, The influence of fibre orientation on the post-cracking tensile behaviour of steel fibre THE INFLUENCE OF FIBRE ORIENTATION ON THE POST-CRACKING TENSILE BEHAVIOUR OF STEEL FIBRE REINFORCED SELF-COMPACTING CONCRETE

Adding fibres to concrete provides several advantages, especially in terms of controlling the crack opening width and propagation after the cracking onset. However, distribution and orientation of the fibres toward the active crack plane are significantly important in order to maximize its benefits. Therefore, in this study, the effect of the fibre distribution and orientation on the post-cracking tensile behaviour of the steel fibre reinforced self-compacting concrete (SFRSCC) specimens is investigated. For this purpose, several cores were extracted from distinct locations of a panel and were subjected to indirect (splitting) and direct tensile tests. The local stress-crack opening relationship (σ-w) was obtained by modelling the splitting tensile test under the finite element framework and by performing an Inverse Analysis (IA) procedure. Afterwards the σ-w law obtained from IA is then compared with the one ascertained directly from the uniaxial tensile tests. Finally, the fibre distribution/orientation parameters were determined adopting an image analysis technique.


INTRODUCTION
Adding fibres to concrete provides several advantages, especially in terms of controlling the crack opening width and propagation, increasing the energy absorption capacity, as well as increasing the post-cracking tensile strength [1,2]. In composites reinforced with low fibre contents, the contribution of the fibres mainly arises after the crack initiation. Crack opening in steel fibre reinforced concrete (SFRC) is restrained by the bond stresses that develop at the fibre / matrix interface during the fibre pull-out. Moreover, one of the most important properties of SFRC is its ability to transfer stresses across a cracked section rather uniformly, which depends on the fibre reinforcement effectiveness, i.e. fibre properties (their strength, bond and stiffness), and fibre distribution/orientation towards the active crack plane [3].
In order to optimize the fibre contribution to the post-cracking behaviour, it is important to enhance the distribution and orientation of the fibres at the crack plane. Since, fibres are more effective fairly aligned along the principal tensile stresses directions [4,5]. The dispersion and orientation of fibres in the SFRC bulk hardened-state results from a series of stages, namely [6]: fresh-state properties after mixing; casting conditions into the formwork; flowability properties; and wall-effect introduced by the formwork. Among the aforementioned parameters, flowability of steel fibre reinforced self-compacting concrete (SFRSCC) is the most important one [7][8][9]. Having in mind that distribution/orientation of the fibres influences significantly the mechanical properties of the SFRSCC, it is important to control and consider both parameters, especially in terms of the design applications.
In this research, the effect of fibre distribution and orientation on the tensile behaviour of a SFRSCC panel is investigated. For this purpose, a total number of 46 cores were extracted from various locations of two panels. These Abrishambaf A. et al., The influence of fibre orientation on the post-cracking tensile behaviour of steel fibre reinforced self-compacting concrete, Frattura ed Integrità Strutturale, 31: 54-66, 2015doi: 10.3221/IGF-ESIS. 31.05 4 Twenty two cores were sawn to produce prismatic specimens to be used in the uniaxial tensile test with the dimensions of 110×102×60 mm 3 . Following the same notching procedure for the splitting test specimens, the prismatic specimens were notched according to parallel (θ = 0°) and perpendicular (θ = 90°) directions regarding the expected concrete flow. The notch was executed on the four lateral faces of the specimen, at their mid-height, with a thickness of 2 mm and a depth of 5 mm.

Splitting tensile test
In a first stage, the σ -w relationship was assessed by carrying out splitting tensile tests. The recommendations of ASTM C-496 standard [12] were followed for this purpose. The tests were executed by closed-loop displacement control. To ensure a proper constant displacement rate, once the crack is initiated, a rather low value of the displacement rate, i.e. 0.001 mm/s, was applied. The crack opening width was averaged from the readouts of five linear variable differential transducers, LVDTs, which were mounted on the surface of the specimen, three on the top and two on the bottom surfaces, see Fig. 2.

