Fibers with characters of high elastic modulus, high elongation rate, and high ultimate tensile strength are usually used to overcome the crack sensibility and intrinsic brittleness of concrete (1, 2). ECC composited by PE or PVA, cement, and fine aggregates exhibit an applausive tensile pseudo strain hardening property, meanwhile, multiple tight cracking with a mean crack width under 100 μm was usually accompanied in the tensile plane (3-6). In addition, the fiber-reinforced concrete containing polyolefin and steel fibers also could exhibit great toughness and crack resistance if the fibers achieve a suitable distribution (7-9), while it is difficult to enhance the ductility obtained by the strain hardening and multiple cracks expanding action compared to PE-ECC and PVA-ECC. Hence, the superior tensile properties of ECC contributed to its wide application prospects in infrastructural construction.

On basis of fracture mechanics and micromechanical principles, ECC was designed by linking from single fiber to single crack scale to achieve the tensile pseudo strain hardening properties (10, 11). The crack bridging model of composites expresses the relation of fiber bridging stress versus crack opening distance, which was of primary importance to implement the multiple crack expanding and tensile pseudo strain hardening characters by energy-based criterion and strength-based criterion (12). Hence, tailoring raw microstructure including fiber, matrix, and fiber/matrix by fiber bridging law can capture a superior tensile property of ECC (13, 14). Moreover, modeling the micromechanical fiber bridging stress vs crack opening width relation also could evolve the macroscopic tensile properties of ECC (15, 16).

To date, continuous efforts have been made to establish the crack bridging relation based on the fiber failure models and interface properties. Li et al. (17) first established the fiber pullout model (FPM) which considered the interfacial frictional stress as a constant during the fiber pullout process. Thereafter, Lin et al. (18) introduce the interfacial slip-hardening coefficient to advance the precision of FPM where interfacial frictional strength is enhanced by the increasing fiber pullout distance. However, fiber fracture was neglected in above two models when fiber bridging stress is greater than the fiber apparent tensile strength. Based on FPM, Maalej et al. (19) first considered fiber rupture in debonding process and proposed the fiber pullout rupture model (FPRM). Afterward, Kanda et al. (20) included a chemical bond stress to advance FPRM where the chemical bond stress and interfacial frictional stress control the fiber-matrix interface properties. Thereafter, Lin et al. (21) further introduced the interfacial slip-hardening coefficient to the above advanced FPRM. Noticeably, fiber bridging stress during pullout stage could exceed the fiber tensile strength due to the slip-hardening character of single fiber (22). However, the situation of fiber rupture in slipping stage was no considered in FPRM and two advanced FPRMs.

In recent years, fiber rupture in slipping stage was gradually introduced to advance the fiber bridging model. Based on the advanced FPRM (21),Yang et al. (23) considered fiber rupture in the debonding and slipping process, matrix micro-spalling, and Cook-Gordon effect to improve the precision accuracy of the fiber bridging model. However, no explicit expression was presented in the model. Yu et al. (24) created the implicit micromechanical modeling of crack-bridging relations of hybrid-fiber strain-hardening cementitious composites considering the interaction between different fibers based on Yang’s model. In addition, based on the advanced FPRM, Huang et al. (14) developed an explicit crack bridging model that included fiber rupture in debonding and slipping stage (21), while the fiber two way pullout was explicated in this model. However, the crack opening width at the fiber complete debonding point was neglected in the fiber limiting embedment and bridging stress, while Cook-Gordon effect was also excluded. Cook-Gordon effect presents a premature fiber/matrix interface debonding normal to the fiber axis caused by a tensile stress located ahead of a blunt matrix crack implementing towards a fiber under remote tensile load, which makes fiber debonding take place ahead of the matrix crack, resulting in stretching of a free fiber segment and additional crack opening (23). Thus, excluding Cook-Gordon effect would reduce the crack width and mitigate the prediction accuracy of fiber bridging model. Based on the reviewed fiber bridging model, it is more reasonable to establish an advanced crack bridging model including the overall properties of the fibers, slip-hardening interfacial coefficient, fiber rupture in debonding and slipping stage, Cook-Gordon effect, and fiber two-way pullout.

