# Chloride diffusivity and electrical resistivity of slag-based concrete reinforced by steel fibers: experimental study and multiscale modeling

Nicolas Reuge, Stéphanie Bonnet, François Bignonnet

## ► To cite this version:

Nicolas Reuge, Stéphanie Bonnet, François Bignonnet. Chloride diffusivity and electrical resistivity of slag-based concrete reinforced by steel fibers: experimental study and multiscale modeling. European Journal of Environmental and Civil Engineering, In press, 10.1080/19648189.2025.2532800. hal-04894736v2

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Submitted on 23 Jul 2025

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# Chloride diffusivity and electrical resistivity of slag-based concrete reinforced by steel fibers: experimental study and multiscale modeling

Nicolas Reuge¹*, Stéphanie Bonnet² and François Bignonnet³

¹ORCiD: 0000-0003-4764-990X

Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183 F-44600, Saint-Nazaire, France

²ORCiD: 0000-0003-0027-344X

Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183 F-44600, Saint-Nazaire, France

³ORCiD: 0000-0002-5279-9112

Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183 F-44600, Saint-Nazaire, France

*Corresponding author, email: reuge@free.fr

## Abstract

This study is part of a project investigating the possible use of slag-based concretes reinforced by steel fibers for various structures of critical interest, exposed to marine environments and aimed at improving the characteristics of the structure compared to the classical solution of plain reinforced concretes. Four concrete mix designs were investigated featuring various fiber contents: 0 (i.e., without fiber), 20 kg.m⁻³, 40 kg.m⁻³ and 60 kg.m⁻³. Mechanical properties and durability indicators were tested, focusing on chloride diffusivity (migration tests) and electrical resistivity (Wenner tests). Migration tests showed that chloride diffusivity increases moderately (13%) between 0 and 40 kg.m⁻³ fiber content and more strongly (36%) between 40 kg.m⁻³ and 60 kg.m⁻³ fiber content. With regard to mechanical properties, the best compromise for fiber content is 40 kg.m⁻³. By using a multiscale homogenization model coupled with an existing hydration model, the effect of fibers on the durability indicators is determined. It successfully reproduces the increases in resistivity as a function of time and its decreases as a function of fiber content. The moderate increase in chloride diffusivity with fiber content (≤ 40 kg.m⁻³) is also well reproduced by accounting for an interphase of 22 µm width around the fibers.

**keywords:** steel fiber-reinforced concrete, blast furnace slag, durability, chloride, corrosion, multiscale homogenization

## 1 Introduction

In marine environments, Reinforced Concrete (RC) structures are subjected to seawater penetration and chloride ingress (Lu et al. 2015, Achour et al. 2018). When chloride ions reach the steel rebars in sufficient quantities, corrosion initiates, ultimately leading to catastrophic cracking of the matrix and drastic loss of mechanical properties of the structure.

From (Lu et al. 2025), adding mineral admixtures (fly ash, silica fume or phosphogypsum) in controlled proportions can improve the resistance of concretes to chloride ingress by modifying the hydration chemistry and resulting in denser materials. In (Dang et al. 2025), the authors showed that adding fly ash and Blast-Furnace Slag (BFS) improved durability of concretes by reducing chloride permeability very significantly while improving their mechanical properties. From (Sosa et al. 2021), adding BFS in concretes results in reducing their permeability to water and oxygen and then to chloride ingress, as also shown in (Kim et al. 2018). From (Fabien et al. 2022), replacing part of the cement by BFS in the concrete mix design modifies the matrix of the porous network in a highly efficient way, slowing down chloride ingress (Ben Fraj et al. 2012; El Achrafi et al. 2023). Moreover, BFS and more generally, mineral
admixtures, are significantly less impactful on the environment than clinker (Kim et al. 2018, Ventura et al. 2020), provide better mechanical strengths to the concretes (Kim et al. 2018) and protect them from various aggressions (Bayrak et al. 2023).

Another possibility for improving service life and properties of RC could consist in adding fibers in the concrete mix design. Presence of fibers can have various effect on the concretes depending on their nature as shown in (Tayeh et al. 2024) regarding polypropylene and basalt fibers. In particular, adding steel fibers seems beneficial (Ali et al. 2023). The corrosion resistance of Steel Fiber-Reinforced Concretes (SFRC) is not a priori obvious since steel fibers can also be subjected to corrosion. However, SFRC have better corrosion resistance than RC, which is attributed to the formation of a macro-cell circuit by fibers in concrete (Ren et al. 2024). The macro-cell inhibits the corrosion of the internal reinforcement due to cathodic protection. From (Song et al. 2023), the corrosion process of fibers differs as a function of the fiber content: at less than 2 vol%, it initiates at fibers edges through matrix defects. Instead, at more than 2 vol%, through a well-connected conductive network, it can initiate at fibers centers under the action of spontaneous current and macro-cell reaction. Note that from Shang et al. (2025), adding an appropriate amount of graphene oxide (between 0.02% and 0.04%) in the SFRC formulation would enhance significantly the corrosion resistance of SFRC.

Furthermore, it is established that the introduction of steel fibers in concrete mix designs results in significant gains in mechanical properties (Aksüt et al. 2024, Ali et al. 2023, Mohamad et al. 2023, Thomas and Ramaswamy 2007) possibly enabling the rebars to be dispensed with. This can be explained by the high mechanical properties of steel fibers compared to bulk concrete and by the partial adherence of the fibers to the concrete. From (Kaklauskas et al. 2014), steel fibers allow enhancing the residual stiffness of the concrete, providing crack control, knowing that this control of crack width is based on the type and amount of steel fibers added. In (Maglad et al. 2023), adding hooked steel fibers to various concrete mixes considerably improved the fracture characteristics, while the brittleness was reduced with increased steel fiber content.

In this study, double-hooked steel fibers were used: they are known to be particularly efficient to provide good ductility to the concrete (Rezakhani et al. 2021). The hooks strengthen cohesion between fibers and bulk concrete. The effect of the aspect ratio of fibers (length/diameter) has been studied in (Chetbani et al. 2023): optimal slump performance of the studied concrete was achieved for an aspect ratio of 63, whereas compressive strength notably increased for an aspect ratio around 70. In (Haido et al. 2021), tests on different lengths of steel fibers (between 25 mm and 50 mm) showed that increasing the fiber length resulted in improved dynamic deflection and ductility of the concrete.

Figure 1 shows an overview of the project focused on Steel-Fiber Reinforced Slag-Based Concretes (SFRSBCs). The aim of the entire project is twofold: (1) determining the optimal content of steel fibers and (2) obtaining a favorable Life Cycle Assessment (LCA). The present investigations deal with the first point only. The optimal content of fibers should provide a maximized durability of the structure with regard to chloride ingress together with suitable mechanical properties. Investigating durability as a function of the mix design requires: (a) measurements of durability indicators; (b) linking modifications of these indicators to the material microstructure which can be done by using multiscale homogenization techniques.

