Adhesion and Sliding Constitutive Relationship between Basalt–Polypropylene Hybrid Fiber-Reinforced Concrete and Steel Bars

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The bonding mechanism between hybrid basalt–polypropylene fiber-reinforced concrete and deformed bars and the bonding performance between hybrid basalt–polypropylene fiber-reinforced concrete and deformed bars were studied by combining experiments and a theoretical analysis.

Abstract

The extreme marine environment of the South China Sea Islands, which features high temperatures, high humidity levels and high salt levels, seriously affects the safety of building structures. The durability of concrete can be significantly improved by adding a basalt–polypropylene hybrid fiber, but its bonding mechanism with deformed bars is complicated. Therefore, the bonding performance of hybrid basalt–polypropylene fiber-reinforced concrete and deformed bars was studied by combining experiments and a theoretical analysis. We designed 38 groups of different concrete strengths, different thicknesses of concrete covers, different anchor lengths and different diameters of rebars. The bond strengths, bond–sliding curves and failure forms of each pull specimen were compared and analyzed. The results showed that the failure forms and bond–slip curves of the basalt–polypropylene hybrid fiber-reinforced concrete specimens and the ordinary concrete specimens were essentially the same. Based on the results of the axial tensile tests, an ultimate bond strength prediction model was developed, and a bond–sliding constitutive model for hybrid fiber-reinforced concrete and steel bars was also established.

1. Introduction

In recent years, fiber or mineral materials have been used to improve concrete performance given the disadvantages of ordinary concrete such as having great brittleness, having poor corrosion resistance and being easy to crack. Adding basalt fibers and polypropylene fibers in concrete cannot only enhance the strength of the concrete matrix but also improves the impact resistance performance and the durability of the concrete, and thus these have good application prospects [1,2,3,4]. Except for their similar line expansion coefficients, the bond performance between steel bars and concrete is the essential reason that the two can work together. In concrete structures, due to the bond performance between concrete and steel bars, the external force loading on this element is resisted by the concrete and steel bars together. Bond performance directly affects stress transfer efficiency, and it also affects the bearing capacities and working performances of reinforced concrete structures and members. Accurate predictions for the nonlinear responses of steel bars and concrete can be made using the finite element method, which depends on the performance of the concrete under multiaxial stress conditions and the bond–sliding constitutive relationship between the steel bars and the concrete. The bond–sliding constitutive relationship is necessary to determine the stiffness matrix and the connecting unit in a finite element simulation.

In recent years, many prediction models for determining the bonding constitutive relationship between steel bars and concrete have been proposed by scholars. The first cyclic load test on axial tensile steel bars in a specimen was conducted by Bresler et al. [5]. They studied the bond degradation mechanism under cyclic loads. Then, they reported the results of the bond’s stress distribution and end slip measurements. Mirza et al. [6] recorded the end slip, embedded rod elongation and crack formation and extension in concentric tensile specimens. A bond stress–sliding prediction model related to the load size, thickness of the concrete cover and concrete strength was derived from the test data. Lin et al. [7] discussed the bond–sliding mechanism in detail, and they proposed a parameter representing the concrete cover and constraint capacity of the stirrup and established a model of the deformation reinforcement bond sliding for crack-breaking and pull-out failures. The model could better predict the bond–slip behaviors of deformed bars with different constraint levels. Using a pull-out test, Chu et al. [8] studied the effect of adding steel fiber on a steel bond’s performance, and they proposed a new steel bond model. For a more realistic bond–slip analysis, the finite initial bond stiffness was considered in addition to the influence of the steel fiber. Harajli et al. [9] evaluated the effects of the concrete cover and the fiber volume fraction on the bond properties of concrete using beam specimens. Based on their experimental results and early research, a bond stress–sliding response model for steel bars embedded in ordinary concrete and in steel-fiber concrete was proposed, and the constitutive relationship describing its characteristic behaviors was established.