The main objective of this paper was to establish an explicit fiber bridging model of ECC that could be easily adopted for all ECC researchers. On basis of Lin et al.’s FPRM model (21), the fiber rupture zone in this study in debonding and slipping stage was separately calculated. Then, the fiber bridging model was derived when complete fiber bridging stress with no fracture subtracted the bridging stress in the fiber debonding and slipping zone. In addition, Cook-Gordon effect and fiber two-way pullout were included to modify the crack bridging model, thus evolving the ultimate σ δ relation of ECC. Finally, the current σ δ relation was verified against experimental fiber bridging behavior in a series of ECCs. This analytical model was expected to favor ECC material design in terms of pseudo strain hardening and multiple crack opening behavior.

To obtain pseudo strain hardening and multiple crack expanding behavior of ECC under tensile load, the fiber bridging theory was fundamentally framed based on the energy-based criterion and strength-based criterion (20, 25). The typical σ δ relation is shown in Figure 1.

Where σ c and σ 0 are the matrix tensile cracking stress and peak fiber bridging stress in σ δ curve, respectively. Strength-based criterion is a prerequisite for multiple cracking development, indicating that fiber bridging stress could be effectively transferred from initial crack to the next crack only when the maximum peak fiber bridging stress exceeds the matrix tensile cracking stress depended on the fracture toughness and flaw size of the matrix (26), which control the start-up of cracks.

Where J t i p and J b ' are the crack tip toughness and complementary energy, respectively. The specific formula of J t i p and J b ' are shown in Figure 1. Energy-based criterion was derived by the reference (27) using Jintegral analysis, which guides saturated multiple cracking development only if the crack tip toughness is lower than the complementary energy. Moreover, crack tip toughness is approximately equal to the formula K m 2 / E m , where E m is tensile elastic modulus. Moreover, matrix cracking can absorb energy in the cracking process, which can affect the stress field intensity factor (KI). With KI increases, the stress at the crack tip is large enough to reach the fracture strength of the material, and the crack will expand and lead to fracture. So, the Km used in this work is the matrix fracture toughness, which is the critical value of KI. The matrix fracture toughness could be obtained by single crack three-point bend test per ASTM E 1820 -05a (28).

Simultaneous requirement of Equation [1] and [2] is indispensable to achieve the characteristics of pseudo strain hardening and multiple cracking of ECC. Both criterions are well employed to guide the materials tailoring to achieve high ductility performances by strain-hardening strength indices P S H s t r e n g t h = σ 0 / σ c and strain-hardening energy indices ( P S H e n e r g y = J b ' / J t i p ) (29). For PVA-ECC, the P S H s t r e n g t h and P S H e n e r g y should exceed 1.5 and 3, respectively, while that of PE-ECC should greater than 1.2 and 3, respectively (13).

Based on the crack plane, the fiber pullout model was designed during debonding and slipping stage. Figure 2 shows the fiber inclined angle and embedment length scheme, where θ is the inclined angle θ ∈ 0, π 2 , L L   a n d   L S are the embedment length of the long end and short end, respectively. δ 0 L   a n d   δ 0 S are the crack opening distance corresponding to the complete debonding point of the long end and short end, respectively. The fiber starts to slip when δ exceeds 2 δ 0 S . The composite bridging stress can be computed by summing all single fiber stressing fibers crossing the crack plane (18). The relation between fiber bridging stress and crack opening distance was established, which mainly included fiber, matrix, and fiber-matrix properties, as shown in Equation [3]. Meanwhile, the crack displacement δ 0 at full-debonding point was defined in Equation [4] (18, 21).

Where σ b b and σ b p presents the fiber bridging stress during debonding and slipping periods, respectively. τ 0 is the matrix/fiber frictional stress, G d is the chemical bond stress, E f is the modulus of elasticity of fiber, β is the slip hardening coefficient, d f presents fiber diameter. L e is the fiber embedment length, which is less than L f / 2 Moreover, f is the snubbing coefficient (30). η = V f E f / ( 1 - V f ) E m , where V f is the fiber volume fraction and E m is the matrix tensile elastic modulus.

Additionally, the fiber apparent tensile strength exhibits a degradation effect, which decreases with the increasing inclined angle (31). For PVA fiber, the strength reduction effect could be accounted by Equation [5].