Regarding mechanical properties, only Ananda et al. (2014) have characterized SFRSBCs: with a slag content by weight of binder of 40% and volume fraction of steel fibers of 1.5%, a compressive strength of 49.8 MPa and a flexural strength of 4.88 MPa have been achieved at day 28. Although studied concrete mix designs contained different slag contents, unfortunately, no slag-based concretes without fiber have been characterized.

Many studies investigating the effects of steel fibers on mass transport in SFRCs focus on liquid permeability. Although there is no universal relationship between liquid permeability and chloride diffusivity, they usually follow the same trends. In (Poorsaheli et al. 2021), all the SFRCs tested (fiber volume fraction between 0.33% and 0.99%) had a lower water permeability (from 50% to 80%)
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Schematic diagram of the project</figcaption></figure>

compared to the plain concrete samples. Previous studies have stated that the presence of fibers reduces crack initiation and propagation in the concrete (Tsukamoto and Wörner 1991, Lawer et al. 2002, Rapoport et al. 2002), resulting in lower permeability.

From the review by Ganesh and Sofi (2022) and from various investigations (Benmokrane et al. 1996; Kamal and Boulfiza 2011; Pakravan et. al. 2017; Stefanoni et al., 2015; Poorsaheli et al. 2021; Farooq and Yokota 2022; Yang et al., 2022; Zhang et al., 2022) and on-site measurements (Ganesh and Sofi 2022), it seems that the presence of steel fibers in concretes can reduce chloride ingress.

However, in the measurements performed by Ortega-Lopez et al. (2022), the use of steel fibers (volume fraction of 0.5%) slightly worsened the durability behavior of the studied SFRC, facilitating chloride ingress. Two studies (Shin et al. 2020; Fattouh et al. 2023) found that the addition of steel fibers degraded the durability of SFRCs in corrosive environments, especially when the concrete cracks despite an initial beneficial effect of restricting the water penetration depth. The Rapid Chloride Migration Tests (RCMTs) performed in (Zhao et al. 2023) showed that steel fibers may cause accelerated chloride ion penetration for a fiber volume fraction greater than or equal to 0.8%.

Therefore, it appears that the effect of steel fibers on chloride ingress in SRFCs is still and open issue.

Transport properties of SFRSBCs have never been studied except by Kim et al. (2021). Their RCMTs showed that adding fibers led to a significant decrease in chloride diffusivity: adding 40 kg.m⁻³ of steel fibers (volume fraction of about 0.5%) leads to a decrease of Dₐpp by about 30%.

Alternatively, concrete resistivity is a property which can be easily measured. It is particularly sensitive to liquid water content which has a low electric resistance. During the process of hydration, water content decreases and resistivity increases. Obviously, in SFRCs, resistivity also depends on steel fiber content.

Multiscale modeling can provide understanding on the effect of the composition and microstructure of SFRCs on their diffusive and resistive properties evolution during hydration. These models rely on a geometrical description of the microstructure and material properties at several length scales. For example, the mechanical properties of SFRCs have successfully been modeled using continuum micromechanics (Sorelli et al. 2008, Qsymah et al. 2017, Chen et al. 2020, Yu et al. 2020, Li et al. 2024). Shen et al. (2023, 2024) modeled the water permeability of hybrid fiber reinforced concrete using the General Effective Media Theory with a three-scale microstructure description. Both the electrical
resistivity and diffusion coefficient of concrete can also be modelled using multiscale homogenization methods and hydration models (Béjaoui et al. 2006, Stora et al. 2008, Damrongwiriyanupap et al. 2017; Achour et al. 2020; Honorio et al. 2020). Yet, it appears that the diffusivity and electrical resistivity of SFRCs has not been modeled using multiscale methods. According to the Nernst-Planck equation, the electrical resistivity and diffusion coefficient of plain concrete are closely related (Baroghel-Bouny et al. 2011; Chidiac and Shafikhani 2020). Hence, a multiscale model validated on resistivity measurements could be used to model the diffusion coefficient.

From a given geometrical model of microstructure (at the nano/micro/macro scales) of the materials, the multiscale approach allows predicting evolution of the different durability indicators under a unified physical framework. Moreover, it allows understanding why and how they evolve as a function of time and fiber content with this slag-based matrix. Thus, it provides basic information about the durability indicators and helps identifying the most favorable concrete formulations for a durable design as done in (Reuge et al. 2025a).

First, the SFRSBC mix designs are detailed and the measurement methods are described: RCMTs and resistivity measurements. Results are presented as a function of time and fiber content and compared to values found in the literature. Beside this, samples are characterized mechanically.

Then, a multiscale model of SFRBCs is developed based on mean-field homogenization methods. The evolution of microstructure and of macroscopic properties of the material will be accounted for by an existing hydration model in combination with a homogenization model. In terms of durability indicators, results are finally compared with measurements.

# 2 Materials and methods

## 2.1 Concrete mix design and fabrication

The base mix design of the reference concrete is given in Table 1. The binder is composed of 50% CEM I cement (Ordinary Portland Cement, OPC) and 50% BFS provided by Ecocem™. Sand and gravels are provided by Groupe CB™ (Boulonnais quarry). Water to binder mass ratio is 0.48. The physical and chemical properties of OPC and BFS are given in Tables 2 and 3 respectively (Ahmad et al. 2024).

The chosen steel fiber is Dramix 5D from Bekaert™ (Figure 2). The physical properties of this double-hooked fiber (0.9 mm diameter, 63 mm length) are given in Table 4.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Double-hooked steel fiber, "Dramix 5D" from Bekaert™</figcaption></figure>
<table><caption>**Table 1:** Constituents of the base concrete</caption><thead><tr><th>Constituents</th><th>Mass content (kg.m<sup>-3</sup>)</th><th>Volume Fraction (%)</th></tr></thead><tbody><tr><td>Gravels, 12 mm – 24 mm</td><td>561.2</td><td>21.6</td></tr><tr><td>Gravels, 4 mm – 12 mm</td><td>432.0</td><td>16.7</td></tr><tr><td>Sand, ≤ 4 mm</td><td>867.8</td><td>33.2</td></tr><tr><td><i>Total Aggregates</i></td><td><i>1861.0</i></td><td><i>71.5</i></td></tr><tr><td>CEM I</td><td>173.5</td><td>5.6</td></tr><tr><td>BFS</td><td>173.5</td><td>6.1</td></tr><tr><td><i>Total Binder</i></td><td><i>347.0</i></td><td><i>11.8</i></td></tr><tr><td>Water</td><td>166.6</td><td>16.7</td></tr><tr><td><i>Total</i></td><td><i>2374.0</i></td><td><i>100</i></td></tr></tbody></table>

