PuPlin^ by Hsevier B.V.

Peer-review under rerponsibi]ity of the Scientific Committee oflCSI 20 171

Kryworcls: writical stresipntaisity factost FRC i^ex^ciisiiins; 2 th^piAtl^CM^ fwo-param etes meedel

1. Inttoduction

Conventional oi" plain concrete is widely used in civil engineering practical applications due to its: (i) low production cost; (ii) excellent mechanical behaviour under compression; (iii) high form-fitting property to be casted

* Corresponding author. Tel.: +39-0521-905709; fax: +39-0521-905924. E-mail address: patrizia.bernardi@unipr.it

Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy

Abstract

2452-3216 © 2017 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of ICSI 2017

10.1016/j.prostr.2017.07.090

in different shapes (Pantazopoulou et al., 2001). The main drawback of concrete is the weakness related to both tensile strength and toughness in presence of cracks.

In order to analyse the concrete fracture behaviour, a fracture mechanics approach different from that used for metals is needed. As a matter of fact, fracture mechanics is non-linear due to the presence of a zone ahead of stressfree crack tip (named fracture process zone, FPZ), where the material shows a non-linear behaviour. The fundamental fracture models available to study the fracture behaviour of plain concrete are: the cohesive crack model (CCM), the crack band model (CBM), the two-parameter model (TPM), the size effect model (SEM), the effective crack model (ECM), and the double-K fracture model (DKFM). More recent proposals are also available in the literature.

An effective way to improve concrete toughness is represented by the dispersion (during mixing) of discontinuous fibres into the concrete mix (Ivanova et al., 2016). As a matter of fact, fibres cross cracks and develop the so-called crack bridging effect on the crack surfaces (Rao et al., 2014).

The application of non-linear fracture models presented for plain concrete has also been extended to FRC. Within this research field, the results of an experimental campaign on FRC specimens with randomly distributed micro-synthetic polypropylene fibrillated fibres (Vantadori et al., 2016) are examined in the present work. The tests concern single-notched beams under three-point bending, where the fibre content varies.

This experimental testing is numerically modelled through non-linear FE analyses by adopting a proper constitutive model for FRC (Cerioni et al., 2008; Bernardi et al., 2013, 2016a). The model, proposed in terms of secant stiffness matrix, is formulated by imposing equilibrium and compatibility conditions both in uncracked and cracked stage. All the fundamental resistant contributions offered by both fibres and concrete are properly taken into account and, in this way, the structural material behaviour is simulated up to failure. Then, the load-crack mouth opening displacement (CMOD) curves numerically obtained are employed to determine fracture toughness for different values of fibre content, according to the two-parameter model (Jenq et al., 1985). Finally, the comparison between such numerical results and those obtained by applying the two-parameter model to the experimental load-CMOD curves is performed.

Nomenclature

a effective critical crack length

a 0 notch length

C initial compliance

Cu unloading compliance

D total stiffness matrix

Dc concrete stiffness matrix

Dc,cr1 cracked stiffness matrix

E elastic modulus

f snubbing coefficient

K-ic critical stress-intensity factor

P peak peak load

S specimen loading span

W specimen depth

a0 = a0/W initial notch-depth ratio

ß=a /W effective notch-depth ratio

8 total strain

8c strain related to fibre reinforced concrete (FRC) between cracks

8cr1 strain related to resistant mechanisms developed in the fracture zone

°f fibre action

To interfacial bond strength

no orientation efficiency factor

2. Analysis of some experimental results

The experimental campaign (Vantadori et al., 2016) consists of three series of specimens: 4 plain concrete specimens, 4 concrete specimens reinforced by randomly-distributed fibres with a content equal to 0.5% by volume, and 4 specimens with 2.5% by volume. The micro-synthetic polypropylene fibrillated (that is, deformed and/or irregular in shape) fibres are made of 100% pure high-density polypropylene. They are generally used as secondary concrete reinforcement. The fibre aspect ratio is equal to 0.003 (length = 18mm), the tensile strength is equal to 450-600MPa, and the elastic modulus is equal to 3.5kN/mm2

The specimen matrix is cementitious, characterised by the following proportions: cement: water: aggregates (by weight) = 1: 0.7 : 3.6. The cement is 42.5 CEM II/A-P, and the maximum aggregate size is 4mm. The experimental compressive strength of the mixture is 42MPa at 28 days after casting.

Three-point bend tests on single edge notched beams were performed at the 'Testing Laboratory of Material and Structures' of the University of Parma. The nominal sizes of each FRC specimen are shown in Figure 1.

