3.5.1. Fibre Spatial Distribution

Detailed mean and standard deviation (STD) of relative magnetic permeabilities for all plates are listed in Table 5. It can be seen that with the increase of fibre content, the relative magnetic permeability also increased. Due to the limited testing depth, the magnetic permeability only increased with the increase of plate thickness from 15 to 35 mm but no obvious difference between thicknesses from 35 to 50 mm.


**Table 5.** Mean and standard deviation (STD) of relative magnetic permeabilities of all plates.

Combining the results from Table 5 and the results from Nunes et al. [22], a relationship between fibre volume content and theoretical relative magnetic permeability μ can be derived as shown in Figure 8. When fibre volume content is 0, the relative magnetic permeability value equals to 1, which represents the magnetic permeability of air. The R-squared value of 0.9987 shows a nearly perfect linear fitting.

**Figure 8.** Relationship between fibre volume content and relative magnetic permeability.

Considering the effective depth of the magnetic probe, if relative magnetic permeability μtest was obtained on a thin specimen, the attenuation factor should be applied to calculate the real fibre volume content. The relationship between real relative magnetic permeability μ<sup>r</sup> and fibre volume content Vf is further developed into

$$
\mu\_{\rm r} = \frac{\mu\_{\rm test} - 1}{1 - \rm AF\_{\rm t}} + 1 \tag{9}
$$

$$
\mu\_{\rm r} = 0.0383 \times V\_{\rm f} + 1 \tag{10}
$$

where AFt represents the attenuation factor when the specimen's thickness is t. Combining Equations (9) and (10), the corrected relationship can be expressed as:

$$\mathbf{V\_{f}} = \frac{\mu\_{\text{test}} - 1}{0.0383 \times (1 - \mathbf{A} \mathbf{F\_{t}})},$$

By using Equation (11), real fibre volume contents can be derived and are indicated in Table 6. For all the 1% vol. plates, the fibre content fulfilled the designed requirement. For 2% and 2.5% vol., some plates possessed lower fibre volume content than the designed value.


**Table 6.** Mean and STD of fibre volume content of all plates.

With the increase of fibre volume content, the standard deviation also increased, which revealed that fibre tended to distribute more non-uniformly. This effect was more severe with the 2.5% vol. UHPFRC plates. A possible reason was the fibre balling or gathering effect. Although there were variations within each plate, the differences were very small compared to the total fibre volume content. Therefore, the plate can be considered almost uniformly spatially distributed.

The first coloured contour plot in Figure 9a describes the fibre distribution of all plates at a unified scale. Colours ranging from blue to red represent the differences of fibre volume content. It can be seen directly that plates 2%–20 mm and 2.5%–15 mm have a lower fibre volume content than the designed fibre volume content. The detailed fibre distribution cannot be visualized, since the range of data was too wide in the coloured contour plots. Thus, a greyscale contour plot is given in Figure 9b as a comparison. The darker shading indicates a lower fibre volume content. It can be seen that there was no obvious fibre spatial distribution trend in the middle area of each plate, only the four boundaries appeared to have a lower fibre volume content. This mainly results from the limitation of the testing area (boundary effect).

**Figure 9.** Fibre volume content distribution of all plates. (**a**) With a scale as the fibre volume content in percentage and (**b**) with a general non-unified scale.

#### 3.5.2. Fibre Orientation Distribution

Based on previous literature by Nunes et al. [22], the fibre orientation was expressed by an orientation indicator ρΔ. The orientation indicator of the red points can be calculated as:

$$\mathfrak{p}\_{\Delta} = \frac{\mathfrak{mu}\_{\text{ij.y}} - \mathfrak{mu}\_{\text{ij.x}}}{2\left(\mathfrak{mu}\_{\text{ij.ave}} - 1\right)}\tag{12}$$

Nunes et al. [22] found the orientation indicator had a sinusoidal relationship with the fibre orientation angle, but an analytical expression between the fibre orientation angle and orientation indicator ρ<sup>Δ</sup> was not presented. Through further derivation, it was found that the orientation indicator is a function of polynomial terms of cos(ϕ). For example, the full expression of orientation indicator 1% vol. UHPFRC in terms of the orientation angle ϕ according to Nunes et al. [22] can be expressed as:

$$\rho\_{\Delta,1\%} = \frac{-1076.68\cos^6(\varphi) + 1615.03\cos^4(\varphi) + 4.03 \times 10^7 \cos^2(\varphi) - 2.02 \times 10^7}{(-0.96\cos^6(\varphi) + 5.68 \times 10^5 \cos^4(\varphi) - 5.68 \times 10^5 \cos^2(\varphi))}\tag{13}$$
 
$$-6.41 \times 10^7 + 0.48 \cos^2(\varphi))$$

After numerical analysis, only the constant terms and the cos2(ϕ) term on the numerator were found to be critical to the value of fibre orientation indicator ρΔ. Therefore, Equation (13) can be further simplified to:

$$
\rho\_{\Delta, 1^{\circledleftarrow}} \approx -0.63 \times \cos^2(\varphi) + 0.315 = 0.315 \times \cos(2\varphi) \tag{14}
$$