<table><caption>**Table 2:** Physical properties of OPC and BFS (Ahmad et al. 2024)</caption><thead><tr><th>Property</th><th>OPC</th><th>BFS</th></tr></thead><tbody><tr><td>Specific gravity (g/cm<sup>3</sup>)</td><td>3.17</td><td>2.9</td></tr><tr><td>Specific surface area (cm<sup>2</sup>/g)</td><td>3635</td><td>4450</td></tr><tr><td>Loss on ignition (950°C)</td><td>1.4</td><td>&lt; 1.5%</td></tr><tr><td>Passing at 32 µm</td><td>--</td><td>≥ 95%</td></tr></tbody></table>

<table><caption>**Table 3:** Chemical compounds of OPC and BFS (Ahmad et al. 2024)</caption><thead><tr><th>% of components</th><th>OPC</th><th>BFS</th></tr></thead><tbody><tr><td>SiO<sub>2</sub></td><td>23.2</td><td>34.9</td></tr><tr><td>Al<sub>2</sub>O<sub>3</sub></td><td>3.1</td><td>10.9</td></tr><tr><td>Fe<sub>2</sub>O<sub>3</sub></td><td>2.24</td><td>0.5</td></tr><tr><td>CaO</td><td>66.4</td><td>42.3</td></tr><tr><td>MgO</td><td>0.76</td><td>6.9</td></tr><tr><td>SO<sub>3</sub></td><td>2.22</td><td>0.1</td></tr><tr><td>K<sub>2</sub>O</td><td>0.18</td><td>0.45</td></tr><tr><td>Na<sub>2</sub>O</td><td>0.11</td><td>0.35</td></tr><tr><td>TiO<sub>2</sub></td><td>0.15</td><td>0.7</td></tr></tbody></table>

<table><caption>**Table 4:** Physical properties of steel fibers</caption><thead><tr><th>Property</th><th>Steel fiber</th></tr></thead><tbody><tr><td>Specific gravity (g/cm<sup>3</sup>)</td><td>7.8</td></tr><tr><td>Length / diameter (-)</td><td>70</td></tr><tr><td>Resistivity (Ω.m)</td><td>1.43·10<sup>-7</sup></td></tr><tr><td>Tensile strength (Mpa)</td><td>2300</td></tr><tr><td>Young Modulus (GPa)</td><td>200</td></tr></tbody></table>

As shown in Table 5, from the base formulation, four mix designs were produced: F0, F20, F40 and F60 with fiber contents of 0, 20 kg.m⁻³, 40 kg.m⁻³ and 60 kg.m⁻³ respectively which corresponds to fiber volume content of 0, 0.26%, 0.51% and 0.77% respectively. Note that a fiber content of 78 kg.m⁻³ would correspond to fiber volume content of 1%. To ensure sufficient and similar workability of the fresh concretes, a polycarboxylate-based superplasticizer (Master-Glenium 201 BASF™) was added, from
2.37 kg.m⁻³ for F0 to 3.95 kg.m⁻³ for F60 (Table 5) to ensure workability. Results of Abrams slump tests have been reported in Table 5 for every mix design. These mix designs were validated by our industrial partner Charier™ and by Bekaert™.

<table><caption>**Table 5:** The four mix designs with different steel fiber contents</caption><thead><tr><th>Mix design</th><th>Fiber content (kg.m<sup>-3</sup>)</th><th>Fiber volume fraction (%)</th><th>Superplasticizer content (kg.m<sup>-3</sup>)</th><th>Slump* (cm)</th></tr></thead><tbody><tr><td>F0</td><td>0</td><td>0</td><td>2.37</td><td>21.0</td></tr><tr><td>F20</td><td>20</td><td>0.256</td><td>2.44</td><td>21.2</td></tr><tr><td>F40</td><td>40</td><td>0.513</td><td>3.00</td><td>16.2</td></tr><tr><td>F60</td><td>60</td><td>0.769</td><td>3.95</td><td>13.0</td></tr></tbody></table>

*Abrams slump test

From every mix design, 13 cylindrical concrete specimens, 11 cm diameter and 22 cm height, were produced. During concrete pouring in the molds, concretes have been vibrated according to NF EN 12390-2 standard (NF EN 12390-2, 2019) as a function of the measured values of slump.

## 2.2 Water porosity and compressive strength at day 28

After 28 days of moist cure, some specimens were cut in slices of 5 cm height. Eight cylindrical slices (2 per mix design), were weighed and dried in an oven at a temperature of 60 °C, a temperature that preserves the microstructural integrity of the concrete. After 5 months of drying, stabilization of the sample masses indicated complete evaporation of the water initially present in the open porosity $\varphi$. Thus, values of $\varphi$ were deduced: whatever the mix design, they were in the range of 11.1%±0.5, as shown in Table 6.

<table><caption>**Table 6:** Properties of the four mix designs</caption><thead><tr><th>Mix design</th><th>Open porosity at day 28 (%)</th><th>Rc at day 28 (MPa); σ(MPa)*</th><th>Rt at day 28 (MPa)**</th><th>Edyn at day 28 (Gpa); σ(GPa)***</th></tr></thead><tbody><tr><td>F0</td><td>11.06</td><td>54.7; 1.60</td><td>5.4</td><td>47.9; 2.77</td></tr><tr><td>F20</td><td>11.66</td><td>54.8; 4.35</td><td>5.3</td><td>46.5; 1.96</td></tr><tr><td>F40</td><td>10.98</td><td>59.9; 2.13</td><td>6.8</td><td>48.1; 2.27</td></tr><tr><td>F60</td><td>10.66</td><td>63.5; 1.75</td><td>7.1</td><td>48.3; 2.15</td></tr></tbody></table>

*Three samples per formulation, $\sigma$ is the standard deviation

**Only one test per formulation

***Eleven samples per formulation

## 2.3 Rapid Chloride Migration Test (RCMT)

The determination of chloride diffusivity in concretes in natural conditions is extremely long. Hence, the technique of RCMT was used, standardized as NT Build 492 (1999) and XP P18-462 v2 (2017): after the curing period, concrete samples were sawn into 50 mm thick cylindrical slices. The slice was placed between the two compartments of a migration cell. The downstream compartment contains a basic solution (NaOH: 3 g.l⁻¹); the same solution is laden with NaCl (35 g.l⁻¹) in the upstream compartment. An electrical potential difference of about 29 V is applied between the extremities of the cell. At the end of the test (duration $\Delta t$: about 48 h here), the specimen is split axially and inner surfaces are sprayed with AgNO₃ solution. Once the silver chloride precipitation on the surface is apparent as blue or white shades, as shown in Figure 3, penetration depths can be measured at regular intervals with a caliper since the position of the silver chloride front is clearly identifiable.
<figure><img src="image_4.png" /><figcaption>Figure 3. F40 sample split after RCMT, chloride fronts delimited by the orange lines</figcaption></figure>

The RCMT migration coefficient $D_{app}$, is given by the following expression:

$$D_{app} = 86.2 \frac{TL}{U \Delta t} \left( \bar{x} - 2.088 \cdot 10^{-3} \sqrt{86.2 \frac{TL\bar{x}}{U}} \right) \quad (1)$$

where $D_{app}$ is also called the chloride ions apparent diffusivity (x10⁻¹² m².s⁻¹), T is the ambient temperature (K), L is the thickness of the specimen (mm), $\bar{x}$ is the average value of the penetration depth (mm) and U is the potential difference (V) across the sample.

