Part II – Mechanical Behaviour of SFRSCC (from meso to macro-scale) 117
6.4 Fibre orientation, efficiency and effectiveness
yi (6.6)
where ¯y and h are, respectively, the coordinate of the centre of gravity of the fibre reinforcement and the height of the analysed cross-section parallel to the vibration direction (Figure 6.4). In the case of ideal fibre distribution, without segregation, the value of ξseg is 0.5. If ξseg tends to 1, the fibres are disposed towards the bottom part of the cross section, whereas if ξseg tends to 0 the opposite occurs.
Figure 6.4: Procedure for inferring the fibre segregation degree.
6.4 Fibre orientation, efficiency and effectiveness
According to Kamerwara Rao (1979), the term fibre efficiency is used to indicate the perfor-mance of an individual fibre with a certain orientation relative to the tensile stress direction.
On the other hand, the term effectiveness is used for the average value of the efficiency, obtained by considering all possible orientations of the fibre.
In Part I of this thesis was studied the efficiency of an individual steel fibre, by means of its pullout behaviour. Additionally it was shown that the fibre efficiency is strongly influenced by the fibre inclination angle towards the pullout load, i.e. relatively to the tensile stress.
6.4.1 Fibre orientation factors One-dimensional case
For the one-dimensional case, the fibre orientation is inevitably aligned with the direction of the applied load. Theoretically, the fibre effectiveness is 1. Several authors (Kooiman 2000, Lappa 2007) suggest that the latter value would be the optimal one, since the fibre efficiency
is always optimal due to preferable fibre orientation regarding the fracture plane. However, in the author opinion this could be incorrect. As previously seen in the chapters regarding the micromechanical behaviour of a single fibre, the fibre efficiency is usually maximum for an orientation angle regarding the pullout load between 0 to 30 degrees. Other researchers found similar findings by conducting pullout tests for fibre inclination angles within 0 and 30o, namely, Banthia and Trottier (1994) for a 15o angle, and Robins et al. (2002) for a 10o angle.
Consequently, the optimal effectiveness factor could not be 1, but a slightly lower value.
Two-dimensional case
In a two-dimensional system, it is assumed that the fibres are randomly oriented in a plane. In Figure 6.5(a) is depicted the 2D case where the fibre orientation angle θ between the embedded fibre length and the crack line (x -axis) can vary between 0 and π. According to Kamerwara Rao (1979) the average efficiency or effectiveness of multiple fibres, η2D, can be determined from projecting the mean fibre length towards the direction of the tensile stress (y-axis). Moreover, it is assumed that all fibres comprised in the plane, represented by “crack line” in Figure 6.5(a), are subjected to a parallel translation until their gravity centre coincide at the intersection of the x and y-axis. Thus the mean effectiveness in a planar random system can be expressed by an orientation efficiency factor computed as follow:
η2D= Z π
0
sin θ · dθ
π = 2
π (6.7)
Three-dimensional case
For the general case, where fibres can have any orientation in a three-dimensional system, the unit sphere model depicted in Figure 6.5(b) reflects the actual isotropic uniform random fibre dispersion in the specimen. Stroeven and Shah (1978) determined the spatial-random effectiveness adopting the geometric probability theory. Similarly to the 2D case, the fibres are imaginary translated from their original positions, while maintaining their orientations, and joined with one end in the sphere with a diameter equal to the fibre length. Assuming the xy plane as the crack plane and the z -axis as the tensile stress direction, the fibre effectiveness or three-dimensional orientation factor is given by (Stroeven and Hu 2006):
η3D = Z π/2
0
cos θ · sin θ · dθ Z π/2
0
sin θ · dθ
= 1
2 (6.8)
where, the trigonometric term cosine represents the relative probability of fibre intersecting with crack, whereas the sine term is the relative spatial frequency as depicted in Figure 6.5(b).