Figure 10a shows the comparison between the original fibre orientation indicator calculated using Equation (13) and the simplified orientation indicator calculated using Equation (14). No obvious difference can be observed from 0 to 90◦. Therefore, the original equation can be replaced with the simplified equation. There also works with other fibre volume percentages.

**Figure 10.** Relationship between fibre orientation angle (unit: degree) and fibre orientation indicator. (**a**) Comparison between simplified and original equation and (**b**) comparison between different fibre content.

The relationship for other fibre percentage is approximately equal (≈) to:

$$
\approx \rho\_{\Lambda, 2\%} \approx -0.299 \times \cos(2\varphi) \tag{15}
$$

$$
\varphi\_{\Delta, \mathcal{Y}\%} \approx -0.29 \mathcal{T} \times \cos(2\varphi) \tag{16}
$$

$$
\approx \rho\_{\Delta, 4\%} \approx -0.282 \times \cos(2\varphi) \tag{17}
$$

This relationship was drawn in Figure 10b. It can be seen that the fibre volume content did not have a significant effect on the fibre orientation indicator, especially when the fibre orientation angle was around 45◦. Generally, the fibre orientation indicator can be expressed as:

$$
\rho\_{\Lambda} = \mathbf{a} \times \cos(2\varphi) \tag{18}
$$

where the fibre orientation indicator coefficient slightly ranges around −0.3 depending on the fibre volume content.

By using Equation (18), the fibre orientation distributions can be characterized. Figure 11 shows the fibre orientation distribution of all plates. Instead of contour plots, the fibre orientation angle is represented by dots in different colours. The fibre orientation ranged from 0 to 90◦ from the horizontal axis. It can be seen that fibres tended to orient at 0◦ at the top and bottom boundaries, while orienting at 90◦ along the left and right boundaries. There are two possible explanations: firstly, fibre tends to align their orientation to the mould due to the wall effect of the plate mould boundary; and secondly, fibre contents at the four boundaries were lower compared to other parts, which resulted from the testing method. The vertical testing value at the top and bottom points and the horizontal testing value at the furthest left and right points were lower.

**Figure 11.** Fibre orientation angle distribution of all plates.

Figure 12 shows the distribution of fibre orientation angle based on 972 (9 × 9 × 12) sample points. It can be seen from the graph that under this specific casting method, the distribution of the fibre orientation angle generally follows a normal distribution and most fibres orient at an angle between 40◦ and 50◦. The normal distribution function was calculated based on the mean and standard deviation values of fibre orientation angle, which can be represented as:

$$f(\boldsymbol{\varphi}) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(\frac{\left(\boldsymbol{\varphi} - \boldsymbol{\zeta}\right)^2}{2\sigma^2}\right) \tag{19}$$

where the mean value σ = 45.60◦ and standard deviation ζ = 15.32◦.

**Figure 12.** Distribution of fibre orientation angles.

Detailed mean and STD value of fibre orientation angles can be seen in Table 7. The increase of fibre content and fibre distribution do not have a direct relationship with the fibre orientation angle. Due to the wall effect, the coefficient of variance for the fibre orientation angle was higher comparing to the fibre volume content.


**Table 7.** Mean and STD of fibre orientation angles of all plates.

#### **4. Mechanical Test Results**

#### *4.1. Compressive Test*

The compressive test was carried out on 0, 1%, 2%, and 2.5% specimens 5 days after curing. Six specimens were tested from each group. With the increase of fibre content, the compressive strength steadily increased in an almost linear trend. Detailed compressive test results can be seen in Table 8.

**Table 8.** Compressive strength of different groups of 100 mm cube specimens.


#### *4.2. Flexural Test*

All the 500 mm × 500 mm × 15 mm, 500 mm × 500 mm × 20 mm, 500 mm × 500 mm × 35 mm, and 500 mm × 500 mm × 50 mm plates were firstly cut into four sections by water blade to maintain the accuracy of dimension (Figure 13a,b). Then, the 250 mm × 250 mm × 15 mm, 250 mm × 250 mm × 20 mm, 250 mm × 50 mm × 35 mm,and250mm×250mm×50mmplateswere furthercutinto200 mm × 50 mm × 15 mm, 200 mm × 50 mm × 20 mm, 200 mm × 50 mm × 35 mm, and 200 mm × 50 mm × 50 mm. The detailed cutting scheme can be seen in Figure 13c. Beam No. 1–8 were tested in this research.

