Mode I fracture toughness of fibre reinforced concrete

Abstract Low fracture toughness of concrete represents a serious shortcoming. An effective way to improve the concrete toughness is represented by the dispersion (during mixing) of discontinuous fibres into the concrete mix. The principal beneficial effect of fibres is the crack bridging in the cementitious matrix, providing resistance to crack propagation before fibre debonding and/or pulling out or failure. In the present paper, the fracture behaviour of FRC (fibre reinforced concrete) specimens is examined, with micro-synthetic polypropylene fibrillated fibres being randomly distributed in concrete. The modified two-parameter model, proposed by the authors to calculate Mode I plain-strain fracture toughness for quasi-brittle material, is able to take into account the possible crack deflection (kinked crack) during stable crack propagation.


INTRODUCTION
Concrete is extensively used as construction material in the civil engineering practice due to: (i) low production costs, (ii) easiness to be cast in different shapes, and (iii) versatility in response to different design requirements [1,2].
Despite its several merits, concrete has low tensile strength and weak cracking resistance. Fracture toughness, which represents the cracking resistance capability, is a fundamental parameter to analyse the fracture behaviour of concrete [3].
As is well-known, concrete needs a fracture mechanics approach different from that for metals [4]. For both metallic and concrete structures, fracture mechanics is nonlinear due to the presence of a zone ahead of stress-free crack tip, where the material shows a nonlinear behaviour [4]. Such a zone is named plastic zone in ductile and brittle metals and is mainly characterised by either hardening plasticity or perfect yielding of the materials, whereas such a zone is named fracture process zone (FPZ) in concrete and undergoes a softening damage (tearing). Concrete, as well as rocks and many other materials, are commonly named quasi-brittle materials.
In ductile metals, the applied energy is dissipated as plastic energy to form plastic zones [2,5,6], and Nonlinear Elastic Fracture Mechanics (NEFM) can be employed to examine fracture behaviour. In brittle metals, the applied energy is consumed as surface energy to form new crack surfaces [2,7], with the surface energy being significantly greater than the plastic energy, and Linear Elastic Fracture Mechanics (LEFM) can be applied to analyse the fracture process and crack propagation.
In quasi-brittle materials, energy is used to form the fracture process zone [2].
Until early 1970s, many investigations were aimed at the application of LEFM in order to examine fracture process and crack propagation [8].
On the basis of such studies, it was well established that LEFM can be employed to large-scale concrete structures only, whereas it cannot be applied to medium/small-mass structures due to the presence of large FPZs [8]. Since late 1970s, starting with the pioneering work by Kaplan [9], concrete fracture has been a subject of great interest.
Many nonlinear fracture models have been proposed by several research groups to study the fracture behaviour of concrete, both to take into account the effect produced by FPZ on concrete nonlinear fracture behaviour and to implement the governing mechanisms in nonlinear fracture [3]. The fundamental fracture models are: the cohesive crack model (CCM) [10], the crack band model (CBM) [11], the two-parameter fracture model (TPFM) [12], the size effect model (SEM) [13,14], the effective crack model (ECM) [15][16][17][18]  Such a deflection is caused by the biaxial stress state due to normal stress produced by bending, and shear stress produced by slippage at interface between cementitious matrix and fibre.
Testing is performed on FRC specimens, where micro-synthetic polypropylene fibrillated fibres (length=18mm, aspect ratio=0.003) are randomly distributed in concrete with different values (0.0, 0.5 and 2.5% by volume) of fibre volume fraction, called fibre content in the following. The fracture toughness is observed to improve by both adding fibres to concrete and increasing the fibre content.

MODIFIED TWO-PARAMETER MODEL
The modified two-Parameter model (MTPM) was proposed by the authors in Ref. [51] to compute the fracture toughness of quasi-brittle materials (for example, bone), when crack propagates under mixed mode loading (Mode I together with Mode II). Such a model can be also employed in the present study, FRC being a quasi-brittle behaviour.
According to the MTPM, the specimen has a prismatic shape and presents a notch in the lower part of the middle cross-section ( Figure 1). As in the case of the TPFM [12], the following specimen sizes are employed: (where W and B are depth and width of the specimen, respectively), notch-depth ratio 0  = W a / 0 = 1/3, and loading-span/depth ratio = W S / = 4 ( Figure 1).

Figure 1.
The tests are performed under three-point bending loading and crack mouth opening displacement control. More precisely, the specimen is monotonically loaded up to the peak load, max P ( Figure   2). When such a load value is achieved, the post-peak stage follows and, as soon as the force equals 95% of max P , the specimen is fully unloaded ( Figure 2). Then the specimen is reloaded up to failure.