In space the fibre orientation is defined by two angles, however, the second angle (the one not related with the tensile stress direction) will lead to the same integration in the nominator and denominator, consequently, there is no effect on the orientation factor (Stroeven and Hu 2006).
(a) (b)
Figure 6.5: Spatial averaging of fibre contribution to stress transfer over the leading crack: (a) two-dimensional case and (b) three-two-dimensional case visualizes by the unit sphere model.
6.4.2 Influence of the geometrical boundaries on fibre orientation
Several researchers have presented theoretical models to take into account the boundary re-strictions imposed by a mould of a structural element in the fibre effectiveness of random short fibre reinforcement (Kamerwara Rao 1979, Soroushian and Lee 1990, Kooiman 2000, Dupont 2003, Stroeven and Hu 2006). In spite of their differences and complexity, commonly to all of them, an average fibre efficiency factor, η, can theoretically be computed considering both the dimensions of the structural element and fibre, and the wall effects.
The fibre orientation is strongly influenced nearby boundaries, especially when the fibre length is relatively large compared to the structural element or specimen dimensions. Accord-ing to an analysis performed by Soroushian and Lee (1990) and Kooiman (2000), the effect of constrained orientation of fibres is mainly more predominant when the element dimensions are smaller than five times the fibre length. When the dimensions of the elements are consider-ably larger than the fibre length, the effect of constraint is reduced, consequently, the fibres’
orientation approaches those of freely random orientated, i.e. without constraints from the boundaries.
Table 6.1 presents some examples of average orientation factors (η), computed from the the-orethical models developed by Soroushian and Lee (1990), Kooiman (2000) and Dupont (2003), considering distinct fibre lengths and number of boundary surfaces. The orientation factor val-ues computed from the distinct methodologies differed. Nevertheless, the differences between values obtained by Soroushian and Lee (1990) and Dupont (2003) are not considerable. The methodology presented by Soroushian and Lee (1990) has flaws if analysed within the scope of the geometrical statistics, geometrical probability theory or integral geometry field (Stroeven 1994, Stroeven and Hu 2006, Stroeven and Guo 2008). On the other hand, in the mathemat-ical model proposed by Dupont (2003), the orientation factor is determined by averaging the orientation factor computed for three distinguished zones in a rectangular cross section. The adopted orientation factors for the bulk, with one boundary (zone 1) and two boundary con-ditions (zone 2) are, respectively, 0.5, 0.6 and 0.84 (see Figure 6.6). Dupont (2003) evaluated his model performance by comparing the calculated number of fibres with the number of fibres crossing a section assessed experimentally on 107 distinct specimens, involving different fibre types. The model provided good predictions of the number of fibres crossing a section.
Figure 6.6: Distinct boundary conditions at a beam’s cross section (after Dupont 2003).
More recently, Stroeven and Hu (2006) presented a three-dimensional model for fibre aniso-tropy supported on the geometrical probability theory, which take into account not only the boundary effect on the fibre orientation, but also the decline of the number of fibres toward the boundary surface. With this model, the authors predicted a decrease (nearby 10%) on the fibres’ stress transfer capability at the boundary zones in comparison to the bulk performance according the same direction.
Table 6.1: Average orientation numbers available in literature, when considering boundaries.
Reference Boundary Specimen dimensions
lf [mm] η [-]
surface b [mm] h [mm]
Soroushian and Lee (1990) 2
150 150 60 ≈ 0.48
4 ≈ 0.57
Kooiman (2000) 2 150 150 60 0.70
30 0.66
Dupont (2003) 4 150 150 60 0.60
30 0.55
6.4.3 Anisometry and anisotropy
As previously discussed, several factors related to the production of fibre reinforced composites, FRC, affect the distribution characteristics of the fibres within a cementicious matrix. Thus it is feasible to assume anisotropic mechanical behaviour for the FRC due to the anisometry in the fibre reinforcement (Stroeven 1979, 1986b, Stroeven and Hu 2006). The actual fibre distribution can be assumed as a mixture of three, two and one-dimensional fibre orientation arrangements, respectively, 3D, 2D and 1D. All the fibres geometrical centres are assumed dispersed randomly.