The tensile strengths, splitting strengths and bending strengths of concrete mixed with polypropylene and basalt fibers have been effectively improved, and the ductility, toughness and impact resistance of concrete have been significantly improved. Therefore, hybrid fiber-reinforced concrete is very different from ordinary concrete. There are obvious differences between the basic theory for hybrid fiber-reinforced concrete and steel bars and the existing theory for ordinary reinforced concrete. The early proposed bond–slip constitutive relationship between steel bars and concrete does not describe all the elements of the bond performance, especially that between steel bars and fiber-reinforced concrete. Therefore, the bond–sliding constitutive relationship applicable to steel bars and basalt–polypropylene hybrid fiber-reinforced concrete is proposed in this paper.

2. Experimental Program

2.1. Test Piece Design

In this paper, four test variables—concrete strength, anchor length, reinforcement diameter and concrete cover thickness—were selected. The test pieces were designed for 38 groups, as shown in

Table 1. Sample list.

Number	Strength Grade	d/mm	la/d	c/mm
C30-14-3d-68	C30	14	3	68
C30-14-5d-68	C30	14	5	68
C30-14-7d-68	C30	14	7	68
C30-14-5d-63	C30	14	5	63
C30-14-5d-42	C30	14	5	42
C30-14-5d-21	C30	14	5	21
C30-16-3d-67	C30	16	3	67
C30-16-5d-67	C30	16	5	67
C30-16-7d-67	C30	16	7	67
C30-16-5d-64	C30	16	5	64
C30-16-5d-48	C30	16	5	48
C30-16-5d-32	C30	16	5	32
C30-18-3d-66	C30	18	3	66
C30-18-5d-66	C30	18	5	66
C30-18-7d-66	C30	18	7	66
C30-18-5d-54	C30	18	5	54
C30-18-5d-36	C30	18	5	36
C30-18-5d-27	C30	18	5	27
C30-20-3d-65	C50	20	5	65
C30-20-5d-65	C50	20	5	65
C30-20-7d-65	C50	20	5	65
C30-20-5d-20	C50	20	5	20
C30-20-5d-40	C50	20	5	40
C30-20-5d-60	C50	20	5	60
C30-22-5d-64	C50	22	5	64
C30-25-5d-62.5	C50	25	5	62.5
C40-14-5d-68	C40	14	5	68
C40-16-5d-67	C40	16	5	67
C40-18-5d-66	C40	18	5	66
C40-20-5d-65	C40	20	5	65
C40-22-5d-64	C40	22	5	64
C40-25-5d-62.5	C40	25	5	62.5
C50-14-5d-68	C50	14	5	68
C50-16-5d-67	C50	16	5	67
C50-18-5d-66	C50	18	5	66
C50-20-5d-65	C50	20	5	65
C50-22-5d-64	C50	22	5	64
C50-25-5d-62.5	C50	25	5	62.5

Note: The specimen number consists of four parts, namely A, B, C and D. One type of specimen is numbered A-B-C-D, where A represents the strength of hybrid fiber-reinforced concrete, B represents the diameter of reinforcement bars, C represents the relative anchoring length and D represents the minimum protective layer thickness.

. In the pieces, the ordinary steel bars embedded in ordinary concrete were named OS-OC, while the ordinary steel bars embedded in hybrid fiber-reinforced concrete were named OS-HFRC. The reinforced concrete specimens were standard cubes with the following dimensions: 150 mm × 150 mm × 150 mm. The lengths of the steel bars were 500 mm, the free-end lengths were 50 mm and the loading end lengths were 300 mm. A test piece is shown in Figure 1.

2.2. Material Properties

In the previous study [4], three grades of concrete strength were designed, each with different admixtures and fiber content. The material tests employed a seven-factor and three-level orthogonal design. We considered the properties of economy, durability, and mechanics to determine the best concrete mix ratio for each strength grade, as shown in

Table 2. Concrete mix proportions (kg/m³).