## 2.4 Electrical Resistivity

Measurements were carried out using a Wenner resistivimeter (see description of the Wenner protocol in (Polder 2001)). From the current I (A) and the measured potential difference ΔV (V), the transfer resistance of this quadrupole can be deduced: R = ΔV/I. This quantity depends very strongly on the 3D geometry and inter-electrode spacing a.

To overcome this, it is necessary to standardize it [Marescot et al. 2006] by introducing a geometrical factor G which can be calculated with Comsol Multiphysics software. For the geometry considered here, G was calculated to be 0.1352 [Palma Lopes et al. 2024].

The apparent resistivity ρ (Ω.m) is then given by the following formula:

$$\rho = G \frac{\Delta V}{I} \quad (2)$$

## 2.5 Mechanical measurements

In order to determine compressive resistances Rc, axial loading was applied on samples (cylindrical specimens, 11 cm diameter and 22 cm height) with a load control system using a hydraulic loading frame with a maximum capacity of 1500 kN. The loading rate was kept constant to a value equal to 2 kN.s⁻¹.

Regarding tensile resistances Rt, the diametrical loading method (i.e. under the Brazilian splitting test configuration) was used: it provides an estimation of the tensile stress through indirect tension (Djerbi et al. 2008).

The dynamic Young's modulus $E_{dyn}$ was determined using the Impulse Excitation Technique (Grindo Sonic equipment) which is a non-destructive measurement simple to perform [Djerbi et al. 2013]. The
Grindo Sonic instrument records the vibration, makes an analysis in the time domain and measures the natural frequency of the dominant vibration mode. The value of E<sub>dyn</sub> was computed based on the sample dimensions, mass and measured frequency according to ASTM E1876-01 [ASTM, E1876-01, 2001].

# 3 Experimental results

## 3.1 Chloride diffusivity

RCMTs were performed after 28 days and 120 days of moist cure. Each time, two tests were carried out per mix design. Fig. 3 shows a tested slice (F40) with the chloride fronts delimited by the orange lines.

Measured D<sub>app</sub> as a function of the fiber content is shown in Figure 4. The discrepancies between the tests are significant, as variability is inherent to this testing method (Aït-Mokhtar et al. 2013). The highest variability is obtained for F40: this can be explained by a less good homogeneity of the mix at this fiber content. Clearly the mixing process has not been efficient enough, and/or the superplasticizer has been under dosed. However, it clearly appears that D<sub>app</sub> increases with the fiber content: at day 28, from about 2.35·10<sup>-12</sup> m<sup>2</sup>.s<sup>-1</sup> (F0) to about 3.6·10<sup>-12</sup> m<sup>2</sup>.s<sup>-1</sup> (F60), and at day 120, from about 2.2·10<sup>-12</sup> m<sup>2</sup>.s<sup>-1</sup> (F0) to about 3.2·10<sup>-12</sup> m<sup>2</sup>.s<sup>-1</sup> (F60). Slight decreases of D<sub>app</sub> seem to occur between day 28 and day 120, but they are too small to be significant with respect to the measurement uncertainties. Increases of D<sub>app</sub> are moderate between fiber contents of 0 and 40 kg.m<sup>-3</sup> and then stronger between fiber contents of 40 and 60 kg.m<sup>-3</sup>. Corrosion affects the fibers from the exposed surfaces to the chloride fronts. In Figure 5a (sample F40), the effects of corrosion on one fiber can be observed in the formation of a preferential pathway for chlorides.

To interpret these observations, it is noteworthy that the potential difference applied during these tests (29 V) inevitably resulted in a decrease in the electrochemical potential of the steel and therefore in a strong acceleration of the corrosion process (Beglarigale and Yazici 2017). This phenomenon can be illustrated by the Pourbaix diagram of the Fe-H<sub>2</sub>O system (Sousa Rivetti et al. 2017): with a decrease in its potential, the Fe element shifts from the passivity to the corrosion zone. Thus, the increases of D<sub>app</sub> with fiber content may have been strengthened by this phenomenon.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Apparent chloride diffusivities measured by RCMTs at day 28 and day 120 as a function of the fiber content</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 5.** (a) Preferential pathway formed by the total corrosion of a steel fiber (F40 sample split) after RCMT, (b) macrophotograph of an exposed steel fiber after natural conditions of chloride ingress (F60 sample split)</figcaption></figure>

In (Kim et al. 2021), the authors applied RCMT (using a potential difference of 30 V) to a similar base concrete (50% cement / 50% BFS) at day 28 and found a $D_{app}$ of about $2.5 \cdot 10^{-12} \text{ m}^2 \cdot \text{s}^{-1}$ (no fiber) which is similar to the value of $2.35 \cdot 10^{-12} \text{ m}^2 \cdot \text{s}^{-1}$ measured here. However, on adding 40 kg.m$^{-3}$ of steel fibers (volume fraction of about 0.5%), they found a decrease of $D_{app}$ by about 30% whereas an increase by about 13% was measured here. Therefore, opposite trends are found between these two studies.

For qualitative observations, a F60 specimen was subjected to natural chloride ingress by immersion in salted water (NaCl: 35 g.l$^{-1}$) during 10 months and then split axially. Figure 5b shows a fiber flush to the exposed face which has been laid bare: some very slight spots of superficial corrosion around the fibers can be observed to a few millimeters in depth. This adds qualitative elements to the above measurements suggesting that the corrosion process is very slow under natural exposure conditions.

Overall, from these measurements, the optimal fiber content is 40 kg.m$^{-3}$ (volume fraction of about 0.5%): the increase by 13% of $D_{app}$ due to the presence of fibers remains quite reasonable up to this fiber content.

## 3.2 Temporal evolution of resistivity

For the measurements of resistivity, the specimens were tested regularly from day 4 to day 166. Results are shown in Figure 6. For every mix design, a point represents the average value of the measurements performed on several specimens and the error bar represents the standard deviation. Error bars tend to increase with time because the number of available specimens decreased (many tests performed in the framework of the project were destructive), from 13 specimens at day 4 (for every mix design) to only one at day 166 (and hence no error bar).

