Resistivity Prediction Model for Basalt–Polypropylene Fiber-Reinforced Concrete

Abstract

As a new cement-matrix composite material, fiber-reinforced concrete performs excellently in freeze–thaw resistance, tensile resistance, and seepage resistance. Since resistivity can be used to evaluate the performance of concrete, the resistivity of basalt–polypropylene fiber-reinforced concrete (BPFRC) was investigated in this study. Various parameters such as fiber type, fiber content, and water–binder ratio were also analyzed. The resistivity of the BPFRC was tested by an improved two-electrode alternating current (AC) method, and the differences in resistivity were analyzed using scanning electron microscopy (SEM) and mercury intrusion porosimetry (MIP). The results showed that adding fiber affected the resistivity of the BPFRC, and the basalt fiber had a more significant effect than the polypropylene fiber. The resistivity of the BPFRC was sensitive to the water–binder ratio, which showed a negative correlation. In addition, a 365-day resistivity model for the BPFRC was established by considering variables such as fiber, water–binder ratio, ambient temperature, and ambient relative humidity. The hydration equations of various cementitious materials were derived, and a time-varying resistivity model for the BPFRC was developed. Finally, the established resistivity model of the BPFRC lays a foundation for the further building of the relationship between resistivity and BPFRC performance.

1. Introduction

The mechanical properties and durability of concrete are directly related to the quality and sustainability of concrete structures. Generally, laboratory destructive tests are used to measure the various performances of concrete, which requires a large number of specimens and is time-consuming and laborious. In view of this, researchers have been trying to find more reasonable property evaluation methods [1,2,3]. The nondestructive testing method has developed rapidly in recent years. This testing method is low-cost and easy to operate, offering the possibility of real-time and continuous evaluation [4,5]. Related research has demonstrated that resistivity, which can be determined through nondestructive testing, can be used as an index to measure the performance of concrete [6,7,8]. After extensive investigation, researchers discovered that the resistivity method could assess various performances of concrete, including compressive strength, elastic modulus, impermeability, and resistance to chloride ions [6,7,8]. For example, Shao et al. [9] measured the resistivity of concrete with various aggregate contents using the four-electrode method and investigated the relationship between resistivity and mechanical properties. A linear regression model for the compressive strength and resistivity was established. Meanwhile, a nonlinear regression model for elastic modulus and resistivity was developed. Andrade [10] and Lu [11] found that the resistivity can be used as an index to evaluate the chloride ion resistance of concrete. Chidiac and Shafikhani [12] analyzed the effect of variables such as coarse aggregates, water–cement ratio, and admixtures on the resistivity and established a model between resistivity and the chloride ion diffusion coefficient. Overall, it is feasible to use the resistivity to evaluate the properties of concrete, and the application prospect of this index is broad [6,7,8,9,10,11,12,13].

To facilitate the application of the resistivity index in engineering, researchers analyzed the influence of different parameters on the resistivity and developed various resistivity prediction models [14,15,16,17]. For example, DuraCrete [14] investigated the resistivity of ordinary silicate cement concrete and developed a prediction model for concrete resistivity considering the water–cement ratio. Gong et al. [15] tested the concrete resistivity using a two-electrode direct current (DC) method and developed an empirical model for concrete resistivity considering the ambient humidity, chloride ion concentration, and water–cement ratio. Yu et al. [16] established a theoretical model for concrete resistivity considering the temperature, humidity, water–cement ratio, and chloride ion concentration.

In recent years, fiber-reinforced concrete has attracted the interest of researchers. It is a new type of cement-based composite material with concrete as the matrix and non-continuous short fibers as the reinforcing material [17,18,19,20,21,22,23]. It has been successfully used in highways, bridges, dams, and residential buildings for its excellent crack resistance and toughening, impact resistance, shrinkage limitation, bending and tensile resistance, freeze–thaw resistance, and seepage resistance [21]. Among them, basalt–polypropylene fiber-reinforced concrete (BPFRC), as a new type of fiber-reinforced concrete, has a wide range of application prospects. The basalt fiber with high elastic modulus can improve the tensile strength and toughness of concrete and reduce the dry shrinkage deformation of concrete [8]. The polypropylene fiber with low elastic modulus can inhibit the formation and development of early plastic shrinkage microcracks in concrete [21,23]. Consequently, a proper mixture of basalt and polypropylene fibers can significantly improve the performance of concrete [23].