Concrete Strength	Cement	Silica Fume	Fly Ash	Slag	Water Reducer	Water	Sand	Stone	Basalt Fiber	Polypropylene Fiber
C30	234.2	22.0	73.2	36.6	3.66	161.0	683.0	1162.9	1.28	0.46
C40	241.6	15.8	79.2	59.4	3.96	150.5	683.4	1163.6	2.56	0.00
C50	333.1	29.0	48.3	72.4	4.83	140.0	774.1	1026.1	1.28	0.00

. The water–binder ratio, sand rate and fiber content of each strength grade are different. This indicates that the influence of fiber on concrete is complex. Reference [4] provides a detailed explanation. The concrete strength tests were conducted in accordance with the guidelines specified in the “Standard for test methods of physical and mechanical properties of concrete (GB50081-2019)” [10]. The final damage modes of the concrete are illustrated in Figure 2. The HRB400 steel bars employed for the test are shown in Figure 3. The basic mechanical properties of the various diameters are presented in

Table 3. Basic mechanical properties of steel bars.

Diameter (mm)	Yield Strength (MPa)	Tensile Strength (MPa)	Yield Strength Ratio	Maximum Total Elongation (%)	Ultimate Elongation (%)
14	460	610	0.73	15.0	29.0
16	445	600	0.74	14.5	24.0
18	465	595	0.78	16.5	35.0
20	440	600	0.73	14.5	24.0
22	425	570	0.75	15.5	23.0
25	435	580	0.75	16.5	23.0

.

2.3. Pull-Out Test Scheme

The pull-out tests were conducted in accordance with the relevant provisions regarding the bond strength tests of concrete and reinforcement in the “Standard for test methods of physical and mechanical properties of concrete (GB50081-2019)” [10]. Central pull-out tests were conducted using the WE-100 universal material testing machine. The loading rate was controlled based on the displacement speed of the test bench, initially at 0.3 mm/min until the drawing force fell to 50% of the peak value, after which it increased to 0.6 mm/min until the end of the test. The displacement of the free end and loading section of the steel bars should be measured using an LVDT with a range of 50 mm, and the load changes during the tests should be directly recorded by the computer. A diagram of the test setup is presented in Figure 4.

3. Test Results and Analysis

3.1. Failure Modes

The tensile strength of concrete is lower than that of steel bars. During the drawing process, the steel bars have not yet reached the yield stage. In contrast, the concrete had already reached its ultimate tensile strength, exhibiting distinctly brittle failure characteristics, as shown in Figure 5a. Figure 5b depicts splitting shear failure, a form of bond anchoring failure between steel bars and concrete. Throughout the loading process, the load escalates rapidly with slip, and concrete splitting failure occurs in the horizontal or descending section of the load–slip curve. However, in contrast to splitting failure, the load does not suddenly plummet to zero, and the specimen retains a specific load-bearing capacity at a later stage. Figure 6c shows the steel bar pull-out failure, an ideal failure mode that allows for the acquisition of the complete process curve of the bond slip. Throughout the loading process, both the free end and loading end slip incrementally with the increasing load, resulting in a relatively steep slope of the bond–slip curve. Upon reaching the ultimate load, the steel bars are gradually pulled out, inducing a shear slip between the steel bars and the concrete along the longitudinal direction of the steel bars, leading to a failure pattern resembling a plough. The pulling process is relatively slow, with the load also reducing to approximately 20% of the peak load and stabilising thereafter.

3.2. Bonding Stress–Slip Curve

Assuming that the bond stress is distributed evenly along the steel bars’ embedded lengths, the average bond stress is calculated as described below:

$$\tau =\frac{F}{\pi d{l}_a}$$

(1)

where τ is the bond stress between steel bars and concrete, F is the mean of the drawing force, d is the steel bar diameter and l_a is the embedded length of steel bars.