Figure 2.
The linear elastic compliance i C (named initial compliance) is used to determine the elastic modulus E [12,51]: where parameter   0  V is given by [52]: The initial compliance value is computed for each test by applying the least squares method. Such an interpolation is performed by fitting the experimental data (in terms of load against CMOD) with a line, aiming to maximise the coefficient of determination in a range of CMOD from 0.00 to 0.01mm. The same procedure is followed to determine the final compliance.
Under mixed mode loading, the effective critical crack length Fig. 1) to determine the critical stress-intensity factor, , is obtained from the following equation [51]: SIFs of a bent crack [53,54].
As is shown in Figure 1, the kinked crack branch consists of two segments, named 1 a and 2 a . If the value of 2 a obtained from Eq. (3) is negative, it means that the effective crack length is . In this case, the effective critical crack length a ( Fig. 1) to determine the critical stress-intensity factor, is obtained from the following equation: Note that, in the case of kinking angle  equal to zero, Eqs (3) and (4) correspond to the unique equation provided by the TPFM [12].
Finally, the critical stress-intensity factor, computed by employing the measured value of the peak load, max P , and considering a straight crack having length equal to the projected length of the effective kinked crack [52]:

MTPM APPLICATION TO FRC SPECIMENS
Specimens are tested under three-point bending ( Figure 1). Testing is performed by means of an Instron 8862 testing machine under crack mouth opening displacement (CMOD) control, employing a clip gauge at an average speed equal to 0.1 mmh -1 . All specimens exhibit a nonlinear slow crack growth before the peak load is reached.
Each specimen consists of a beam 40mm x 40mm x 200mm ( S =160mm), and presents a notch of 13.3mm in the lower part of the middle cross-section. Note that the depth (W ) is taken equal to the width ( B ) of the specimen, having previously verified that the B size will not influence the fracture behaviour of FRC specimen.
The specimen matrix is cementitious and characterised by the following proportions: cement: water: aggregates (by weight) = 1: 0.7 : 3.6. The maximum aggregate size is 4mm, and the cement is a 42.5 CEM II/A-P. This mixture presents a compressive strength of 30MPa at 28 days.
Three types of specimens are tested: plain concrete specimens (from P-1 to P-4 in Table 1), concrete specimens reinforced by randomly-distributed micro-synthetic polypropylene fibrillated fibres with a content equal to 0.5% by volume (from R05-1 to R05-4 in Table 1) or 2.5% by volume (from R25-1 to R25-4 in Table 1).

RESULTS AND DISCUSSION
The fracture toughness The load -CMOD plot for specimen R05-1 is shown in  Table 1.
Further, the values of according to the TPFM are also computed (by assuming   0) and listed in last column of Table 1. Table 1 The elastic modulus E , the kinking angle  (Fig.1), the peak load max P , and the effective critical crack length a are also displayed in Table 1 for each tested specimen.

Figures 3, 4, 5
When the kinking angle  is not constant along the crack path (see Figure 6), the orientation of the first deflected segment   is plotted in Figure 7 for each type of specimen. Such data are well interpolated by the following three equations (see the continuous lines in Fig.7), one for each specimen type:   For fibre content FC equal to 3%, the above equations estimate a decrease of E equal to about 6% and an increase of max P and S C II I K ) (  equal to about 14% and 34%, with respect to the values related to plain concrete specimens.

EXPERIMENTAL EVALUATION OF FPZ
The value of the effective critical crack length is compared with that experimentally deduced through the field of longitudinal displacements.
The Digital Image Correlation (DIC) technique is used to extract 2D full-field maps of experimental displacements in the mid-span zone of each specimen.
Specimens are irregularly spray-painted before testing in order to get a well-contrasted grey-scale speckle pattern. A full-format Nikon D3X (6048x4032 pixels) digital camera is employed for data acquisition. The camera captures images of the suitably illuminated specimen surface at a rate of one frame every 15 seconds for plain concrete specimens and every 30 sec for FRC specimens.
The sequence of images has been treated by means of the software Ncorr developed in MATLAB environment [56]. The horizontal relative displacements are analysed between two parallel reference vertical lines symmetrically located with respect to the mid-span section, when the peak load is achieved. By examining, for example, the contour plot of the horizontal displacements for specimen R25-3 shown in Figure 9, a discontinuity of such displacements ahead of the initial notch length can be observed, indicating a stable propagation of the crack prior to failure.

Figure 9
In Figures (10)- (12), the horizontal relative displacements determined through the DIC technique at the mid-span cross-section are displayed against the x -coordinate (shown in Fig. 1), for each tested specimen. The experimental displacement profile is plotted in thick line.  Table 1 and represented by the horizontal dashed lines in Figs (10)-(12).

CONCLUSIONS
In the present paper, the fracture behaviour of FRC specimens, characterized by micro-synthetic polypropylene fibrillated fibres randomly distributed in the cementitious matrix, has been examined.
The modified two-parameter model has been employed in order to take into account the possible crack deflection (kinked crack) during stable crack propagation.
Three types of specimens have been analysed: (a) plain concrete specimens, (b) concrete specimens with fibre content equal to 0.5% by volume or (c) fibre content equal to 2.5% by volume.
The concrete fracture toughness has been observed to improve by increasing the volume fraction of fibres, and an equation has been proposed to estimate such a parameter for a given fibre content.
The overestimation of the fracture toughness determined by considering crack propagation under pure Mode I loading has been also evaluated, and an equation has been proposed to compute such an increment for a given kinking angle value.