Uniaxial tensile test
The stress -crack opening width (σ -w) relationship was also directly ascertained through uniaxial tensile tests, which were executed according to the RILEM TDF-162 [13]. This test was carried under closed-loop displacement control, adopting the following displacement rates during the test: 0.005 mm/min up to a displacement of 0.05 mm, 0.02 mm/min up to a displacement of 0.1 mm, 0.08 mm/min up to a displacement of 0.5 mm, and 0.1 mm/min until the completion of the test. The test was controlled by the averaging signal received from the four LVDTs installed on the lateral surface of each prismatic specimen, see  Fig. 4 depicts both the envelope and average force -crack mouth opening response (F -w) obtained from the splitting tensile tests, when the notch direction was parallel (θ = 0°) and perpendicular (θ = 90°) to the concrete flow direction. During the first phase of the test, the F -w relationships were almost linear up to the load at the crack onset, since the LVDTs recorded the elastic deformation of the SFRSCC specimen. Therefore this deformation should have been removed from the F -w curves, however since this deformation was marginal it could be neglected. After the crack onset, two distinct behaviours were observed for the θ = 0º and 90° series. Regarding the θ = 0º series ( Fig. 4(a)), the composite exhibited a non-linear hardening behaviour until the peak load was attained, followed by a softening phase. On the other hand, for θ = 90º series it was observed just a softening phase immediately after the crack onset. Briefly, these differences between the θ = 0º and 90° series could be ascribed to rather distinct number of fibres intersecting the crack plane at the specimens' notch, due to a preferential fibre alignment perpendicular to the concrete flow direction. The higher number of fibres at the crack plane of the θ = 0º series specimens will promote a higher stress transfer grade between the crack surfaces and, consequently, higher post-cracking residual forces. The aspects related to the fibre distribution / orientation will be detailed further ahead.

Splitting tensile test
In general, the F -w relationship showed a high scatter. However, in the case of SFRSCC the scattering in the results was expected since the mechanical behaviour of this material was significantly dependent on the fibre dispersion/orientation, even for specimens with same geometry and from the same batch. Moreover, in this particular case the relatively high scatter could also be enhanced because the specimens were extracted from different locations of the panels, i.e. with distinct distances from the casting point. Note that during the casting of the panels, the fibre distribution /orientation was influenced by the viscosity and velocity of the fresh concrete along the flowing process.
Figs 5(a) and 5(b) show three average F -w relationships obtained with the splitting tensile test for specimens from the θ = 0º and 90° series, respectively. The "Average" curve was obtained by averaging the readouts of the five LVDTs mounted on the specimen, whereas the "Top" and "Bottom" curves were obtained by averaging only the readouts of the LVDTs mounted, respectively, at the upper and lower specimen's surface. From Fig. 5  6 that the readouts of the LVDTs mounted on the upper surface of specimens showed a relatively higher crack opening width compared to the one recorded by the LVDTs at the lower specimen surface. This denotes that the crack opened asymmetrically, which could be ascribed to the variation of effective fibres along the depth of the panel due to segregation. This aspect will be detailed further ahead, when the fibre distribution parameters are endorsed.