F0 shows a three-stage evolution: from day 4 to around day 28, resistivity increases fast until it reaches 200 Ω.m. Between day 28 and day 96, resistivity continues to increase but more slowly to reach 276 Ω.m and then, it seems to stabilize to finally reach 287 Ω.m at day 166: these results are in accordance with those found in (Eid et al., 2023) on concrete specimens with Portland cement excepted the kinetics as the binders contained only Portland cement. These evolutions are due to the hydrations of cement and slag which induce reduction in porosity and thus in saturation rate as shown in (Ali Ahmad et al. 2024, Atallah et al. 2024a). The hydrated products progressively block the passage of ions, as the interconnectivity between the pores and their size decreases over time.
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Temporal evolution of concrete resistivity measured for the four mix designs</figcaption></figure>

For mix designs containing fibers, the behavior is similar: a significant increase in resistivity is measured in the first 28 days. At this stage, the concrete is very porous and a large part of the conductivity is induced by the interstitial solution. However, thereafter, the resistivity increases less and finally seems to stabilize. Conductivity is both electrolytic and electronic. As the pores begin to close, conductivity becomes predominantly electronic and is therefore less dependent on the interstitial solution. Obviously, a high dosage of long fibers makes the material highly conductive and lessens the impact of the characteristics of the pore solution and therefore of the hydration of the material (Cleven et al. 2022). That being said, for a given content, steel fibers have a constant contribution to the concrete resistivity and presumably do not have any effect on the evolution of the hydration degree.

## 3.3 Mechanical properties

From mechanical tests performed at day 28, compressive strength $R_c$ was found to increase gradually with fiber content, from 54.7 MPa for F0 to 63.5 MPa for F60 (see Table 6). This is quite remarkable because adding steel fibers in concrete does not necessarily result in better compressive strengths (Fattouh et al. 2023). Furthermore, tensile strength $R_t$ progressively increases with fiber content, from 5.4 MPa for F0 to 7.1 MPa for F60.

For F40, compared to F0, the increase in tensile / compressive strengths $R_t$ and $R_c$ are about 26% / 10% respectively at day 28. A fiber content of 60 kg.m⁻³ leads to even better mechanical properties but presents serious disadvantages: a significant increase of $D_{app}$, and the difficulty of the mixing process despite the addition of superplasticizer. Fiber "urchins" can form, preventing fibers from anchoring in the cementitious matrix (Dinh et al. 2016) and hence propitious to chloride ingress.

Grindosonic measurements performed at day 28 led to a dynamic Young's modulus $E_{dyn}$ of from 47.9 GPa for F0 to 48.3 for F40 and therefore did not evidence a clear effect of the presence of fibers.

# 4 Modeling by multiscale homogenization methods

## 4.1 Theoretical background

Concrete is a heterogeneous material, containing aggregates, sand, hydration products (including C-S-H gel), capillary pores, anhydrous particles, and optionally fibers. The characteristic sizes of this heterogeneity range from the nanometer to the centimeter (Bernard et al. 2003). Its properties (diffusivity, conductivity, ...) at the structural scale, here referred as the macroscopic scale, depend on its microstructure. Homogenization methods aim at determining the material properties at the
macroscopic scale (for example, the homogenized conductivity $\sigma^{\text{hom}}$), to be used by engineers for the structural design, from the properties of its constituents at microscopic scale (e.g. local conductivities $\sigma_i$) and their geometrical organization. This section provides an overview of homogenization methods to determine the effective properties of heterogeneous materials. For further developments, see (Buryachenko 2007, Kachanov and Sevostianov 2018).

The presentation focuses on electrical conductivity. Note that the homogenization of ionic diffusion proceeds by analogy with that of electrical conductivity, by replacing the local current density by the ionic flux, the electrical conductivity by the diffusion coefficient and the electrical field by the ion concentration gradient.

Homogenization methods assume the separation of scale principle: the characteristic size $L$ of the fluctuation of the fields at the structure scale is assumed much larger than that of a Representative Volume Element (RVE) of the material. In turn, the RVE is assumed much larger than the characteristic sizes $\ell$ of microstructure heterogeneity. Under these assumptions, a distinction can be made between:

- the macroscopic fields used at the macroscopic scale in engineering design (the macroscopic current density $J(X)$ and electrical field $E(X)$), which vary with a slow space variable $X$ scaling as $L$;

- the microscopic current density and electrical field fluctuations $(j - J)(x)$ and $(e - E)(x)$, which vary with a fast space variable $x$ scaling as $\ell$ used at the microscopic scale.

Considering a Representative Volume Element (RVE) $\Omega$ of volume $V_\Omega$ in the studied medium, the macroscopic constitutive law is:

$$ J = \sigma^{\text{hom}} E \qquad (3) $$

where $\sigma^{\text{hom}}$ is the homogenized conductivity. This is the classical Ohm's law. At the microscopic scale, the conductivity is a heterogeneous field. In a material phase $i$ which occupies a subdomain $\Omega_i$ of volume $V_i$ within the RVE, the local expression of Ohm's law relates the local current density $j_i$ to the local electrical field $e_i$ by the local electrical conductivity $\sigma_i$:

$$ j_i = \sigma_i e_i \qquad (4) $$

From two scale series expansion of the fields in the scale separation parameter $\epsilon = \ell/L \ll 1$, the homogenized conductivity $\sigma^{\text{hom}}$ can be obtained from the various $\sigma_i$ by defining a homogenization problem on the RVE. In that problem, $E$ appear as a loading term at the boundary of the RVE, $j$, $e$ are unknown fields related by the local ohm law, and satisfying the current conservation law $\text{div } j = 0$ and compatibility equation $\text{curl } e = 0$ (Sanchez-Palencia 1974, Jikov et al. 1994). The macroscopic variables $J$ and $E$ are related to their microscopic counterparts $j$ and $e$ by volume averaging over the RVE:

$$ J = \langle j \rangle_{\Omega} = \frac{1}{V_{\Omega}} \int_{\Omega} j dV = \sum_{i} f_{i} \langle j \rangle_{i} \qquad (5) $$

where: $\langle j \rangle_i = \frac{1}{V_i} \int_{\Omega_i} j_i dV$, $\langle i \rangle_i = \frac{1}{V_i} \int_{\Omega_i} j_i dV$, and: $V_\Omega = \sum_i V_i$

$$ E = \langle e \rangle_{\Omega} = \frac{1}{V_{\Omega}} \int_{\Omega} e dV = \sum_{i} f_{i} \langle e \rangle_{i} \qquad (6) $$

Solving formally the homogenization problem yields the local electrical current $j$ field and, by averaging, the macroscopic current $J$ as a linear function of $E$, and hence the sought homogenized conductivity $\sigma^{\text{hom}}$ such that: $J = \sigma^{\text{hom}} E$. Two main type of homogenization methods exist. Full field homogenization defines explicitly an RVE of the material, and usually solve the homogenization
problem using numerical simulations. Full field methods are not used here because of the high computational cost and the difficulty to represent faithfully the microstructure of concrete (and in particular, of cement paste) over several length scales. Instead, mean field homogenization methods used here have a negligible computational cost and rely on reduced morphological information on the microstructure. In mean field homogenization, only estimates of the average of j and e in each type of constituent, and eventually J, are sought for any value of E, which is sufficient to provide an estimate of $\sigma^{hom}$.