The average bond stress is defined as the ultimate bond strength, τ_u, when the bond specimen is damaged. The corresponding free-end slip amount represents the peak slip amount. As illustrated in Figure 6, the average bond stress–slip curve demonstrates a consistent trend.

Figure 6a shows that an increase in concrete strength results in a corresponding rise in mechanical biting force, leading to a steeper slope in the curve and a higher peak value of bond stress. Bond stress begins to decline as the bonding interaction weakens. Concrete strength also directly impacts the relative slip, with higher strengths corresponding to smaller relative slips and steeper curves. When the relative slip value of the steel bars and concrete exceeds one rib distance and the mechanical bite force is completely eradicated, the friction between the steel bars and concrete generates bonding stress. The strength of the concrete serves as a constraint on the steel bars, influencing the friction generated and consequently, the residual bonding stress.

The curves of the different steel bar diameters are presented in Figure 6b. A higher bond stress of reinforced concrete is observed with increasing steel bar diameter; a larger diameter leads to a smaller surface area of ribs, thereby reducing the bond anchorage effect. Concurrently, the relative slip amount decreases with an increase in steel bar diameter. This decrease is primarily attributed to the reduced tensile deformation of the steel bars in the non-anchored section, resulting in a smaller slip amount at the loading end, and ultimately leading to a lower slip of the specimen.

The curves for the different steel bar diameters are illustrated in Figure 6c. The thickness of the reinforcement protective layer dictates the restraint effect of concrete on reinforcement. A thicker protective layer results in a less pronounced crack development. When the thickness of the protective layer is minimal, the restraint on the expansion of cracks is not effectively inhibitory, allowing the cracks in the rib to propagate to the concrete surface prematurely, leading to a splitting–shear failure. Conversely, a thicker protective layer exhibits a more significant inhibitory effect on crack development, and accordingly, experiences a higher bonding stress.

The variations in the curves of the different anchoring lengths are presented in Figure 6d. The figure shows that the bond stress reduces as the bond anchorage length increases. Generally, a longer anchorage length corresponds to a deterioration in bond performance, primarily due to the non-uniform distribution of bond stress along the anchorage length axis. The stress value at the loading end increases, while the stress value at the free end decreases. Furthermore, a longer anchorage length results in a lower average bond stress.

3.3. Bonding Properties between Ordinary Steel Bars and Different Concretes

Figure 7 depicts the failure morphologies at the interfaces between steel bars and ordinary reinforced concrete (OC) and hybrid fiber-reinforced concrete (HPBRC) specimens. The ultimate failure mode of the conventional concrete specimens predominantly involves splitting failure (S-type), characterised by a loud splitting sound, swift load reduction, and the specimens being divided into two or three pieces. In contrast, the hybrid fiber-reinforced concrete specimens exhibited superior toughness and ductility, with their failure mode primarily manifesting as splitting–shear failure (S-S type). Upon concrete splitting failure, the load rapidly decreases but the specimens’ primary maintain integrity and retain a certain degree of load-bearing capacity.

The tensile behaviour of HPBRC significantly improves. Prior to loading, the stress between the cement-based matrix and the fiber was uniformly distributed (Figure 8a). Upon loading, the tensile load is transferred from the cement-based matrix to the fiber through the shear transfer mechanism, with the fiber bearing a portion of the load due to its high elastic modulus. The stress distribution along the fiber embedding length exhibited an uneven pattern, with higher shear stresses at the ends and lower in the middle (Figure 8b). The tensile stress was higher in the middle and lower at both ends. The stress distribution within the cement-based matrix is depicted in Figure 8c.

The primary cause of the impairment of the bond between steel bars and OC is the radial component of oblique extrusion pressure q, which induces cyclic tensile stress σ_c in the surrounding concrete. Once σ_c surpasses the concrete’s ultimate tensile strength, the concrete fractures immediately, resulting in a rapid decline in bond stress (Figure 9a). The radial component q of transverse costal oblique compression pressure is jointly resisted by the annular tensile stress σ_c of the concrete and the shear stress τ of the fiber and cement base (Figure 9b). As previously reported [11], adding fibers can effectively enhance the concrete’s ultimate tensile strength, deflection deformation capacity and peak ductility.