Uniaxial tensile test
Figs 6(a) and 6(b) illustrate the average and envelope force-crack mouth opening relationship obtained from the uniaxial tensile test for the θ = 0º and 90° series, respectively. The value of the crack opening was determined by averaging the readouts of the four LVDTs. For both series (θ = 0° and 90°), the F -w curve was almost linear up to the crack initiation. After the crack onset, two distinct behaviours were observed for the θ = 0º and 90° series similarly to what was observed in splitting tensile tests, which could also be ascribed to the distinct number of fibres at the crack plane.
Regarding the θ = 0º series, fibres start to be slowly pulled-out being observed a semi-hardening response after the crack onset. Afterwards a plateau response was observed until a crack width of about 0.6 mm, and finally it was followed by a smooth reduction in the residual forces. Actually, during the uniaxial tensile testing of the θ = 0º specimens, once the peak load was achieved the sound of the fibre rupturing was clearly heard, which caused a rapid reduction in the value of the residual forces. This was also confirmed by inspecting the fracture surface visually after the specimen testing. Based on pull-out tests of hooked end fibres [14], fibre rupture may occur between the slip intervals of 0.6-1.0 mm, for the fibre inclination angle with the loading direction of 30°. As it will be discussed further ahead, for this series the most probable fibre orientation angle towards the cracking plane was around about 35º, this value was derived from the orientation probability distribution ascertained from an image analysis procedure.
On the other hand, the θ = 90º series, after the crack initiation, shown a sudden force decrease up to a crack width of nearby 0.07 mm followed by a plateau. Cunha et al. [5] assessed the micro-mechanical behaviour of hooked end fibres by performing fibre pull-out test. It was verified that after a fibre sliding of nearby 0.1 mm, the fibre 7 reinforcement mechanism was mainly governed by the hook plasticization during the fibre pull-out process.
Additionally, in some specimens of the θ = 90º series, in particular those located closer to the casting point, shown a pseudo-hardening behaviour as it can be observed by the upper bound of the experimental envelope ( Fig. 6(b)).
Afterwards, beyond a crack width of about 0.9 mm, a reduction of the residual force was observed, which corresponded to the fibre rupture.

Evaluation of fibre distribution parameters
In order to assess the distribution and orientation of fibres, an image analysis technique was implemented due to its simplicity and relatively low cost [15,16]. This technique comprised of four main stages: in the first stage, the fracture surface of the specimen was grinded. Then the surface was polished by acetone in order to increase the reflective properties of steel fibres. Secondly, by using a high resolution digital photograph camera, a coloured image of the grinded surface was obtained. Finally, the achieved image was analysed using ImageJ [17] software to recognize steel fibres. The analysis procedure of an image was depicted in Fig.7. After analysing the results, the following parameters were derived out: ii) fibre orientation factor,   : determined as the average orientation towards a certain plane surface by Eq. 2: where, f T N is the total number of fibres that can be determined by counting all the visible ellipses and circles at the cross section, θ is the out-plane angle that is defined as the angle between the fibre's longitudinal axis and a vector orthogonal to the plane.
iii) fibre segregation parameter, seg  : to calculate the location of the steel fibres gravity centre, an average value of the coordinates in the Y axis of entire fibres should be determined in the analysed cross-section.  (3) In this equation, y is the coordinate in the Y axis of the fibre's gravity centre, and h is the height (or depth) of the analysed cross-section, iv) number of effective fibres per unit area, f eff N : the summation of fibres with deformed hooked in a unit area.
Since this parameter cannot obtain from the image analysis results, therefore it was executed by visual inspection of the fracture surface. Table 2 includes the results of the image analysis performed on a plane surface of the uniaxial tensile specimens, see 254%, respectively, when compared to the θ =90° series. The differences between these series corroborate that the fibres were preferentially reoriented due to the concrete flow. This could be clearly noticed if the orientation parameter,   , for each series was compared (see Table 2). On the other hand, when the notch plane was parallel to concrete flow direction, a higher orientation factor observed comparing to specimens with θ =90°. Therefore, in θ = 0° series, a higher number of the effective fibres in the fracture surface was appeared which increased the concrete fracture parameters. Regarding the fibre segregation factor, the obtained average values of seg  were insignificantly higher than 0.5 meaning that a slight fibre segregation occurred through the panel's depth. This could also justify the asymmetric crack opening width observed in the splitting tensile specimens. Fig. 9 illustrates the orientation profiles obtained for each average orientation factor in comparison to both the two- [18] and the three-dimensional (3D) isotropic uniform random fibre distribution,   = 0.5, [19]. It was shown that the fibre orientation profile followed a Gaussian distribution [20,21].
The orientation profile for θ= 0°series was represented by a distribution shifted to the left side, while for θ= 90° series the distribution profile tends to the right side. Therefore, within the specimens of θ= 0° series, fibres have a tendency to be aligned more perpendicular to the studied plane. The obtained distribution profile for the θ= 0° series was completely distinct from either the 2D or 3D theoretical isotropic uniform random distributions. Meanwhile, the distribution obtained for the θ= 90° series was very similar to the 2D theoretical distribution. In conclusion, for SFRSCC laminar structures, assuming a 2D or 3D uniform fibre random distribution may be far from the reality, since the influence of fibre orientation due to the concrete flow also needs to be taken in to the account. Fig. 10 shows the exponential relationship between the number of fibres, f N , and effective fibres, f eff N .