Estimates of the homogenized conductivity are built from a series of auxiliary problems, the so-called Eshelby problems. For each type i of heterogeneity with conductivity $\sigma_i$, one defines a problem featuring an ellipsoidal inclusion of uniform conductivity $\sigma_i$ in a uniform reference material with conductivity $\sigma_0$, on which an effective uniform electrical field $E_0$ is applied. Based on Eshelby (1957), the electrical field is uniform inside the ellipsoidal inclusion i, with the following expressions:

$$ \mathbf{j}_i = \mathbf{A}_0^i \cdot \mathbf{E}_0, \text{ with: } \mathbf{A}_0^i = \left[ \mathbf{I} + \mathbf{P}_0^i \cdot (\boldsymbol{\sigma}_i - \boldsymbol{\sigma}_0) \right]^{-1} \quad (7) $$

where $\mathbf{I}$ is the identity tensor and $\mathbf{P}_0^i$ is the Hill tensor of the ellipsoidal inclusions i which can be expressed as a function of $\sigma_0$ and of the shape of the inclusion.

In the case where the conductivity within the inclusion is not uniform, equation (7) can still be used, provided $\sigma_i$ is replaced by an equivalent conductivity $\sigma_i^{eff}$ of the heterogeneous inclusion (Barthélémy and Bignonnet 2020). This will be used in the following to model a fiber surrounded by an interphase as a heterogeneous inclusion. In the case where the inclusion is made of a spheroidal core of conductivity $\sigma_c$ surrounded by a spheroidal shell of conductivity $\sigma_s$, which is confocal with the core, its equivalent conductivity is obtained by:

$$ \boldsymbol{\sigma}_i^{eff} = \boldsymbol{\sigma}_s + \phi_c \left[ \left( \boldsymbol{\sigma}_c - \boldsymbol{\sigma}_s \right)^{-1} + \mathbf{P}_s^c - \phi_c \mathbf{P}_s^s \right]^{-1} \quad (8) $$

where $\phi_c$ is the ratio of the volume of the core to the total volume of the heterogeneous inclusion (core + shell), and $\mathbf{P}_s^c$ (respectively $\mathbf{P}_s^s$) is the Hill tensor of the spheroidal shape of the core (respectively of the whole heterogeneous inclusion) in a reference medium with conductivity $\sigma_s$.

For matrix-inclusion composites, the RVE is now considered as composed of a matrix phase m in which are embedded several families of inclusions i. In the Mori-Tanaka scheme (Mori and Tanaka 1973), the electrical field localization in each inclusion phase i is estimated from the Eshelby problem in which $\sigma_0 = \sigma_m$ and the effective field e is taken equal to the intrinsic average of the local electric field over the matrix phase. Then, from the averaging rules (5) and (6), $\sigma^{hom}$ can be estimated by the following expression:

$$ \boldsymbol{\sigma}^{hom} = \left( \sum_{i} f_{i} \boldsymbol{\sigma}_{i} \mathbf{A}_{m}^{i} \right) \cdot \left( \sum_{i} f_{i} \mathbf{A}_{m}^{i} \right)^{-1} \quad (9) $$

This scheme can be used whatever the number n of families of ellipsoidal inclusions i, each of them being associated with a given ellipsoidal shape, orientation and tensor of electrical conductivity. The family of spheroidal inclusions i is defined by its aspect ratio $\omega_i$ (axial length / diameter) and by its volume fraction $f_i (= V_i/V_\Omega)$ in the RVE. In this work, when the inclusion i is not a sphere, an isotropic distribution of orientations of the ellipsoids is assumed (see (Achour et al. 2020), Appendix A for practical details). In the case where the conductivity within the inclusion i is not uniform, eq. (9) can still be used, provided $\sigma_i$ is replaced by the equivalent conductivity $\sigma_i^{eff}$ defined in equation (8) (Barthélémy and Bignonnet 2020).
For composites which do not have a matrix phase, the so-called "self-consistent scheme" (Hill 1965) is used instead. Its expression is an explicit equation similar to equation (9), in which $\sigma_m$ is replaced by the sought for $\sigma^{\text{hom}}$ and $P^i_m$ by $P^{i\text{hom}}$.

Finally, the resistivity of the homogenized material $\rho^{\text{hom}}$ is simply given by $1/\sigma^{\text{hom}}$. By analogy, these schemes can be applied to determine the effective chloride diffusivity in the material $D_{\text{eff}}$ (Achour et al. 2020).

## 4.2 The different scales of homogenization

The description of the cement paste is inspired from the multiscale "engineering" models (Pivonka et al. 2004; Pichler and Hellmich 2011; Damrongwiriyanupap et al. 2017). Three main phases are distinguished: anhydrous clinker, hydration products and capillary pores. Moreover, as shown by Figure 7, the description of the cement paste is decomposed in two scales (Achour et al. 2020):

- Level 1: a hydrate foam is described as an aging disordered assemblage of hydration products and capillary pores. To simplify, the hydrates are considered as a single porous phase, which includes gel porosity. Hydrates are modelled by oblate spheroids and the capillary pores by prolate spheroids with a finite aspect ratio, so that the connectivity of these phases will depend on the porosity of the hydrate foam. A self-consistent scheme is used over an isotropic distribution of orientations of both types of spheroids. Aging is accounted for by the reduction of porosity with time obtained from a hydration model.

- Level 2: the cement paste is described as a composite with a matrix of hydrate foam from level 1 in which spherical anhydrous grain inclusions named "clinker" are embedded, using the Mori-Tanaka scheme.

At the scale of mortar (level 3), the Mori-Tanaka scheme is used to represent the mortar with a matrix of cement paste embedding sand particle inclusions, which are considered as spherical.

At the scale of concrete (level 4), the Mori-Tanaka scheme is used. Inside the matrix represented by the mortar, two types of inclusions are considered. Gravels are considered as spherical inclusions. Fibers are considered as prolate spheroidal inclusions. Their distribution of orientation is assumed isotropic. In the case of RCMTs, fibers are assumed surrounded by an interphase. This interphase represents steel which has been completely corroded and assumed as a shell completely devoid of any solid matter. Its equatorial thickness is noted $\Delta r_{\text{int}}$.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** The four levels of homogenization, from the hydrate foam to the concrete</figcaption></figure>
For the different constituents at the different levels, the values of properties, volume fractions and aspect ratios are reported in Table 7. Note that $F_h$, $F_s$ and $F_g$ are formation factors of the hydrate, sand and gravel phases that will be defined later and VCCTL is the hydration model (see next section).