Figure 1. Three-point bend notched specimen (sizes in mm).

An Instron 8862 testing machine under crack mouth opening displacement (CMOD) control was used, employing a clip gauge (average speed = 0.1 mmh-1) and measuring the load through a load cell.

Each specimen was monotonically loaded. After the peak load was achieved, the post-peak stage followed and, when the force was equal to about 95% of the peak load, the specimen was fully unloaded (up to a force value equal to about zero), by proceeding under load control. Then, the specimen was reloaded up to failure. All specimens exhibited a non-linear slow crack growth before the peak load was reached, as is shown in Figure 2.

3. The two-parameter model

The two-parameter model (Jenq et al.,1985) requires the registration of the applied load (P ) against the crack mouth opening displacement. The initial compliance, Ci, is employed to determine the elastic modulus, E , whereas the unloading compliance, Cu, is used to estimate the effective crack length, a (Figure 1). Firstly, the elastic modulus is computed through the following expression:

E = 6 • S • a0-^(a)

c -w 2 • b

where S is the specimen loading span, a0 is the initial notch length, Ci is the linear elastic compliance, W and B are the specimen depth and thickness, respectively, and V1 (a) is given by :

V1(a) = 0.76-2.28a + 3.87a2 -2.04a3 + 066 (2)

being a = a0/W .

OPENING DISPLACEMENT, [mm]

Figure 2. Experimental data and numerical curves in terms of load-CMCD for: (a) plain concrete specimens; (b) FRC specimens with a fibre content equal to 0.5% by volume; (c) FRC specimens with a fibre content equal to 2.5% by volume.

Then, the effective critical crack length, a = a0 + A ac (being A ac the stable crack growth at peak load) can be determined from the following relationship:

E = 6S• a -^(a) Cu -W 2B

where V (a) is obtained from Eq. (2) by replacing a0 with the critical quantity a , and Cu is the unloading compliance at about 95% of the peak load.

Finally, the critical stress-intensity factor, ksic, is computed by employing the measured peak load, Pmax , and the effective critical crack length, a :

3P . S

^=-pWk^a •F № <4)

F(p) = 1 1.99 -0(1 -0) (2.15 - 3.930 + 2.102)

^ (1 + 20) (1 -0)3/2 (5)

being 0 = a/W .

4. Fracture toughness by means of both the 2D-PARC model and the two-parameter model

4.1. A non-linear numerical model: 2D-PARC

The non-linear constitutive model 2D-PARC, able to represent the behaviour of FRC up to failure, is herein adopted. The model is expressed in the form of a secant stiffness matrix, which makes it suitable to be adopted in conjunction with Finite Element (FE) technique. This stiffness matrix, which is defined for each integration point, is able to consider material cracking by following a smeared cracking approach. Its theoretical formulation, deduced for an ordinary reinforced concrete (RC) membrane element subjected to general in-plane stresses, is explained in detail in Cerioni et al. (2008). Extensions of 2D-PARC model to Fibre Reinforced Concrete (FRC) have recently been proposed to include the resistant contribution offered by fibres dispersed in the concrete mix, both before (Bernardi et al. 2016a) and after cracking development (Bernardi et al. 2013, 2016b). The model has also been extended to RC structures externally retrofitted with fabric reinforced cementitious matrix composites (FRCM) in Bernardi et al. (2016c).

In the present work, the attention is focused on the case of fibre-reinforced concrete elements without ordinary steel reinforcement. In the uncracked stage, concrete is modelled following the non-linear elastic formulation originally proposed by Ottosen (1980). The non-linear stress-strain relationships for concrete under a general biaxial state of stress are determined by properly updating the secant values of the elastic modulus and Poisson ratio, inserted into the concrete stiffness matrix Dc. The effect of fibres, which stiffen the material response in compression, is also included, as is described in Bernardi et al. (2016a).

When the maximum principal stress exceeds the adopted failure envelope in the tension region, for the considered integration point a stiffness matrix accounting for cracking contributions is computed. Cracking is assumed to develop at right angle with respect to the principal tensile stress direction, and a strain decomposition procedure is adopted. The total strain e is subdivided into two components Ec and e«-1, related to fibre reinforced concrete (FRC) between cracks and to all the resistant mechanisms that develop in the fracture zone, respectively. The behaviour of FRC between cracks is described through the matrix Dc, derived for the uncracked stage, with

slight modifications (i.e. by applying a proper reduction to the concrete strength envelope) to account for the degradation induced by cracking.