NUMERICAL SIMULATION
The most suited test to derive the mode I fracture parameters is the uniaxial tensile test. However, the latter test involves some difficulties such as: the necessity of specialized and expensive equipment; sophisticated test set-up to avoid detrimental interferences, like load eccentricity, since it decreases the stress at the onset of crack initiation [22]. On the other hand the splitting tensile test could be considered as an alternative option for this purpose, because it is cheaper, less sophisticated testing equipment is needed, and can be executed on both cubes and extracted cores.
In this section a methodology to predict the stress -crack width (σ -w) relationship of SFRSCC using an inverse analysis, IA, procedure based on the results of the splitting tensile test will be presented and discussed. For this purpose, numerical simulations of the splitting tensile tests were carried out with a nonlinear 3D finite element model. In order to confirm the accuracy of the proposed methodology, the σ -w response obtained through the IA of the splitting test results was compared to the σ -w response obtained from the uniaxial tensile test.

Modelling and simulation
The average experimental force -crack width responses of the splitting tensile tests (Fig. 4) were simulated using ABAQUS ® finite element software [23]. Eight-node hexahedral shape solid elements with 8-integration points were used. The concrete damage plasticity model was implemented in order to simulate mechanical properties of concrete [24,25]. Because of the symmetry in the specimen geometry, supports and test loading applied in the splitting tensile test, only a quarter of the core was simulated, see Fig. 11(a). Since the specimen had distinct thicknesses it consists of two main parts namely: Notch and Un-notch (flush). After meshing each part individually, the assembled 10 mesh is shown in Fig. 11(b) with a total number of 5674 elements. Similar to the performed in the experimental procedure, in the numerical simulation a prescribed displacement was applied on top of the notch.

Concrete constitutive model
The concrete damage plasticity, CDP, model was used to simulate the mechanical performance of concrete because it is proficient to model the cracking of concrete in tension and crushing in compression. On the other hand, this model uses the concept of isotropic damage elasticity in combination with isotropic compression and tension plasticity to simulate the inelastic behaviour of concrete under compressive and tensile stresses. The CDP model uses a yield surface that is defined as the loading function proposed by Lubliner et al. [26]. The evaluation of the yield surface is controlled by two hardening variables, namely, the plastic strain in tension ( pl t  ) and the plastic strain in compression ( pl c  ). In the case of the effective stress, the yield function is determined as follow: where: In these equations, p and q are two stress invariants of the effective stress tensor, namely, the hydrostatic stress and the Von Mises equivalent effective stress, respectively, max  stands for the maximum principal effective stress and is the algebraic maximum eigen value of the effective stress  [27], x represents Macauley bracket Parameter c k is physically assumed as a ratio of the distances between, respectively, the compressive meridian and the tensile meridian with hydrostatic axis in the deviatoric cross section. If this ratio tends to 1, the deviatoric cross section of the failure surface becomes a circle similar to the Drucker -Prager yielding surface. However, definition of this parameter is only possible if the full triaxial compressive tests are executed on concrete specimens [28].
Tables 3 include the adopted initial parameters for the CDP model used to simulate the response of the splitting tensile tests.