The fiber shear stress shoulders a portion of the circumferential tensile stress of the concrete, leading to an increase in concrete strength. Consequently, the bond strength between steel bars and concrete is improved. The prominent impact of hybrid fibers on the damage to the bond between steel bars and concrete is the enhanced concrete ductility after cracking, with a better post-peak residual adhesion capacity than OC (Figure 10). Following the initiation of concrete cracking, the rapid progression of cracks is mitigated by the fiber bridging action, resulting in a relatively mild damage process. During the intercostal concrete crushing or cutting process, the hybrid fiber concrete can still provide better constraint through the bridge action. Thus, the residual bond of hybrid fiber concrete and steel bars exhibit superior performance.

4. Bonding–Sliding Constitutive Model

4.1. Ultimate Bond Strength

The ultimate bond strength of steel bars and hybrid fiber-reinforced concrete is calculated as follows:

$${\tau }_u=F_u/\pi d{l}_a$$

(2)

where F_u is the maximum drawing force, d is the diameter of the steel bars and l_a is the anchoring length.

The calculation results of the ultimate bond strength are shown in

Table 4. Ultimate bond strength test values.

Number	Fu/kN	τu/MPa
C30-14-5d-68	57.76	18.76
C40-14-5d-68	64.50	20.95
C50-14-5d-68	77.46	25.16
C30-14-3d-68	38.00	20.57
C30-14-7d-68	59.44	13.79
C30-16-5d-67	69.13	17.19
C40-16-5d-67	79.74	19.83
C50-16-5d-67	90.34	22.46
C30-16-3d-67	47.22	19.57
C30-16-7d-67	71.67	12.73
C30-14-5d-21	46.00	14.94
C30-14-5d-42	53.79	17.47
C30-14-5d-63	57.36	18.63
C30-16-5d-32	57.30	14.25
C30-16-5d-48	62.17	15.46
C30-16-5d-64	69.13	17.19
C30-18-3d-66	59.76	19.57
C30-18-5d-66	82.19	16.15
C30-18-7d-66	90.70	12.73
C30-18-5d-27	69.72	13.70
C30-18-5d-36	72.20	14.58
C30-18-5d-54	77.71	15.27
C30-20-3d-65	70.50	18.71
C30-20-5d-65	96.57	15.37
C30-20-7d-65	97.29	11.06
C30-20-5d-20	60.70	9.66
C30-20-5d-40	89.35	14.22
C30-20-5d-60	106.88	17.01

Note: The specimen number consists of four parts, namely A, B, C and D. One type of specimen is numbered A-B-C-D, where A represents the strength of hybrid fiber-reinforced concrete, B represents the diameter of reinforcement bars, C represents the relative anchoring length and D represents the minimum protective layer thickness.

.

Because national regulations consider the impact of hybrid fibers on bond strength, most of the predicted values tend to be conservative. Therefore, statistical regression analysis was carried out on the test data, and the calculation formula was fitted by establishing a custom function. The ratio of the ultimate bond strength τ_u to $\sqrt{f_c}$ is linearly related to the anchoring depth and the relative protective layer thickness, respectively. Therefore, τ_u satisfies Equation (3).

$$\left\{\begin{array}{l}u(x,y)=ax+c\\ u(x,y)=by+d\end{array}\right.$$

(3)

where x is l_a and y is c.