Where σ f u is fiber tensile fracture strength, f ′ is the fiber tensile strength reduction coefficient.

Fiber two way pullout was first reported with respect to PVA fiber due to slip-hardening interface behavior (34). The σ e f f e c t i v e δ exclude the fiber two-way pullout situation. In this study, the so-called nominal fiber was adopted to supply a reasonably approximate crack opening width contributed by fiber two-way pullout (14, 23).

The single fiber bridging stress is the function of embedment length L e and inclined angle θ , thus Equation (10) can be rewritten as Equation [31] (23) Where ξ b is the efficiency of fiber bridging, which represents the fiber contributing stress as fibers increasingly fracture across a crack plane. Thus, ξ b will gradually reduce to zero. It is noteworthy that the efficiency fiber contributing stress for fibers containing the rupture fiber in debonding and slipping stage is defined as ξ e f f e c t i v e b which is given in Appendix I. Hence, according to Equation [31], σ b is presented in Equation [33]. For fibers having a long embedment length l e   ( l e ∈ ( L f 2 , L f ) ) , under the effective bridging stress σ b , the displacement of fiber in debonding and pullout stage can be deduced (14), as given in Equation [34] and [35], respectively.

Where σ b is the nominal fiber bridging stress, ξ b can be derived in Equation [32].

Where   σ 0 = τ 0 V f L f 2 d f , γ = 8 G d E f d f ( 1 + η ) τ 0 L f ,   u 0 = τ 0 L f 2 E f d f ( 1 + η ) , α = 2 d f β L f . The displacement of δ L is the function of maximum fiber bridging stress σ b m a x and embedment length of the long side l e in the loading process, as shown in Equation [36].

Assuming l e is a uniform random distribution in the interval from L f 2 to L f , thus the probability density function of l e is defined as f l e by Equation [37]. Afterwards, the remedial crack opening width δ r is approximately averaging δ L by the integral of l e , as presented in Equation [38].

Moreover, it is known from Equation [34], the maximum fiber bridging stress σ b m a x in debonding stage has a critical value (see Equation [39]), which deduces the current fiber critical embedment length l e 0 , as given in Equation [40].

Where ζ = V f σ b m a x 4 σ 0 . A long embedment length of fibers with higher than l e 0 is still at the debonding stage.

According to the above definition, two scenarios could be generated according to l e 0 .If l e 0 ≤ L f 2 , namely σ b m a x ≤ 4 σ 0 V f 1 + γ 2 , implying that the long embedded side of fibers is in debonding process, δ r is given by Huang et al. (14).

If L f 2 < l e 0 ≤ L f , namely 4 σ 0 V f 1 + γ 2 ≤ σ b m a x < 4 σ 0 V f 2 + γ 2 , which indicates that the long embedded side of fibers is in the debonding and pullout process, δ r is given as follows [42]

Cook-Gordon effect will contribute the additional width to the σ e f f e c t i v e δ , which describes a precocious fiber debonding normal process. Meanwhile, the fiber-matrix is separated under the fiber bridging stress related to the elastic crack tip field of the adjoining matrix crack in the horizontal plane, thus leading to an attached crack opening δ c g generated from the fiber elastic stretching of the fiber segment (23). Therefore, the fiber debonding process initiates ahead of the matrix cracking, leading to stretch a free fiber segment δ c g , as given in Equation [43] (23).

Where α is the Cook-Gordon parameter. In general, α = 2 d f for PVA fiber was suggested, while α is set as 15 d f for polyethylene (PE), and polypropylene (PP) fibers (35).

Overall, the total crack opening width including the contribution of fiber two-way pullout and Cook-Gordon effect is given as [44]:

Figure 9 shows the computation flow scheme of final σ δ . The experimental data including fiber, matrix, and fiber/matrix parameters from Yang et al. (23) were employed to verify the final σ δ . The micro-parameters for PE-ECC and PVA-ECC are listed in Table 1. It is well known that PVA fiber has a strong chemical bond strength and slip hardening behavior in hardened pastes. Table 1 illustrates the micro-parameters for PVA-ECC and PE-ECC.