This study aims to investigate the resistivity of BPFRC and analyze the effect of fiber type, fiber content, and water–binder ratio on the resistivity. Based on previous research results, the BPFRC resistivity was tested in this study using an improved two-electrode AC method [24,25,26,27,28]. Additionally, the effects of fiber type, mixing method, admixture dosing, and water–binder ratio on the BPFRC resistivity were analyzed. The effect was positive with a low volume content of fiber (e.g., 0.1%) and negative with the high one (e.g., 0.2%). The differences in the BPFRC resistivity were analyzed by MIP and SEM. In addition, a 365-day resistivity model for BPFRC was established, considering various factors. Meanwhile, a time-varying resistivity model of BPFRC was established in consideration of the hydration process of cementitious material. On this basis, it is hoped that the resistivity model of the BPFRC can be developed to lay a foundation for further establishing the relationship between resistivity and BPFRC properties.

2. Materials and Methods

2.1. Experimental Materials

The gelling materials included ordinary silicate cement (SC), fly ash (FA), silica fume (SF), and slag powder (SP). The chemical compositions and physical properties of various gelling materials are shown in

Table 1. Chemical compositions of gelling materials (kg/m³).

Materials	CaO	SiO2	Al2O3	Fe2O3	MgO	SO3	Na2O	K2O	TiO2
SC	57.7	18.48	4.82	3.26	1.51	1.83	0.33	1.07	0.23
FA	21.14	35.71	16.57	8.92	1.41	1.94	-	-	-
SF	0.4	76.35	0.20	0.55	0.53	-	0.51	-	-
SP	35.75	33.50	6.45	1.34	5.78	1.47	0.36	-	-

and

Table 2. Physical properties of gelling materials.

Material	Water Requirement of Normal Consistency (%)	Specific Surface Area (cm2/g)	Density (g/cm3)	Fineness	Moisture Content (%)
SC	25.8	390	3.09	2.79	-
FA	101	23,000	2.16	5.48	0.06
SF	115	248	2.35	2.85	0.2
SP	113	440	2.9	0.3	0.7

, respectively. The fibers used in the experiment included basalt and polypropylene fibers. The fibers were monofilament fibers. The morphology of the fibers is shown in Figure 1, and their physical properties are listed in

Table 3. Physical properties of basalt fiber and polypropylene fiber.

Types	Diameter (μm)	Length (mm)	Density (g/cm3)	Tensile Strength (MPa)	Elastic Modulus (GPa)	Elongation (%)
Basalt fiber	15	18	2.56	≥2400	≥40	≤3.1
Polypropylene fiber	30	19	0.91	>270	>0.3	≤40

. The coarse aggregate was obtained by crushing limestone, and the particle size was 5~20 mm. The bulk density and apparent density of the coarse aggregates were 1.45 g/cm³ and 2.65 g/cm³, respectively. The fine aggregate was river sand with fineness modulus, bulk density, and apparent density of 2.8, 1.44 g/cm³, and 2.65 g/cm³, respectively. Based on previous studies on the mechanical properties and durability of the BPFRC [24,25,26], the optimal mix ratio of the BPFRC was obtained, as shown in

Table 4. Mixture ratio of BPFRC (kg/m³).