The partial derivative and integral of x and y were calculated, respectively, to obtain Equations (4) and (5):

$$u(x,y)=ax+b+\phi (y)$$

(4)

$$u(x,y)=ax+cy+e$$

(5)

Combined with the test data, double-parameter curve fitting was conducted, and then the ultimate limit bond strength calculation formula was developed:

$$\frac{{\tau }_u}{\sqrt{f_c}}=0.21\frac{c}{d}-0.32\frac{l_a}{d}+3.56$$

(6)

where τ_u is the ultimate bonding strength/MPa; f_c is the compressive strength/MPa of concrete; c is the thickness of the protective layer/mm; l_a is the anchoring length/mm; and d is the diameter of the steel bars/mm.

The comparison between fitting and test results is shown in Figure 11, and compared with the prediction model [12,13,14,15]. The fitting results agree well with the test results, and the vast majority of errors are within ±10%, as shown in Figure 11. It indicates that the ultimate bond strength calculation type is well fitted, which can effectively predict the ultimate bond strength of deformed bars and hybrid fiber-reinforced concrete.

4.2. Bond–Slip Constitutive Relationship

4.2.1. Ascending Segment

The curve between the average bond stress and the amount of slip is called the bond–slip whole-process curve. According to its characteristics, the process curve is divided into four segments: ascending segment (0–s₁), peak segment (s₁–s₂), descending segment (s₂–s₃) and residual segment (s₃–).

According to the research of Eligehausen and others [16], by combining the characteristics of the bond–slip curve in the rising section, the curve of the rising section is described by Equation (7):

$$\tau ={\tau }_u{\left(\frac{s}{s_1}\right)}^{\alpha }\left(0\le s\le s_1\right)$$

(7)

where τ is the bonding strength; s is relative slip; and α is the correlation coefficient between 0 and 1.

The magnitude of the upward slope of the bond–slip curve is significantly influenced by the parameter α. When α = 0, the bond stress reaches its maximum value instantaneously. In contrast, when α = 1, a linear relationship exists between the bond stress and slip. Researchers have conducted extensive studies on the coefficient α, proposing various values based on specific experimental outcomes. The variation in the value of α can be attributed to differences in concrete strength, the geometry and the type of reinforcing bars. According to the method suggested in Fib’s recent report, the coefficient α is derived through the process of nonlinear curve fitting. By combining previous research findings and experimental data, a value of α = 0.4 is obtained.

The peak slip s1 is significantly affected by numerous factors, including concrete strength, embedded length, protective layer thickness, stirrup constraints and loading rate, as corroborated by a wealth of previous test results. In previous studies, the test value of s1 demonstrates minimal variation. However, given the variegated nature of each situation, s1 must be determined based on specific test data. For steel bar shear pull-out failure, s1 is assumed to be inversely proportional to the square root of concrete strength, as proposed by Alsiwat et al. [17]. However, Harajli et al. [18] discovered no correlation between s1 and concrete strength or constraints, but rather a primary relationship with the spacing of the steel bars. For bars with a diameter less than 25 mm, s1 can be approximately equated to 0.2 times the spacing of the transverse ribs. Zhao et al. [19] also found that s1 is not particularly sensitive to changes in concrete strength but rather increases with the increase in the spacing of the cross ribs (C_clear). The results of a statistical regression analysis revealed that the value of s1 was approximately 0.1 C_clear. Murcia et al. [20] conducted a bond test on large-diameter steel bars and observed that s1 was nearly 0.07 d, which is more aligned with the findings of Zhao et al.

The adequacy of the concrete protective layer to provide restraint or equivalent functionality to stirrups determines the mode of bond failure, which commonly presents as pull-out failure. In this scenario, the concrete between the steel bars is completely severed or crushed. The bond stress is primarily determined by the shear strength of the concrete, rather than the splitting capacity of the protective layer. Consequently, the additional restraint provided by the concrete cover or stirrups has a limited impact on the bonding behavior. Thus, the variable s1 in the failure mode of steel bar shearing and pull-out is dictated by the spacing of the steel bars’ transverse ribs. The modifications in peak slip s1 and the spacing C_clear of the steel bars’ transverse ribs based on the previous test data are depicted in Figure 12.