**Figure 13.** Concrete plate cutting. (**a**) Water blade cutting; (**b**) concrete plate after cutting; and (**c**) schematic graph.

The flexural test were conducted 5–7 days after curing. The experiment was carried out on a 3-ton universal testing machine at a constant deflection control speed of 0.3 mm/min. Taking a 15 mm thick beam as an example, the three-point bending test setup can be seen in Figure 14. The effective span was 150 mm. The experiment was terminated once the load dropped below 50% of the peak load. The displacement movement of the machine was used to plot the load–displacement diagram

**Figure 14.** Three-point bending test for the UHPFRC beam. (**a**) Schematic graph (unit: mm) and (**b**) experimental set-up.

The flexural strength for a three-point bending test can be calculated from Equations (20) to (22).

$$\mathbf{M} = \frac{\mathbf{F}\_{\mathbf{f}} \times \mathbf{L}}{4} \tag{20}$$

$$\mathcal{W} = \frac{\text{BH}^2}{6} \tag{21}$$

$$\mathbf{f\_{f}} = \frac{\mathbf{M}}{\mathbf{W}}\tag{22}$$

where


Average first crack flexural strength and peak flexural strength of eight beams of each plate can be seen in Table 9. The merged cell on the right side shows the average value of four plates. For UHPC, no micro or macrocracks were observed before reaching peak load. The first crack strength equalled the peak flexural strength.


**Table 9.** First crack flexural strength and peak flexural strength (unit: MPa).

#### **5. Correlation between Magnetic Probe Test and Mechanical Test**

To correlate the results between the fibre distribution test and the mechanical test for each beam, the relative inductance values of the blue points in Figure 13c are also needed. The value is derived from the nearby two red points. For the blue points of Beams 1–4, the average relative magnetic permeability can be expressed as:

$$
\mu\_{\rm ij,ave} = \frac{\mu\_{\rm ij,x} + \mu\_{\rm ij,y} + \mu\_{\rm i(j+1),x} + \mu\_{\rm i(j+1),y}}{4} \tag{23}
$$

For Beams 5–8, it can be calculated by:

$$
\mu\_{\rm ij,ave} = \frac{\mu\_{\rm ij,x} + \mu\_{\rm ij,y} + \mu\_{(i+1)j,x} + \mu\_{(i+1)j,y}}{4} \tag{24}
$$

For the red points in Figure 13c, the fibre orientation indicator can be calculated straightly using Equation (12). For the blue points, the orientation indicator can be derived from the nearby two red points. For the blue points on Beams 1–4, the orientation indicator ρ<sup>Δ</sup> can be derived as:

$$\rho\_{\Lambda} = \frac{\mu\_{\text{i}\downarrow\text{y}} + \mu\_{\text{i}(\text{j}+1),\text{y}} - \mu\_{\text{i}\downarrow\text{x}} - \mu\_{\text{i}(\text{j}+1),\text{x}}}{\mu\_{\text{i}\downarrow\text{y}} + \mu\_{\text{i}(\text{j}+1),\text{y}} + \mu\_{\text{i}\downarrow\text{x}} + \mu\_{\text{i}(\text{j}+1),\text{x}} - 4} \tag{25}$$

$$\rho\_{\Delta} = \frac{\mu\_{\text{i}\text{j},\text{y}} + \mu\_{\text{(i+1)}\text{j},\text{y}} - \mu\_{\text{i}\text{j},\text{x}} - \mu\_{\text{(i+1)}\text{j},\text{x}}}{\mu\_{\text{i}\text{j},\text{y}} + \mu\_{\text{(i+1)}\text{j},\text{y}} + \mu\_{\text{i}\text{j},\text{x}} + \mu\_{\text{(i+1)}\text{j},\text{x}} - 4} \tag{26}$$

Figure 15 shows four typical load-deflection curves for 1% and 2% vol. UHPFRC. Both load-softening and load-hardening behaviours can be observed. Statistically, only 9 out of

32 1% vol. UHPFRC beams had obvious strain hardening behaviour. For 2% vol. UHPFRC, 27 out of 32 beams had load-hardening behaviour after the first crack. For 2.5% vol. UHPFRC beams, 29 out of 31 beams had load-hardening performance.