Stress -strain relationship for modeling the SFRSCC uniaxial compressive behaviour
In CDP model, once the concrete compressive strength ( cu  ) attained, the concrete shifts to the non-linear phase as depicted in Fig. 12 When unloading during the softening stage, the CDP model uses Eq. (10) to convert inelastic strain to plastic strain ( pl c  ). In this equation, c d , which is known as damage parameter in compression (Fig. 12), depends on the concrete damage level, and ranges from zero to one, respectively for undamaged and fully damage material.
By assuming 0 E as the concrete initial modulus of elasticity in the undamaged phase, it can be determined the compressive stress and also the effective cohesive stress by Eqs. (11) and (12) Table 4 includes the values of the model parameters used in the numerical simulation of the splitting tensile tests.

Stress -strain relationship for modeling the SFRSCC uniaxial tensile behaviour
The stress -strain response under uniaxial tension had a linear elastic behaviour until the material tensile strength ( 0 t  ) was attained. Afterward, the tensile response shifted to the post-cracking phase where a non-linear response was assumed, Fig. 13. The SFRC post-cracking strain, ck t  , can be determined by subtracting the elastic strain corresponding to the undamaged part from the total strain: From the stress vs. cracking strain response (  , as well as the effective cohesive tensile stress, t  , are derived using an approach similar to the one adopted for the compressive behaviour, as detailed in section 3.2.1.

Inverse analysis procedure
The σi and wi values that define the tensile stresscrack width law were computed by fitting the numerical loadcrack width curve to the correspondent experimental average curve. From the nonlinear finite element analysis, the numerical loadcrack width response, FNUM -w, was determined, and compared to the experimental one, FEXP -w.
At last the normalized error, err, was computed as follows: where F iEXP and F iNUM were the experimental and the numerical load value at i th crack width value, respectively. The final σ -w relationship was defined by the parameters set that lead to the lowest normalised error between the experimental and numerical compressive force versus crack width curves.

Numerical results
Fig. 14 illustrates the numerical response obtained from the inverse analysis of the splitting test results (NUM SPLT ), as well as the average and envelope (experimental force -crack width curves, EXPSPLTAvg and EXPSPLTEnvelope, respectively. As shown in Fig. 14, a good accuracy between the experimental and numerical simulation was achieved, even though a slight difference was observed, but the estimated error (err) was lower than 5 %. Fig. 15 depicts the σ -w relationship obtained from the inverse analysis that leads to the smallest error. The numerical tensile strengths for the θ = 0° and θ = 90° series were 3.6 and 3.2 MPa, respectively. By comparing the response for both series, similar to the uniaxial tensile test, the post-cracking residual stresses in θ = 0° series were also significantly higher due to the fibre tendency to be reoriented perpendicular to concrete flow direction as previously discussed. Therefore, in θ = 0° specimens there are more effective fibres to bridge the crack plane than in θ = 90° series.