The relationship between s₁ and the spacing C_clear of the cross ribs is approximately linear. This can be obtained using Equation (8).

$$s_1=0.10976C_{clear}$$

(8)

4.2.2. Peak Segment

It can be seen from Figure 13 that the peak bond strength is close to the ultimate bond strength, that is, τ = τ_u(s₁ < s ≤ s₂). The amount of slippage in the peak segment is Δ_s = s₂ − s₁. It can be found from the test data that the value of Δ_s is always between 0.1 and 0.3 mm. Therefore, it is taken as 0.2 mm in this article.

4.2.3. Descending Section

It can be seen from Figure 14 that the bond stress of the specimen drops significantly after the peak. Due to the rapid crack development after the peak, the bond stress decreases exponentially. The hoop force generated by the balanced mechanical occlusion is hardly assisted by the lateral restraint when the crack propagates through the concrete cover. However, since the internal cracking of the concrete does not immediately lead it to lose tensile properties, the specimen can still maintain a certain residual bond strength. Therefore, the relationship between bond stress and slip after the peak value can be found by adopting Equation (9):

$$\tau ={\tau }_u{\left(\frac{s}{s_2}\right)}^{\beta }\left(s_2<s\le s_3\right)$$

(9)

The effect of c/d on β is shown in Figure 14. We can see that the data have a large discreteness. This is mainly due to the fact that the descending section of the specimen without the transverse descends quickly, and the data points are not easily captured, which will lead to test errors. Generally, β increases slightly with the c/d increase. Despite the cracking inside, the concrete does not immediately lose its tensile properties, thereby sustaining a specific residual bond strength. It can be determined from the test data that β = −0.7.

4.2.4. Residual Section

When the slip amount is close to the cross-rib spacing, the bond stress maintains a stable value. It hardly decreases until the reinforcement is pulled out and destroyed. Therefore, the expression of the residual section of the whole stick–slip curve is

$$\tau ={\tau }_r=\phi {\tau }_u\left(s_3<s\right)$$

(10)

where, φ is obtained by linear fitting of the residual bond strength to the ultimate bond strength (φ = 0.3).

In summary, the constitutive model of the whole-process curve of local bond–slip for hybrid fiber-reinforced concrete and deformed steel bar is expressed as follows.

$$\tau =\left\{\begin{array}{cc}{\tau }_u{\left(\frac{s}{s_1}\right)}^{\alpha }& \left(0<s\le s_1\right)\\ {\tau }_u& \left(s_1<s\le s_2\right)\\ {\tau }_u{\left(\frac{s}{s_2}\right)}^{\beta }& \left(s_2<s\le s_3\right)\\ \phi {\tau }_u& \left(s_3<s\right)\end{array}\right.$$

(11)

where the ultimate bond strength is represented by τ_u. s₁ is calculated according to Equation (4). s₂ = s₁ + 0.2 mm; s₃ = C_clear; φ = 0.3; α = 0.4; β = −0.7. The proposed bond–slip model is shown in Figure 15.

4.3. Comparison and Verification of Constitutive Models

The predicted stress–slip curve was compared with the measured average bond stress–slip curve. It can be observed that the ascending section of all prediction models aligns well with the corresponding section of the measured curve, as illustrated in Figure 16a. According to the literature [12], bonding stress in the ascending section primarily relies on chemical bonding and friction forces. The strength of concrete cement-based structures can be enhanced by incorporating basalt and polypropylene fiber, without increasing the frictional resistance between the steel bars and the concrete. Furthermore, hybrid fibers improve bonding performance by hindering crack development, and reducing crack width to ensure the fiber is in a fully tensioned state. However, there is only limited sliding between the steel and concrete in the ascending section, resulting in most fibers being in an untensioned or marginally tensioned.