Groups	SC	FA	SF	SP	Water Reducer	Water	Coarse Aggregate	Fine Aggregate	BF	PF
NC-30	234.2	73.2	22	36.6	3.66	161	1162.3	683	0	0
BF-30-0.1	234.2	73.2	22	36.6	3.66	161	1162.3	683	2.56	0
PF-30-0.1	234.2	73.2	22	36.6	3.66	161	1162.3	683	0	0.91
HF-30-0.1	234.2	73.2	22	36.6	3.66	161	1162.3	683	1.28	0.455
HF-30-0.2	234.2	73.2	22	36.6	3.66	161	1162.3	683	2.56	0.91
BF-40-0.1	241.6	79.2	15.8	59.4	3.96	150.5	1163.6	683.4	1.28	0
BF-50-0.05	333.1	48.3	29	72.4	4.83	140	1026.1	774.1	1.3	0

. BPFRC specimens were given the notation a-b-c, where a represented the fiber type (BF for basalt fiber, PF for polypropylene fiber, and HF for hybrid fiber); b represented the compressive strength (30 for 30 MPa, 40 for 40 MPa, and 50 for 50 MPa); and c represented the volume content of the fiber (0.05 for 0.05%, 0.1 for 0.1%, and 0.2 for 0.2%). Note that for specimens with compressive strengths of 30, 40, and 50 MPa, the water–binder ratios (mass ratio of water to gelling materials) were 0.44, 0.38, and 0.29, respectively.

2.2. Experimental Methods

In this study, an improved two-electrode alternating current (AC) method was applied to measure the resistivity of the BPFRC. Meanwhile, a scanning electron microscope (SEM) was used to observe the microstructure, and mercury intrusion porosimetry (MIP) was selected to measure the pore structure.

2.2.1. Measurement of Resistivity

The DC and AC methods were commonly used to test the concrete resistivity [26,27,28]. In the case of the DC method, the spatial properties of the cement-based materials were altered due to the migration of the ions, resulting in a poor characterization of the resistivity [26,27]. For the traditional AC method, the conductive solution was usually heated after energization, which reduced the accuracy of the test. In view of this, an improved AC method was proposed to measure the resistivity of the BPFRC in this study. Figure 2 shows the specific testing device. To avoid the polarization reaction and high temperature, the voltage and frequency of the AC were set to 1 V and 1000 Hz, respectively. The testing process was as follows:

Three specimens were made for each mixing ratio, and the size of the specimens was 100 mm × 100 mm × 50 mm. The measurement of the resistivity was conducted after the vacuum saturation of the specimen was completed. First, clamp the specimen between the test tanks with the end contacting the electrode. Then, fix the test tanks with bolts, and seal the gap between the specimen and test tank with quick-drying sealant. After the sealant was cured, a 3% NaCl solution was injected into the test tanks. Finally, the test tanks were connected to the resistance measurement device with wires, and then the AC resistance value could be obtained. The resistivity of the specimen (ρ) can be calculated according to Equation (1):

$$\rho =\frac{RA}{L}$$

(1)

where L represents the height of the specimen, taking the value of 50 mm; R represents the value of the resistance; and A represents the cross-sectional area of the specimen, taking the value of 100 mm × 100 mm.

2.2.2. Observation of Microstructure

In this study, an SEM was used to observe the microstructure of the BPFRC specimens. First, one specimen in each group was broken into fragments, and one fragment with a diameter of about 5 mm was selected randomly. Then, the fragment was attached to the sample holders using double-sided conductive adhesive, and vacuum gold spraying was performed. Subsequently, the microscopic morphology of the fragment was observed in the SEM.

2.2.3. Measurement of Pore Structure

Considering the insufficient accuracy of the saturation weighing method, the MIP was selected to measure the pore structure of the BPFRC specimens in this study. Similarly, one specimen in each group was first broken into fragments, and one fragment with a diameter of 3–5 mm was randomly selected to measure the pore structure. Next, the selected fragment was stored in ethanol solution. Before the measurement of the pore structure, the fragment was dried in a drying oven until the mass remained constant. Then, the Auto Pore IV 9500 mercury porosimeter was used to measure the pore structure of the fragment [29].