However, fibres from the 1D portion are oriented parallel to the so-called orientation axis, while fibres from the 2D portion are oriented uniformly random parallel to the so-called orientation plane (Stroeven and Hu 2006).
For partially planar-oriented fibre structures, Stroeven (1978, 1979) demonstrated, based on the geometric probability theory, that the number of fibres per unit cross-sectional area in a slice of concrete, of thickness t, projected on a plane parallel to the filling direction can be computed as (see Figure 6.3):
Nf||= LV [(t/lf + 0.5) + (2/π − 0.5) · ω] (6.9)
where LV is the total fibre length in a unit volume of concrete (= 4Vf/πd2), lf is the length of each single fibre and ω is the degree of orientation defined in section 6.3.3. In equation 6.9, Vf is the fibre content in volume and d is the fibre diameter. On the other hand, for the case of a plane perpendicular to the filling direction, the number of fibres projected on it is given by (Stroeven 1978, 1979):
Nf⊥= LV [(t/lf + 0.5) − 0.5 · ω] (6.10)
When only the data of the number of fibres crossing a specific plane is available, equations 6.9
and 6.10 can be simplified for the limiting case of a thin slice (t → 0):
Nf|| = LV [0.5 + (2/π − 0.5) · ω] (6.11)
Nf⊥= 0.5 LV (1 − ω) (6.12)
Hence, this anisometry of the fibre distribution is directly reflected in a anisotropic be-haviour of the fibre reinforced composite. Stroeven (1986b) derived the following constitutive relationships for estimating the composite strength:
σ|| = σm (1 − Vf) +1
6λ τ Vf· (1 + 0.5 c Vf) (6.13)
σ⊥= σm (1 − Vf) +1
6λ τ Vf· (1 − c Vf) (6.14) where λ and τ are, respectively, the fibre aspect ratio and interface bond strength, whereas c is a constant (computed from ω = cVf) in which ω is the degree of orientation. σm and Vf are, respectively, the mortar strength, the fraction of the cross-sectional area transmitting tensile strengths and the fibre volume fraction. On the other hand, in the case of a random uniform fibre dispersion can be found:
σ||= f⊥= σm (1 − Vf) +1
6λ τ Vf (6.15)
According to Stroeven (1986b), the strengths computed from equations 6.13 and 6.14, and using the values of ω determined by stereological computations from the image analysis pro-cedure, are in agreement with experimental test results performed in different orthogonal di-rections. Analysing the latter equations, can be drawn out that the rearrangement of fibres in a specimen can either enhance fibre reinforcement effectiveness up to Vf(1 + 0.5 c Vf) in a favourable cross-section (see equation 6.13), or reduce the fibre reinforcement effectiveness to Vf(1 − c Vf) (see equation 6.14).
More recently, Stroeven and Hu (2006) presented a complete three-dimensional solution to boundary cases in fibre reinforced composites, where the strength relationships in bulk and in boundary zones (i.e. having into account the wall-effect) are derived for prismatic specimens of fibre reinforced concrete. Assuming that, in most practical cases, fibre reinforcements develops a partially planar structure in bulk, the stress components parallel to the orientation plane of the 2D portion can be computed by:
σ||= σm (1 − Vf) +1
3λ τ Vfη¯b3· (1 + 0.5 ω) (6.16) where ¯ηb3 is a stress transfer efficiency parameter in the boundary layer, which for prismatic elements with surfaces parallel to orientation plane of the 2D portion yields 0.203. The deriva-tion of this parameter can be found elsewhere (Stroeven and Hu 2006). On the other hand, the stress components perpendicular to the orientation plane can be determined in a similar fashion by:
σ⊥= σm (1 − Vf) + 1
3λ τ Vfη¯b3· (1 − ω) (6.17)