Consequently, during the initial ascending stage of the curve, the hybrid fiber makes a negligible contribution to the bond stress between the steel bars and the concrete. The descending section of the forecast model differs from the corresponding section of the measured curve. The falling portion of the measured curve is underestimated, as illustrated in Figure 16b. Cracks emerge in the concrete at this stage, and the mixed fiber at the crack is in tension. After concrete cracks, the hybrid fiber can provide a certain degree of lateral restraint, inhibiting the rapid expansion and extension of cracks. Consequently, the bonding stress does not plummet abruptly due to concrete cracking, resulting in a smoother curve. Therefore, in the descending phase of the curve, the contribution of hybrid fibers to the bond stress between steel and concrete is underestimated. As a result, the descending segment of the prediction model differed from the descending segment of the measured curve.

Figure 17a compares the average bond stress–slip curve obtained from Huang’s experiment with the prediction model [21]. This represents the average bond stress–slip curve in comparison with the predicted model. The model demonstrates substantial agreement with the ascending section of the test curve. However, the descending section exhibited a discrepancy from the test curve. Increasing the steel fiber content results in higher bond strength, peak slip and bond toughness. Polypropylene fiber has a low modulus of elasticity; thus, it has a minimal impact on bond strength and peak slip. The prediction model presented in this study is based on basalt and polypropylene hybrid fiber concrete, while the measured curve pertains to steel fiber and polypropylene hybrid fiber concrete. Steel fiber enhances the strength and toughness of concrete more significantly than basalt. The restraint effect of steel fiber and polypropylene hybrid fiber on concrete cracking is underestimated by the prediction model.

Consequently, the prediction model differs from the measured curve. To further validate the accuracy of the model, the measured curves from Chu [8], Byung [22], and Xiao [23] are employed for comparison. As illustrated in Figure 17b, the quantity of steel fiber has a pronounced impact on bond strength and bond toughness. The inhibitory effect of steel fiber on concrete cracks is significantly underestimated by the prediction model. The descending segment of the predicted curve lies below the measured curve, as shown in Figure 17c,d. Despite some differences, the prediction model exhibited considerable agreement with the measured curve. Thus, the prediction model proposed in this article differs from the curve obtained from the experiment; however, the discrepancy is limited. The proposed prediction model demonstrates higher accuracy and is a valuable reference.

5. Conclusions

The bond property between deformed steel bars and hybrid fiber-reinforced concrete was examined in this study through experimental methods. The effects of concrete strength, protective layer thickness, embedded length and steel bar diameter on the bond property were analyzed. Additionally, a calculation formula for the characteristic bond strength value between deformed steel bars and hybrid fiber-reinforced concrete was proposed. Based on the test results, a constitutive relationship for the average bond stress–slip curve of basalt–polypropylene hybrid fiber-reinforced concrete and steel bars was established. The following main conclusions are drawn:

• The failure modes between the bond of basalt–polypropylene hybrid fiber-reinforced concrete and steel bars are essentially identical to those of the bond between conventional reinforced concrete and steel bars, which include three primary failure modes: concrete split failure, split–shear failure and steel bar pull-out failure.
• The bond–slip curve characteristics between basalt–polypropylene hybrid fiber-reinforced concrete and steel bars are essentially equivalent to those of the ascending section in ordinary reinforced concrete. However, upon concrete cracking, the basalt–polypropylene hybrid fiber provides additional lateral restraint, resulting in a higher ultimate bond strength for basalt–polypropylene hybrid fiber-reinforced concrete and steel bars. The descending section of the curve is gentler.
• Through a comparative analysis of the influence of concrete strength, protective layer thickness, embedded length and steel bar diameter on the ultimate bond strength, a prediction model for the ultimate bond strength of steel and hybrid fiber-reinforced concrete was established.
• Based on experimental data, the effects of bond strength and slip amount on the bond–slip curve were comprehensively considered, and a mean bond stress–slip constitutive relationship between basalt–polypropylene hybrid fiber-reinforced concrete and steel bars was established through statistical regression analysis.