3. Analysis of BPFRC Resistivity

Figure 3 presents the resistivity of the BPFRC specimens. Compared with NC-30, the resistivity of PF-30-0.1, BF-30-0.1, HF-30-0.1, and HF-30-0.2 increased by 11.54%, 20.66%, 29.57%, and −22.94%, respectively, as shown in Figure 3. Obviously, the addition of fibers had a significant effect on the resistivity of the specimens, and the effect of the basalt fibers on the resistivity was higher than that of the polypropylene fibers. It can also be found that the effect of the hybrid basalt and polypropylene fibers on the resistivity was higher than that of the basalt fibers or polypropylene fibers when the fiber content was 0.1%. However, the resistivity of the specimens was reduced when the content of the hybrid basalt and polypropylene fibers was 0.2%. The reasons for the effect of fibers on the resistivity of the specimen were as follows:

In general, the resistivity of the cement-based materials was positively related to their compactness [28]. According to the SEM observation, there were many capillary pores and cracks inside NC-30 (Figure 4a). When the basalt fibers or polypropylene fibers with a volume content of 0.1% were added to the BPFRC specimen, the number of capillary pores and cracks decreased (Figure 4b,c). This was mainly due to the fact that the basalt fibers could effectively inhibit the formation and development of the micro-cracks and segregation cracks in early dry shrinkage, thereby greatly reducing the shrinkage cracks in the specimen [23,25]. Additionally, polypropylene fibers acted as a “support” for the aggregates when they were evenly distributed in the specimen. They reduced the precipitation of the concrete surface and the segregation of the aggregates and reduced the content of small harmful pores [24,30]. Therefore, the addition of the fiber would increase the compactness of the specimen and further enhance the resistivity of the specimen. However, when the volume content of the fibers was increased to 0.2%, the dispersion of the fibers was greatly affected, which resulted in some fibers stacking in the specimen (Figure 4d). In this case, more pores and cracks appeared in the specimen, so the resistivity of the specimen was significantly reduced.

Figure 5a shows the total porosity of the specimen. It can be seen that the total porosities of NC-30, BF-30-0.1, PF-30-0.1, HF-30-0.1, and HF-30-0.2 were 13.51%, 14.05%, 15.38%, 14.59%, and 15.62%, respectively. Obviously, the total porosity of the specimen was slightly increased after the addition of the fibers. To investigate the effect of different pore sizes on concrete, Wu and Lian [31] divided the concrete pores into four categories according to the pore size, including harmless pore (pore size ≤ 20 nm), less-harmful pore (pore size ≤ 50 nm), harmful pore (50 nm < pore size < 200 nm), and more-harmful pore (pore size > 200 nm). Figure 5b presents the pore size distribution of the specimen. It can be found that the specimen possessed fewer harmless pores and more harmful and less-harmful pores when the fiber addition was 0.1%. The addition of fibers optimized the internal pore structure of the specimen, thereby increasing the concrete resistivity.

Additionally, compared with BF-30-0.1, the resistivity of BF-40-0.1 and BF-50-0.05 increased by 18.24% and 44.78%, respectively (Figure 3). The compressive strength of the specimen was mainly determined by the water–binder ratio, so it can be inferred here that the water–binder ratio had a significant impact on the resistivity of the specimen. By analyzing the resistivity of these three specimens, it can be found that the resistivity increased with the decrease in the water–binder ratio. This trend can be explained by the results of the pore structure measurement. As shown in Figure 5a, the total porosities of BF-30-0.1, BF-40-0.1, and BF-50-0.05 were 14.05%, 11.59%, and 8.05%, respectively. Obviously, as the water–binder ratio decreased, the internal structure of the specimen became denser, which would increase its resistivity.