<table><caption>**Table 7:** Input parameters for the multiscale model</caption><thead><tr><th>Constituent</th><th>Electrical conductivity (Ω<sup>-1</sup>.m<sup>-1</sup>)</th><th>Chloride diffusion coefficient (m<sup>2</sup>.s<sup>-1</sup>)</th><th>Volume fraction (%)</th><th>Aspect ratio (-)</th></tr></thead><tbody><tr><td colspan="5"><strong>Level 1: Hydrate foam</strong></td></tr><tr><td>Capillary pores</td><td>σ<sub>cp</sub> from VCCTL</td><td>D<sub>cp</sub> = 2.1·10<sup>-9</sup>*</td><td>from VCCTL</td><td>8 (adjusted)</td></tr><tr><td>Hydrate</td><td>σ<sub>cp</sub> / F<sub>h</sub>, with F<sub>h</sub> = 1.25·10<sup>4</sup> (adjusted)</td><td>D<sub>cp</sub> / F<sub>h</sub></td><td>from VCCTL</td><td>0.013</td></tr><tr><td colspan="5"><strong>Level 2: Cement paste</strong></td></tr><tr><td>Hydrate foam</td><td>homog. from level 1</td><td>homog. from level 1</td><td>from VCCTL</td><td>- (matrix)</td></tr><tr><td>Clinker</td><td>0</td><td>0</td><td>from VCCTL</td><td>1</td></tr><tr><td colspan="5"><strong>Level 3: Mortar</strong></td></tr><tr><td>Cement paste</td><td>homog. from level 2</td><td>homog. from level 2</td><td>≈ 46</td><td>- (matrix)</td></tr><tr><td>Sand</td><td>σ<sub>cp</sub> / F<sub>s</sub></td><td>D<sub>cp</sub> / F<sub>s</sub></td><td>≈ 54</td><td>1</td></tr><tr><td colspan="5"><strong>Level 4: Concrete</strong></td></tr><tr><td>Mortar</td><td>homog. from level 3</td><td>homog. from level 3 or adjusted to 2.2·10<sup>-12</sup></td><td>62</td><td>- (matrix)</td></tr><tr><td>Gravels</td><td>σ<sub>cp</sub> / F<sub>g</sub>, with F<sub>g</sub> = F<sub>s</sub></td><td>D<sub>cp</sub> / F<sub>g</sub></td><td>38</td><td>1</td></tr><tr><td>Steel fiber with interphase</td><td>steel core: 7.10<sup>6</sup><br>(Δr<sub>int</sub> = 0)</td><td>steel core: 0<br>interphase: D<sub>cp</sub></td><td>between 0 and 0.77</td><td>equivalent inclusion: 70 steel core: depending on Δr<sub>int</sub></td></tr></tbody></table>

## 4.3 The hydration model

The Virtual Cement and Concrete Testing Laboratory (VCCTL) software (Bentz and Garboczi 1991; Garboczi and Bentz 1991; Bentz 2005; VTTCL online documentation 2022) was used. It is based on the CEMHYD3D system, widely used in the literature for many cement types and more specifically for Portland and slag blended cements (Chen et al. 2007; Atallah et al. 2023). The hydration model of VCCTL, along with other hydration models, has been compared to experimental results on cement pastes with BFS in (Atallah et al. 2023). It is here selected as it showed a reasonable accuracy regarding the evolution of the composition of cement paste during hydration and provides an explicit dependence on time, although slag hydration rate is sometimes underestimated as compared to an extensive experimental database (Atallah et al. 2024a and 2024b).

Input parameters are the water to cement ratio, the clinker and slag composition and ratios. Outputs include the volume fractions of anhydrous phases, hydration products and capillary pores within the cement paste, the pore solution concentrations for various ions and the resistivity of the pore solution as a function of time: results are shown in Figures 8a and 8b.

The evolutions of the volume fractions are inputs for the homogenization schemes (see e.g., eq. (9)). The pore solution conductivity is input as the conductivity of the capillary pore phase at level 1, and is used to compute the conductivity of the other porous phases (hydrates, sand, gravels, see Table 7).

<footnote>*Cl ions diffusivity in free water</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 8. Results of the hydration model as a function of time, (a) volume fractions of the different phases and (b) resistivity of the poral solution</figcaption></figure>

## 4.4 Applications

### 4.4.1 Model input parameters

- **Geometrical parameters:** the aspect ratio (length over diameter) of intact steel fibers is taken as 70 from the geometrical dimensions provided by the fiber producer (0.9 mm diameter, 63 mm length). The aspect ratio of hydrates $\omega_h$ is set to 0.013. This value has been selected in (Achour et al. 2020) to correctly reproduce the setting of cement paste with the self-consistent scheme, due to the percolation of the solid phase when the hydrate volume fraction to capillary porosity exceeds an $\omega_h$-dependent threshold. Based on these authors, the aspect ratio of capillary pores $\omega_{cp}$ was considered equal to 6 for CEM I cement pastes, but it is considered here as an adjustment parameter for slag blended cement pastes to properly account for the connectivity of capillary pores, which is $\omega_{cp}$-dependent in the self-consistent scheme (see Achour et al. 2020, Appendix).

- **Volume fractions:** at the scale of the paste, the temporal evolution of volume fractions of capillary pores, anhydrous clinker and hydrates was calculated by the hydration model. At the scale of the mortar, $f_p$ (cement paste) and $f_s$ (sand particles) given in Table 7 are deduced from Table 1. For fibers, $f_f$ are between 0 and 0.77% depending on the mix design. Depending on $f_f$, $f_m$ and $f_g$ were readjusted proportionally.

- **Local properties:** the temporal evolution of the electrical conductivity of the poral solution in the capillary pores $\sigma_{cp}$ was calculated by the hydration model. The diffusion coefficient in the interphase, when considered, is set to the one in the capillary pores. The conductivity (and the chloride diffusivity) in the porous phases i (hydrates, sand, gravel) is related to the poral solution by their formation factors $F_i$ defined as (Baroghel-Bouny et al. 2011; Chidiac and Shafikhani 2020):

$$ \sigma_i = \sigma_{cp} / F_i \qquad (10) $$

The formation factors account for the porosity, connectivity and tortuosity of the porous phases. For the aggregates, by assuming that the chloride diffusion coefficient of the aggregates is similarly related to the chloride diffusion coefficient in capillary porosity, it is estimated by: $D_{agg} = D_{cp} / F_{agg}$. From chloride migration tests carried out in (Achour 2018) on a sample of rock coming from the Boulonnais quarry, the formation factor of the sand and gravel is found to be $F_s = F_g = 1.84 \cdot 10^4$.

The formation factor of the hydrates $F_h$ was considered as the second adjustment parameter. Electrical conductivity and chloride diffusivity in anhydrous clinker is assumed negligible (see Table 7).
### 4.4.2 Modeling of temporal evolution of resistivity

The resistivity measurements did not trigger corrosion, hence $\Delta r_{int}$ is considered equal to 0. Model results and measurements are compared in Figure 9a. Adjustments of $\omega_{cp}$ and $F_h$ were performed in such a way that results and measurements are in perfect accordance for the F0 mix design at days 28 and 96: it led to values of $\omega_{cp}$ and $F_h$ equal to 8 and $1.25 \cdot 10^4$ respectively. As shown by Fig. 9a, calculations and measurements show an agreement whatever the fiber content. Most of the numerical results are within the error bars of the measurements. This indicates that the morphological description proposed here is adequate.