**Figure 15.** Load–deflection curves for 200 mm × 50 mm × 20 mm UHPFRC beams. (**a**) 1%-15-s6; (**b**) 1%-15-s4; (**c**) 2%-20-s8; and (**d**) 2%-20-s5.

Apart from the tensile property of the UHPC matrix, both the fibre orientation and fibre volume content can determine whether the beam performs a load-hardening behaviour or not. Initially in the uniaxial tensile test, fibres were fully bonded, and the tensile load was mainly carried by the concrete matrix. Thus, the first crack tensile strength mainly depended on the tensile properties of the concrete matrix, which agrees with previous researchers [28,29]. After the concrete cracks, concrete hardly sustained any loads, but fibres were not pulled-out yet due to the static frictional force τ between fibres and the concrete matrix. With the increase of uniaxial tensile load, the static frictional force also increased. Whether the beam had a load-hardening or load-softening performance depends on whether the whole fibre–concrete interfaces can provide enough frictional force, while the frictional force is related to the roughness of the interface, number of fibres (fibre volume content), and effective embedment length (fibre orientation). As can be seen in Figure 16, the effective static frictional force fst,eff carried by one fibre can be estimated as:

$$\mathbf{f\_{st}} = \boldsymbol{\pi} \times \mathbf{d\_f} \times \mathbf{l\_{em}} \times \boldsymbol{\pi} \tag{27}$$

$$\mathbf{f}\_{\rm st,eff} = \mathbf{f}\_{\rm st} \times \cos \boldsymbol{\uprho} = \boldsymbol{\pi} \times \mathbf{d}\_{\rm f} \times \mathbf{l}\_{\rm em} \times \boldsymbol{\pi} \times \cos \boldsymbol{\uprho} \tag{28}$$

#### where


**Figure 16.** Effective embedment length of fibres inside the concrete matrix in the uniaxial tensile test in fibre activation stage.

The total effective static frictional force fst,n can be calculated as:

$$\mathbf{f\_{st,n}} = \mathbf{f\_{st,eff}} \times \mathbf{n\_f} = \boldsymbol{\pi} \times \mathbf{d\_f} \times \mathbf{l\_{em}} \times \boldsymbol{\pi} \times \cos \boldsymbol{\varphi} \times \mathbf{n\_f} \tag{29}$$

Assuming the static frictional strength is constant in Equation (29), it can be seen that the total effective static frictional force has a linear relationship with Vf cos(ϕ) and number of fibres across the crack plane. Figure 17 shows the relationship between Vf cos(ϕ) and the peak flexural strength. For 1% UHPFRC, there is no obvious trend, mainly owing to the insufficient frictional force to support load-hardening performance, therefore, the peak flexural strength was determined by the properties of the concrete matrix. A linear relationship can be found for 2% and 2.5% UHPFRC. If the concrete matrix w consistent, the relationship between the uniaxial tensile strength ft,ϕ1, ft,<sup>ϕ</sup><sup>2</sup> at different fibre orientation angles ϕ1, ϕ<sup>2</sup> can be calculated as:

$$\frac{\mathbf{f\_{t,q}}}{\cos(\varphi\_1) \times \mathbf{V\_{f1}}} = \frac{\mathbf{f\_{t,q}}\_{\mathbf{t}, \mathbf{q}} \mathbf{2}}{\cos(\varphi\_2) \times \mathbf{V\_{f2}}} \tag{30}$$

**Figure 17.** Relationship between Vf × cos(ϕ) and peak flexural strength. (**a**) 1% vol. UHPFRC; (**b**) 2% vol. UHPFRC; and (**c**) 2.5% vol. UHPFRC.

#### **6. Conclusions**

This research focused on investigating fibre distribution of UHPFRC using C-shape magnetic probe. The following conclusions can be drawn.

• The effective testing depth of the C-shape magnetic probe was firstly determined by curve fitting analysis using MATLAB. An exponential equation was derived to alter the relative magnetic permeability value, and to correlate with the real fibre volume content for elements with different thicknesses.


**Author Contributions:** Conceptualization, J.X. and L.L.; methodology, J.X. and L.L.; formal analysis, L.L. and J.X.; writing—original draft preparation, L.L.; writing—review and editing, J.X., C.C., and S.J.; supervision, J.X., C.C., and S.J.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Suzhou Municipal Construction Research Centre, grant number 2019-13 and the APC was funded by Xi'an Jiaotong-Liverpool University.

**Acknowledgments:** The authors would like to thank Jiangxi Beirong Circular Materials Company for providing the dry-mix UHPFRC material in this investigation. The views and findings reported here are those of the writers alone, and not necessarily the views of sponsoring agencies.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