4. A 365-Day Resistivity Model for BPFRC

There are numerous variables that affect the resistivity of the concrete, such as the ambient temperature, ambient relative humidity, water–binder ratio, and admixtures [32,33,34,35]. Researchers have developed predictive models for resistivity, taking into account various factors [14,15,16,17,32,33,34,35]. For instance, the Song model [32] mainly considered the effect of the mineral admixtures, and the Solgaard model [33] mainly considered the effect of the steel fiber. Nevertheless, the resistivity model considering the basalt fiber and polypropylene fiber has not been studied. Additionally, Sun et al. [28] concluded that the resistivity variation of concrete after 365 days of curing was small, which was crucial for the investigation of concrete properties. Therefore, in this study, parameters such as α, β, γ, χ, and ε were introduced to establish a 365-day resistivity model by considering fiber, water–binder ratio, ambient temperature, and ambient relative humidity, as shown in Equation (2):

$$\rho =\alpha {\left(\frac{\omega }{b}\right)}^3+\beta V_B^2+\gamma V_B{V}_P+\chi V_p+\epsilon$$

(2)

where ω/b is the water–binder ratio of the BPFRC; α, β, γ, and χ are the influence factors of the water–binder ratio, basalt fiber, hybrid effect, and polypropylene fiber, respectively; ε is a constant; and V_B and V_P are the volume of the basalt fiber and polypropylene fiber, respectively.

The resistivity was influenced by the ambient temperature and pore water saturation in the actual environment. Hope [34] obtained the temperature influence coefficient of the concrete resistivity after experimental study and theoretical derivation, as shown in Equation (3). Gjoerv et al. [35] investigated the differences in concrete resistivity at various pore water saturations. Based on the experimental data in this literature, the relationship between concrete resistivity and pore water saturation can be obtained, as shown in Equation (4). The pore water saturation of the concrete was related to ambient relative humidity. Equations (5) and (6) were used by Sun et al. [28] to estimate the relationship between pore water saturation and ambient relative humidity.

$$k_T=\exp \left[2900\left(\frac{1}{T}-\frac{1}{293}\right)\right]$$

(3)

$${\rho }_S={\rho }^{S^{-1.7567}}$$

(4)

S = RH − 0.15 (if RH = 50~90%)

(5)

S = 5RH − 3.85 (if RH > 90%)

(6)

where k_T represents the temperature influence coefficient; S represents the pore water saturation; RH represents the ambient relative humidity; ρ_s represents the resistivity considering the pore water saturation; and ρ represents the resistivity of the saturated concrete.

Finally, based on the data of the water–binder ratio and fiber volume in this study, a 365-day resistivity model for the BPFRC considering ambient temperature and ambient relative humidity can be established, as shown in Equations (7) and (8):

$$\begin{gathered} {\rho }_{m,RH,T}=k_T{\left\{-354.92{\left(\frac{\omega }{b}\right)}^3+434.08V_B^2+2522.69V_B{V}_P+30.51V_p+63.52\right\}}^{{\left(RH-0.15\right)}^{-1.7567}} \\ \left(\mathrm{if}RH=50~90\%,V_B+V_P\le 0.1\%\right) \end{gathered}$$

(7)

$${\rho }_{m,RH,T}=k_T{\left\{-354.92{\left(\frac{\omega }{b}\right)}^3+434.08V_B^2+2522.69V_B{V}_P+30.51V_p+63.52\right\}}^{{\left(5RH-3.85\right)}^{-1.7567}}\left(\mathrm{if}RH>90\%,V_B+V_P\le 0.1\%\right)$$

(8)

5. Time-Varying Resistivity Model for BPFRC

It has been shown that in addition to the ambient temperature, ambient relative humidity, and water–binder ratio, the hydration process of gelling material also has a significant effect on the concrete resistivity [32]. This section mainly deduced the hydration process of various gelling materials under the multiple-gelling material system and finally established a time-varying resistivity model for the BPFRC.