All the resistant mechanisms related to the kinematics developed at crack location (i.e. aggregate bridging and interlock, fibre effects) are included in matrix Dc,cr1, as a function of two main local variables, namely crack opening wi and sliding vi, by using effective laws available in the technical literature to individually represent each physical phenomenon.

The procedure adopted for modelling of aggregate bridging and interlock is described in Cerioni et al. (2008). For such a reason, the attention is herein focused on the improvement provided by fibres to the post-cracking behaviour of concrete structures. This improvement is mainly related to the so-called fibre-bridging, which consists in the transmission of additional tensile stresses across crack surfaces. This mechanism is taken into account by modifying the cracked stiffness matrix Dc,cr1 through the introduction of an additional term (due to fibre effects across the crack), which is summed up to aggregate bridging contribution.

Fibre action a, which depends on fibre geometric and mechanical characteristics, as well as on their content in the admixture, is evaluated by adopting the micro-mechanical formulation proposed by Li et al. (1993). It is worth noticing that the evaluation of fibre action ay requires the knowledge of three main parameters: the snubbing coefficient f the interfacial bond strength t0, and the orientation efficiency factor rjo (Li et al. 1993; Bernardi et al. 2013). Their values are here selected within the range of variability given in the literature for polypropylene fibres.

Considering both the equilibrium and compatibility conditions for the cracked material, the stress field in the cracked stage can be written as follows by developing some mathematical steps:

o = (dc-; + Dc,crl-; = D£ (6)

D being the total stiffness matrix.

4.2. Numerical simulation of experimental tests by means of the 2D-PARC model

The 2D-PARC model is applied in order to numerically simulate the fracture behaviour of FRC specimens described in Section 2. To this aim, a FE mesh constituted by triangular 6-node membrane finite elements is adopted. Figure 3 shows the adopted discretization, which refers only to one half of the notched specimen, taking advantage of the symmetry of the problem. As can be seen, the mesh is properly refined around the notch, where a stress concentration is expected. Numerical analyses are performed under displacement control, by imposing an increasing vertical displacement to the central loaded point, in order to achieve a better numerical convergence.

Figure 3. FE mesh of FRC specimens (sizes in mm).

The load-crack mouth opening displacement (CMOD) curves, numerically obtained and shown in Figure 2, highlight that the numerical model employed is able to correctly describe the fracture behaviour of all the analysed specimens, characterised by different values of fibre content.

□y employing such numerical curves and applying the two-parameter model (see Section 3), the elastic modulus

(Enum ), the peak load (Ppeaknum), the effective critical crack length (anum) and the critical stress-intensity factor,

ksicnum, for the above three specimen series are computed and listed in Table 1. Note that the numerical unloading

compliance Cu is approximately computed as the secant compliance at peak load, that is, according to the alternative procedure of the TPM to be applied when tests are performed employing an actuator which cannot control unloading.

The corresponding averaged values, computed from the experimental load-crack mouth opening displacement (CMOD) curves (Figure 2), are also reported in Table 1.

Table 1. Numerical and experimental results for the three specimen series in terms of: elastic modulus, peak load, effective critical crack length and critical stress-intensity factor.

Series E aum E exp p peak,aum p peak,exp a aum a exp Ks IC,aum Vs ^IC ,exp

[MPa] [MPa] [N] [N] [mm] [mm] [MPa m1/2] [MPa m1/2]

P 17073.06 17952.35 713.01 658.00 19.29 18.82 0.673 0.553

R05 16783.24 17647.56 634.83 734.75 17.39 17.82 0.520 0.569

R25 16115.84 16945.80 649.85 755.43 17.76 19.02 0.547 0.720

From Table 1, it can be remarked that the ksicnum values are in a satisfactory agreement with the averaged experimental ones, and the scatter between them may be attributable to the different procedure employed to compute the parameter Cu .

5. Conclusions

In the present work, an experimental campaign performed on concrete specimens reinforced with polypropylene fibres has been examined. The tests concern single-notched beams under three-point bending. Such an experimental testing has been numerically modelled through non-linear finite element analyses, where a proper constitutive law for fibre-reinforced concrete is implemented. The numerical load-crack mouth opening displacement (CMOD) curves have been employed to determine the critical stress-intensity factor, according to the two-parameter model. The comparison between such numerical results and those obtained by employing both the experimental load-CMOD curves and the two-parameter model has shown a quite satisfactory agreement.

Acknowledgements

The authors gratefully acknowledge the financial support provided by the Italian Ministry for University and Technological and Scientific Research (MIUR), Research Grant PRIN 2015 No. 2015JW9NJT on "Advanced mechanical modelling of new materials and structures for the solution of 2020 Horizon challenges".

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