Figure 10 shows theoretical σ δ relation of PVA-ECC analyzed by the current model. When only one way was considered, the predicted peak bridging stress and corresponding crack opening width were 64 μm and 1.50 MPa, respectively. If two ways are accounted, the same peak stress is obtained at the crack opening width of 165.8 μm. Moreover, the Cool-Gordon effect can add 2.1 μm to the crack opening width and reach 167.9 μm, indicating that the contribution of Cool-Gordon effect to crack opening width is faint due to the small assuming Cook-Gordon parameter α = 2 d f for PVA fiber.

Figure 11 illustrates the predicted composite σ − δ relation of PVA-ECC. The predicted and experimental composite characteristics of PVA-ECCs and PE-ECCs are listed in Table 2. From Figure 11, it was difficult to research the variation of pseudo elastic properties of three models in the elastic part. The approximate first crack strength from three models was presented, which was lower than the experimental data. Afterward, cracks gradually develop during strain in three models, which show a different maximum fiber bridging stress and corresponding crack opening width. Moreover, the accuracy coefficients of the two key parameters by three models are listed in Table 2. For PVA-ECC with a volume fraction of 0.1%, the experimental curves scattered inside the grey zone referenced by (23), where its peak stress and corresponding crack opening width was 0.295 MPa and 165.0 μm, respectively, , while that of Huang et al. (14) presented a peak stress of 0.31 Mpa at the crack width of 131.0 μm, and that of Yang et al. (23) has a peak stress of 0.35 Mpa at the crack width of 107.0 μm. Hence, employing the current model could further improve the accuracy of crack opening width that reaches 175.1μm with a proximal peak bridging stress of 3.0 MPa. In addition, for PVA-ECC with a volume fraction of 0.5 %, the maximum fiber bridging stress obtained by Huang’s model, Yang’s model, and the current model were 1.52 MPa, 1.72 MPa, and 1.50 MPa, respectively, while the corresponding crack opening width were 141.5μm, 125.6 μm, and 171.9 μm, respectively. Hence, compared to the real experimental maximum fiber bridging stress of 1.65MPa and crack opening width of 160.1 μm, the current model has a higher prediction accuracy than the other two models (see Table 2). It was clearly found that a proximate crack opening presented higher precision by using the current model than the other two predicted models. In fact, the current models use the opposite logic calculation compared to the Huang’s model, in which fiber bridging stress with no rupture subtracts the fiber fracture stress in debonding and pullout stage. In Huang’s model, the operative partial fiber bridging stress in the debonding and pullout stage are severalty picked, thereafter summing those partial stresses to form the fiber bridging stress of composites. Therefore, the current model provides an effective accuracy compared to the experimental data, which is also easy to understand for researchers focusing on the fiber reinforced cement composites field.

Based on the micro-parameters in the previous study (36), Figure 12 demonstrates the comparison of the predicted σ − δ relation for PE-ECC by three models. It can be seen from Figure 12 and Table 2 that, the maximum bridging stress and corresponding crack opening width of the M-C group by using Yang’s model are around 12.99 MPa and 99.0 μm, respectively, while that of Huang’s model presented a maximum stress of 13.1 Mpa at the crack width of 56.8 μm. Noticeably, the Cool-Gordon effect was significant in verifying the real width of PE-ECC due to the big Cool-Gordon effect of α = 15 , which was not included in Huang’s model from Figure 12 (b), (c) and (d). Employing the current model could further improve this accuracy of crack opening width that reaches 97.9 μm with a slightly bigger peak bridging stress of 14.0 MPa. Hence, Yang’s model and the current model have a higher average crack width than Huang’s model for PE-ECC since the Cool-Gordon effect of Huang’s model could not effectively be implemented. On the other hand, Yang’s model and current model show a proximal value for M-25, M-50, and M-100 compared to the real average crack width calculated from the crack patterns of the tensile test. A near-peak bridging stress was illustrated for three models. However, the current model is easy to calculate compared to the implicit Yang’s model. However, the corresponding crack opening width at maximum bridging stress by three models is lower than the experimental data for PE-ECC since there is amount of inelastic strain produced by micro-cracks near the main crack (15).

This crack bridging model for ECC was developed with an opposed calculation program based on FPRM model (21), which improves the predicted precision by subtracting the fiber rupture stress in debonding and pullout stage. Based on the results obtained, the following conclusions can be drawn.