5.1. Single-Gelling Material System

The reactivity equations of the four gelling materials were established by analyzing the factors influencing the reactivity of SC, FA, SF, and SP in the single-gelling material system [28], as shown in Equations (9)–(12):

$$R_c=\frac{\omega /\mathrm{c}}{0.38}\left[1-\exp \left(-k_c{t}^{1/3}\right)\right]$$

(9)

$$R_f=k_{f1}\times {\gamma }_f\times {\left(\frac{\omega }{b}\right)}^{k_{f2}}\times {\left(\frac{C_c\left(1-p_1\right)}{C_f{p}_1}\right)}^{k_{f3}}\times \left[1-\exp \left(-k_f\times {\left(\frac{B_f}{1000}\right)}^{k_{f4}}\times t_e^{n_f}\right)\right]$$

(10)

$$R_{\mathrm{g}}=k_{\mathrm{g}1}\times {\gamma }_g\times {\left(\frac{\omega }{b}\right)}^{k_{\mathrm{g}2}}\times {\left(\frac{C_c\left(1-p_2\right)}{C_{\mathrm{g}}{p}_2}\right)}^{k_{\mathrm{g}3}}\times \left[1-\exp \left(-k_{\mathrm{g}}\times {\left(\frac{B_{\mathrm{g}}}{1000}\right)}^{k_{\mathrm{g}4}}\times t_e^{n_{\mathrm{g}}}\right)\right]$$

(11)

$$R_{\mathrm{s}}=k_{\mathrm{s}1}\times {\gamma }_{\mathrm{s}}\times {\left(\frac{\omega }{b}\right)}^{k_{\mathrm{s}2}}\times {\left(\frac{C_c\left(1-p_3\right)}{C_s{p}_3}\right)}^{k_{\mathrm{s}3}}\times \left[1-\exp \left(-k_{\mathrm{s}}\times t_e^{n_{\mathrm{s}}}\right)\right]$$

(12)

where ω/c represents the water–cement ratio (mass ratio of water to cement); p₁, p₂, and p₃ represent the percentages of FA, SP, and SF in cement-based composites, respectively; B_g and B_f represent the fineness values of SP and FA, respectively; k_c represents the kinetic parameter of SC hydration reaction; k_fi, k_gi, and k_si represent the reaction degree factors of FA, SF, and SP, respectively; and R_c, R_f, R_g, and R_s represent the reaction degree of SC, FA, SF, and SP at time t, respectively.

The hydration of SC is a continuous process, and the hydration process of SC has been simplified by Vagelis et al. [36]. The calcium hydroxide produced after the completion of the reaction of 1 g of SC clinker is shown in Equation (13). Additionally, the admixture of FA, SF, and SP undergoes a secondary hydration reaction [36]. According to the reaction process, the calcium hydroxide absorbed by 1 g of FA, SP, and SF in the complete reaction process can be obtained, respectively, as shown in Equations (14)–(16):

$$C_{\mathrm{c}}=1.321\left(m_{C,c}-0.7m_{O,c}\right)-\left(1.851m_{S,c}+2.182m_{A,c}+1.392m_{F,c}\right)$$

(13)

$$C_f={\gamma }_f\left(1.851m_{S,F}+2.907m_{A,F}\right)-1.321m_{C,FH}$$

(14)

$$C_g={\gamma }_g\left(1.851m_{S,B}+2.907m_{A,B}\right)-1.321m_{C,B}$$

(15)

$$C_{\mathrm{s}}=1.851m_{S,S}{\gamma }_s$$

(16)

where m_i,c (i = C, O, S, F, A) represents the mass fractions of CaO, SO₃, SiO₂, Fe₂O₃, and A1₂O₃ in SC, respectively; m_i,F (i = S, C, A) represents the mass fractions of SiO₂, CaO, and A1₂O₃ in FA, respectively; m_C,FH represents the mass fraction of CaO in high-calcium FA; m_i,B (i = S, A, C) represents the mass fractions of SiO₂, A1₂O₃, and CaO in SP, respectively; m_s,s represents the mass fraction of SiO₂ in SF; and γ_f, γ_g, and γ_s represent the pozzolanic activity of FA, SP, and SF, respectively, and their values can be obtained according to Ref. [32].