<figure><img src="image_10.png" /><figcaption>Figure 9 (9a,9b). Temporal evolutions of (a) concrete resistivity, measurements (points) and numerical simulations (curves) and (b) effective chloride diffusivity obtained by the numerical simulations, for the four mix designs</figcaption></figure>

From the hydration model, the hydration degrees of clinker and slag at day 166 are 93.5% and 25.3% respectively. Thus, the kinetics of hydration of slag is clearly slower than the kinetics of hydration of the clinker. After day 96, the stabilization of the hydration observed experimentally is only apparent, the hydration of slag continues slowly.

### 4.4.3 Modeling of chloride diffusivity

The diffusivities $D_{app}$ shown in Fig. 4 are apparent ones. Effective diffusivity $D_{eff}$ (at the scale of the concrete) can be deduced from $D_{app}$ by the following relation (DuraCrete 1998):

$$D_{eff} = (\varphi + slp) D_{app} \qquad (11)$$

where slp is the slope of the chloride binding isotherm. According to investigations by Ben Fraj et al. (2012) for a Slag-Based Concrete (SBC) similar to F0, slp is estimated at 0.45: this is very close to the value of 0.46 proposed in Reuge et al. (2023). Thus, knowing that $\varphi$ of the concretes studied here is about 0.11, $D_{eff}$ is approximately equal to $0.56D_{app}$.

From Achour et al. (2018), chloride diffusivity in sand and gravels was measured as $1.1 \cdot 10^{-13} m^2 \cdot s^{-1}$.

Considering no fiber (F0), model results are shown in Figure 9b: they lead to values of $D_{eff}$ of $3 \cdot 10^{-12} m^2 \cdot s^{-1}$ and of $1.95 \cdot 10^{-12} m^2 \cdot s^{-1}$ at days 28 and 120 respectively. Such a temporal evolution cannot be really validated by the experimental results due to the measurement uncertainties. From the measurements, at day 120, $D_{eff}$ is $1.23 \cdot 10^{-12} m^2 \cdot s^{-1}$: the relative discrepancy of the calculations with the measurements (45%) is rather limited considering the measurement uncertainties and the fact that no adjustment parameter was used here for the model regarding diffusion properties. Moreover, model results (Fig. 9b) show the increase in $D_{eff}$ as a function of the fiber content, considering the three mix
designs F20, F40 and F60 and considering $\Delta r_{int}$ equal to 22 µm. When $\Delta r_{int} = 0$, the curves for these fiber contents are superposed on the one for F0. It is noteworthy that from the results of Fig. 9b, the following deduction can be made: if the measurements are not carried out in the same period of time after cast, a decrease of $D_{app}$ can be obtained as a function of the fiber content; this could explain some inconsistent results found in the literature.

In the following developments which focus on the effect of the fiber reinforcement on $D_{eff}$ at day 120, chloride effective diffusivity at the scale of the mortar $D_{Cl,m}$ is adjusted to $2.2 \cdot 10^{-12} m^2.s^{-1}$ to allow a perfect adjustment of $D_{eff}$ with the measurements on the fiber free samples as shown in Figure 10.

From Fig. 10, for F20 and F40, adjustment on measurements is good considering $\Delta r_{int}$ equal to 22 µm. $D_{eff}$ increases almost linearly as a function of the volume fraction of fiber $f_v$ since the latter is very small (below 1%). Experimentally, the trend is also linear for fiber contents between 0 and $40 kg.m^{-3}$ and then increases further at $60 kg.m^{-3}$: this may be due to an increasing connectivity of the preferential pathways at such a high fiber content. They develop not only along the corroded fibers but also in the potential crack networks. For F60, $\Delta r_{int}$ has to be taken equal to 32 µm. Note that Fig. 11 also shows the results obtained with $\Delta r_{int}$ equal to 0 and $50 \mu m$.

High contents of steel fibers may promote development of cracks; therefore, it is not recommended to use fiber contents greater than $40 kg.m^{-3}$ (or about 0.5 in volume content).

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Effective chloride diffusivities as a function of the fiber content at day 120, measured by RCMTs (points) and obtained by the numerical simulations (lines)</figcaption></figure>

## 5 Conclusion

Steel fiber-reinforced slab-based concretes with the same base formulation of the concrete (binder: 50% cement, 50% slag), one formulation without fiber and three formulations with different steel fiber contents, $20 kg.m^{-3}$, $40 kg.m^{-3}$ and $60 kg.m^{-3}$, were cast and studied. The main conclusions are:

- Rapid chloride migration tests revealed an increase in the chloride ions diffusivity as a function of the fiber content, moderately (up to 13%) between 0 and $40 kg.m^{-3}$ and more significantly (53%) for $60 kg.m^{-3}$. This is due to the corrosion of the steel fibers induced by the decrease in the electrochemical potential of the steel fibers and by the presence of fiber "urchins" for high fiber content ($> 40 kg.m^{-3}$).
- The optimal fiber content proved to be 40 kg.m⁻³ (corresponding to a volume fraction of about 0.5%) with still a relatively low chloride ions diffusivity and significantly better mechanical strengths than without fibers: +26% / +10% for tensile / compressive strengths respectively at day 28.

- The temporal evolution of resistivity proved to be slow relatively to concretes based on ordinary Portland cement. This is a known effect of the presence of slag which slows down the hydration process. Moreover, as expected, the presence of steel fibers tends to lower the resistivity of the concrete in proportion to the fiber content. The hydration and resistivity are correlated.

- A multiscale homogenization scheme was developed. In terms of temporal evolution of resistivities, calculations and measurements showed an agreement within experimental error bars whatever the fiber content.

- Regarding chloride diffusivities, the model reproduced measurements for fiber contents between 0 and 40 kg.m⁻³ by accounting for the presence of corroded fractions of steel fibers of 22 µm width due to the migration test itself.

- The adequacy of the model with experiments suggests that both the multiscale homogenization scheme and the hydration model are valid and shows the robustness of the approach.

Thus, experimental and modeling points of view are quite complementary and have enabled the mechanisms involved in the case of the steel fiber-reinforced slab-based concretes studied here to be identified.

In practice, in future considerations, the knowledge of effective chloride diffusivities (as functions of time and corrosion degree of the fibers) will allow performing service life estimations of a structure exposed to chloride ingress induced by drying/wetting cycles.

The effects of natural exposure conditions to chloride on the concrete properties remain to be investigated. This would further improve the assessment of the interest of using steel fiber-reinforced slab-based concretes for structural purposes in marine environments.

Subject to a cost analysis, galvanized or stainless-steel fibers could be an option.

## Acknowledgements

This study was carried out within the framework of the WEAMEC program through the B2FE project. WEAMEC and its local partners, Région Pays de la Loire (France), FEDER, CARENE and Nantes Métropole, are gratefully acknowledged for their support. Moreover, progress has been achieved within the framework of the DEMCOM Project (grant ANR-20-CE22-0008-01), funded by the Agence Nationale de la Recherche (ANR). For the purpose of open access, the author has applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

## Preprint

A preprint of this manuscript has been published (Reuge et al. 2025b) prior to the submission of this manuscript to European Journal of Environmental and Civil Engineering.

## Declaration of interests

The authors report there are no competing interests to declare.
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