Figs 16(a) and (b)
shows for each series the uniaxial σ -w relationships obtained from the inverse analysis procedure of the splitting tensile test (NUMSPLT), the envelope and average curves from uniaxial tensile test (EXP UTT Envelope, EXP UTT Avg.) executed according to the RILEM TDF-162 recommendations [13]. Moreover, the σ -w response for the splitting test was determined from Eq. 16 as recommended by ASTM C-496 standard (EXP SPLT ) [12] and is also represented in Fig. 16.
where F is the applied line load, d is the diameter of the cylinder (150 mm) and l is the thickness of the net area in the notched plane (50 mm). The σ -w relationship obtained by the inverse analysis procedure rendered a relatively good approximation of the uniaxial tensile response, principally, for the series θ = 90°. Regarding the θ = 0° series, NUM SPLT and EXP SPLT methods showed a very close numerical tensile strength, which were higher than EXPUTT Avg. However, as expected, splitting tensile test tends to slightly overestimate the tensile strength compared to the uniaxial tensile test.
At the early cracking stages (w < 0.6 mm) NUMSPLT and EXPSPLT methods rendered σ -w responses nearby upper bound limit of the EXP UTT Envelope, this overestimation could be correspondent to the effects of the compressive stress along the loading plane. For higher crack opening widths, since in EXPSPLT method, stress was determined from Eq. 16 that assumes a linear elastic stress distribution even after cracking of the matrix, this approach was unable to predict post-cracking tensile response with enough accuracy. On the other hand, NUMSPLT started to get closer to the response obtained from the uniaxial tensile test.
Regarding the θ = 90° series (see Fig. 16(b)), the inverse analysis procedure of the splitting tensile tests also overestimated the tensile strength when compared to the tensile strength obtained from the uniaxial tests, although it was within the experimental envelope. Based on the EXPUTT results, a sharp stress reduction happened once the crack initiated due to the brittle nature of the matrix and lower number of effective fibres at the fracture plane. The sudden stress decay occurred until the beginning of the hook mobilization, which happened at a crack width of around 0.3 mm. The result of the inverse analysis method reproduced the EXP UTT response with a good accuracy since unlike the θ = 0° series that showed higher residual stresses, the θ = 90°series had lower post-cracking residual stresses, therefore the load bearing capacity of the specimen has decreased and the compressive stresses were not so preponderant in the overall response. 15 hence larger dissipated energy than the specimens with θ = 90º. The observed variation in the post-cracking parameters could be ascribed to a preferential fibre orientation at the crack surface. On the other hand, in the casting process of the panels from the centre, since the wall effects are negligible, the flow velocity is uniform and diffuses outwards radially from the casting point, see Fig. 17. Therefore, the fibres have a tendency to reorient perpendicular to the concrete flow direction. Consequently, in the θ = 0° series because of the high number of effective fibres with favourable orientation, the composite showed a semi-hardening response while in the θ = 90° series, since fibres were rotated due to the concrete flow velocity, the number of the effective fibres was reduced and lower residual forces were achieved.  (Table 2), the CoVs of f eff N and ηθ for the θ=90° series were considerably higher than for the θ=0° series.

CONCLUSIONS
In this study, the influence of fibre dispersion/orientation on the tensile post-cracking parameters of steel fibre reinforced self-compacting concrete panel was investigated. The σ -w law was determined indirectly from inverse analysis of the splitting tensile test results, as well as directly derived from the uniaxial tensile test. According to the experimental and numerical investigation, the following conclusions could be derived out: 2. Roughly, a linear relationship between number of the effective fibres, orientation factor and post-cracking parameters were observed. It was shown that by increasing the number of effective fibres as well as their orientation, fracture parameters tend to raise. This strong dependency could explain that in θ=0° series due to the appearing higher number of effective fibres which were mainly perpendicular to the crack plane, the concrete represented a semi-hardening behaviour, while in the other series a high stress decay was achieved.
3. In the case of casting panels from the centre, fibres have a tendency to align perpendicular to the radial flow, mainly due to the uniform flow profile velocity that diffuses outwards radially from the centre of the panel.
Consequently, the total number of the effective fibres was higher in crack planes parallel towards the concrete flow (θ=0°) when compared to the other case of an orthogonal crack plane towards the concrete flow (θ=90°).
4. In this study it was also represented that the fibre orientation tendency in a panel is completely reverse in comparison to a prismatic specimen. On the other hand, in a panel, due to the reason explained earlier, fibres have a tendency to re-orient perpendicular to concrete flow direction while in the prismatic specimens a fibre orientation tendency parallel to the flow direction was reported [29,30]. Therefore, although determination of tensile response of SFRC by performing three-point bending test on the prismatic specimens was recommended by the design codes such as RILEM 162-TDF and EN-1465 [31,32], but due to the different fibre orientation profiles in prismatic and planar structural elements, it could lead to an unrealistic tensile behaviour of planar structural elements like panels, shells or walls.
5. The inverse analysis of the splitting tensile response can estimate with a relatively good accuracy the uniaxial tensile behaviour, in particular, for low fibre contents. In general, in the case of using a relatively high content of fibres which lead to a partially or completely tensile hardening behaviour (as it was the case of the θ=0° series), the proposed methodology could somehow overestimate the σw law. In this case it is preferable to use the modified version of splitting tensile test similar to the one proposed by di Prisco et al. [33].

FIGURES AND TABLES
List of Figures:                  Table 1: Mix composition of steel fibre reinforced self-compacting concrete per m 3 .