The empirical equation of the reactivity of a single FA can be obtained by the regression of 225 sets of data in Refs. [37,38,39,40,41,42,43,44], as shown in Equation (17). The empirical equation of the reactivity of a single SP can be obtained according to Refs. [39,40,45,46,47,48], as shown in Equation (18). Similarly, the empirical equation for the reactivity of a single SF can be obtained from the regression of Refs. [40,48], as shown in Equation (19):

$$R_f=0.8181\times {\gamma }_f\times {\left(\frac{\omega }{b}\right)}^{0.2876}\times {\left(\frac{C_c\left(1-p_1\right)}{C_f{p}_1}\right)}^{0.2716}\times \left[1-\exp \left(-0.0376\times \left(\frac{B_f}{1000}\right)\times t_e^{0.7689}\right)\right]$$

(17)

$$R_{\mathrm{g}}=0.9393\times {\gamma }_g\times {\left(\frac{\omega }{b}\right)}^{0.4576}\times {\left(\frac{C_c\left(1-p_2\right)}{C_{\mathrm{g}}{p}_2}\right)}^{0.1260}\times \left[1-\exp \left(-0.5022\times \left(\frac{B_{\mathrm{g}}}{1000}\right)\times t_e^{0.3465}\right)\right]$$

(18)

$$R_{\mathrm{s}}=1.0564\times {\gamma }_{\mathrm{s}}\times {\left(\frac{\omega }{b}\right)}^{0.5524}\times {\left(\frac{C_c\left(1-p_3\right)}{C_s{p}_3}\right)}^{0.1719}\times \left[1-\exp \left(-0.1139\times t_e^{0.6406}\right)\right]$$

(19)

5.2. Multiple-Gelling Material System

In a multiple-gelling material system, the reaction process of the admixtures (SC, FA, SF, and SP) is complex. The reaction degree of a certain admixture is affected by the proportion of various admixtures. Consequently, in the multiple-gelling material system, the reaction degree of the admixture can be obtained by modification of Equations (17)–(19). The specific Equations are shown in (20)–(22) [32,49,50]:

$$R_f=0.8181\times {\gamma }_f\times {\left(\frac{\omega }{b}\right)}^{0.2876}\times {\left(\frac{C_c\left(1-p_1-p_2-p_3\right)}{C_f{p}_1+C_{\mathrm{g}}{p}_2+C_s{p}_3}\right)}^{0.2716}\times \left[1-\exp \left(-0.0376\times \left(\frac{B_f}{1000}\right)\times t_e^{{}_{0.7689}}\right)\right]$$

(20)

$$R_{\mathrm{g}}=0.9393\times {\gamma }_g\times {\left(\frac{\omega }{b}\right)}^{0.4576}\times {\left(\frac{C_c\left(1-p_1-p_2-p_3\right)}{C_f{p}_1+C_{\mathrm{g}}{p}_2+C_{\mathrm{s}}{p}_3}\right)}^{0.1260}\times \left[1-\exp \left(-0.5022\times {\left(\frac{B_{\mathrm{g}}}{1000}\right)}^{}\times t_e^{0.3465}\right)\right]$$

(21)

$$R_{\mathrm{s}}=1.0564\times {\gamma }_{\mathrm{s}}\times {\left(\frac{\omega }{b}\right)}^{0.5524}\times {\left(\frac{C_c\left(1-p_1-p_2-p_3\right)}{C_f{p}_1+C_{\mathrm{g}}{p}_2+C_s{p}_3}\right)}^{0.1719}\times \left[1-\exp \left(-0.1139\times t_e^{0.6406}\right)\right]$$

(22)

The water–binder ratio can be replaced by the equivalent water–binder ratio, as shown in Equation (23). Substituting Equation (23) into Equation (9), the final hydration degree of the cement in the multiple-gelling material system can be obtained, as shown in Equation (24).

$${\left(\frac{\omega }{b}\right)}_e=\frac{\omega /b}{\left[\left(1-p_1-p_2-p_3\right)+R_f{p}_1+R_g{p}_2+R_s{p}_3\right]}$$

(23)

$$R_{{}_c}^{\infty }=\frac{\omega /b}{\left[\left(1-p_1-p_2-p_3\right)+R_f^{\infty }{p}_1+R_{\mathrm{g}}^{\infty }{p}_2+R_s^{\infty }{p}_3\right]}\times \frac{1}{0.38}$$

(24)

where $R_{{}_c}^{\infty }$, $R_{{}_f}^{\infty }$, $R_{{}_g}^{\infty }$, and $R_s^{\infty }$ represent the final reactivity of SC, FA, SP, and SF, respectively.

5.3. Time-Varying Resistivity Model of BPFRC

Combined with the 365-day resistivity model developed in this study, a time-varying resistivity model for the BPFRC can be established by considering the hydration process of the gelling material, as shown in Equations (25) and (26). This model can be used not only to predict the development of the BPFRC resistivity, but also to predict the development of the concrete resistivity. To verify the reliability of the model, the predicted values calculated by this model were compared with the experimental values in Refs. [19,28,49], as shown in Figure 6. It can be seen that the error between most predicted values and experimental values was less than 20%. Therefore, this resistivity model had a certain accuracy.

$$\begin{gathered} {\rho }_{\mathrm{m},RH,T}\left(t\right)={\mathrm{k}}_T\times {\left\{\frac{R_c}{R_c^{\infty }}\times \left[-354.92{\left(\frac{\omega }{b}\right)}^3+434.08V_B^2+2522.69V_B{V}_P+30.51V_p+63.52\right]\right\}}^{{\left(RH-0.15\right)}^{-1.7567}} \\ \left(\mathrm{if}RH=50~90\%,V_B+V_P\le 0.1\%\right) \end{gathered}$$

(25)

$$\begin{gathered} {\rho }_{\mathrm{m},RH,T}\left(t\right)={\mathrm{k}}_T\times {\left\{\frac{R_c}{R_c^{\infty }}\times \left[-354.92{\left(\frac{\omega }{b}\right)}^3+434.08V_B^2+2522.69V_B{V}_P+30.51V_p+63.52\right]\right\}}^{{\left(5RH-3.85\right)}^{-1.7567}} \\ \left(\mathrm{if}RH>90\%,V_B+V_P\le 0.1\%\right) \end{gathered}$$

(26)

6. Conclusions

In this study, the resistivity of BPFRC was measured, and the effects of fiber type, fiber content, and water–cement ratio on resistivity were analyzed. The specific conclusions are as follows:

(1)

An improved two-electrode AC method was proposed to measure the BPFRC resistivity, which overcame the shortcomings of the DC method in the polarization reaction and heat generation under the action of long-time voltage.

(2)

Adding fiber affected the resistivity of the BPFRC. The effect was positive with a low volume content of fiber (e.g., 0.1%) and negative with the high one (e.g., 0.2%) due to fiber stacking. The effect of the polypropylene fiber, basalt fiber, and hybrid fiber on improving the resistivity increased in turn. The resistivity of the BPFRC showed a negative correlation with the water–binder ratio.

(3)

The microscopic morphology and pore structure parameters of the BPFRC were extracted by SEM and MIP, and the differences in the resistivity of BPFRC were analyzed.

(4)

A 365-day resistivity model for the BPFRC, considering the temperature, humidity, fiber, and water–binder ratio, was developed in this study. On this basis, a time-varying resistivity model for the BPFRC considering the hydration process of the gelling material was established.

(5)

The effect of chloride ions was not considered in the developed BPFRC model, which needs to be fixed in the follow-up study. Additionally, quantifying the relationship between the BPFRC’s resistivity and the mechanical and durability performance indicators would be an essential research topic.