A Modified Fiber Bridging Model for High Ductility Cementitious Composites Based on Debonding-Slipping Rupture Analysis

Modified micromechanical bridging model is established with consideration of the fiber rupture effect at debonding and slipping stages. The bridging model includes the debonding and slipping rupture of fibers and establishes the fiber/matrix interfacial parameters (friction τ 0 , chemical bonding force G d , slip-hardening coefficient β ). A different interfacial bonding can cause fiber rupture. The influence of the interfacial conditions on the fiber rupture risk was investigated. In the modified bridging model, the effective bridging stress, the debonding rupture stress, and the slipping rupture stress were clearly identified. Finally, single-fiber pullout tests with different embedded lengths were carried out to validate the bridging model. The relationship between the fiber bridging stress and the crack opening predicted by the bridging model was consistent with the experimental results. This modified micromechanical bridging model can be used to quantitatively calculate the actual fiber bridging capacity and to predict the ductility of the high ductility cementitious composites reinforced by different types of fibers.


Introduction
High ductility cementitious composites (HDCCs) exhibit an excellent tensile ductility ability accompanied by closely spaced multiple cracks appearing before the nal failure [1,2]. e tensile ductility can reach several hundred times that of traditional cementitious composites, and the average tight crack opening is typically less than 100 μm [3,4]. ese extraordinary characteristics can be designed by the "bridging law," which give the ber bridging stress versus crack opening relation for short randomly distributed bers [5][6][7]. With a proper design, the bers bridging stress can be transferred e ciently back to the HDCC matrix after the rst cracking event, which enables the composite to undergo multiple cracking and have a high ductility behavior [8,9]. e bridging model has been extensively investigated in the past as a key theory for HDCC [10]. Nevertheless, this existing ber bridging model is usually limited to speci c application conditions that do not account for chemical bonding or ber slipping rupture analysis during crack propagation.
In the ber bridging model design, handling ways of the ber/matrix interfacial parameters (friction stress τ, chemical bonding force G d , and slip-hardening coe cient β) determine the accuracy and the applicable conditions. In simpli ed bridging model [10], the G d and β are ignored. Besides, the friction τ was assumed to be constant and equal to an τ 0 . Based on those assumptions, only hydrophobic high-strength bers, such as PE ber and carbon ber, are quali ed. However, some hydrophilic bers, such as PVA ber, can form a high chemical bond and strong sliphardening e ect with the surrounding matrix [11][12][13], and the simpli ed model cannot be used in this case because of its restriction. Lin and Li replaced the constant friction by linearly increasing the interfacial friction and considered the parameters of G d and β to improve the bridging model [5]. erefore, most synthetic fibers can be treated with the model whether they are hydrophilic or hydrophobic.
Another essential element of analysis in the bridging model is the fiber rupture phenomenon during crack propagation. Without considering of fiber rupture, the fiber bridging ability will be seriously overestimated [8]. In fact, fibers may rupture at the debonding stage or the slipping stage once the fiber stress exceeds fiber tensile strength. Maalej et al. [14], Kanda et al. [9], and Lin et al. [15] extended the bridging model by including the chemical bonding G d , the slip-hardening effect β, and the fiber debonding rupture analysis. Moreover, the bridging stress versus crack opening relation for a single-fiber pullout with a normal and an inclined angle was derived to explain the fiber strength reduction when an inclined angle is used. Wang [16] and Yang et al. [17] established a two-way fiber bridging model by considering the matrix micro-spalling effect. Based on Yang's two-way model, Huang et al. [8] devoted to account for fiber rupture phenomenon including debonding rupture and slipping rupture at the fiber pullout stage. e accuracy of the predicted composite bridging stress versus crack opening relation has greatly been improved compared to previous models. Nevertheless, the slipping rupture analysis is not clearly clarified in Huang's model. Besides, the existing crack opening δ 0 at the debonding stage was ignored in the slipping rupture analysis, which will lead to an inaccurate evaluation for the slipping rupture of fibers. Lu and Leung [18] elaborated on the cracking process and the stress-strain relation by considering the variation of matrix strength along the member, the increased crack bridging stress in the hardening regime, and the possibility of fiber debonding rupture. As a result, the fiber bridging law was gradually perfected for high ductility cementitious composites. However, the bridging model still needs to be improved to more accurately analyze fiber rupture.
In this study, some noteworthy details including the fiber/matrix interfacial parameters and the fiber rupture growth during crack propagation were investigated. e complex slipping rupture phenomena of the entire rupture process of the fiber in the matrix were accurately analyzed.
is work presents the current model for the evolution of the fiber bridging stress with crack opening based on fiber rupture analysis, which included the friction τ 0 , the chemical bonding force G d , and the slip-hardening coefficient β. Compared to existing bridging models, this model considers the fiber state change during the pullout process and is more realistic. In bridging model, the effective bridging stress, the debonding rupture stress, and the slipping rupture stress were clearly identified. Finally, single-fiber pullout tests with different embedded fiber lengths were carried out to validate the bridging model. Fibers in HDCC are assumed to be randomly distributed in all 3 dimensions, and the spatial locations of the fibers are expressed as f(z) and f(θ) [19]. Detailed descriptions are given in section 2.1. e fiber/matrix micromechanical interfacial parameters can be obtained by the single-fiber pullout test. Typical single-fiber pullout curves of PVA-HDCC are shown in Figure 1.

Modified Micromechanical Bridging Model
As can be seen in Figure 1, the slip-hardening phenomenon occurs after debonding stage. e slip-hardening effect describes the phenomenon that the pullout load increases continuously after the debonding stage of the fiber. e main reason is that the fiber surface is abraded during pullout process. e pullout channel is blocked by the fiber residue ( Figure 2), resulting in an increase in the pullout load. Consequently, an intact pullout curve without fiber rupture can be divided into two major regimes: a debonding stage and a slippage stage. Actually, fibers may rupture in the debonding stage or the slipping stage. Rupture in the debonding stage (RD type), rupture in the slipping stage (RS type), and complete pullout without rupture (CP type) are illustrated with an example in Figure 1. For the PVA fiber, serious abrasion and delamination are observed during the pullout process (Figure 2), which can cause slip-hardening effect and increase the risk of slipping rupture. Furthermore, the fiber tensile strength will also decease due to the abrasion effect, even if the fibers are embedded vertically. e interfacial parameters (G d , τ 0 , β) are calculated by equations (1) to (3) [20].
where G d is the chemical debonding energy value (J/m 2 ), τ 0 is the frictional bond strength (Pa), β is the slip-hardening coefficient, P a is the peak load of the pullout curve in the debonding stage (N), P b is the load after the sudden drop following P a (N), E f is the fiber modulus of elasticity (Pa), d f is the fiber diameter (m), L e is the fiber embedment length (m), (ΔP/Δs)| S⟶S 0 is the slope of the pullout curve after full debonding, and S 0 is the displacement corresponding to a full debonding (m).

Single-Fiber Bridging Stress.
A theoretical single-fiber bridging pullout model was derived by Lin and Kanda [15]. needs a greater applied force to complete the debonding process whereas the shorter embedded side can be sliding. According to aforementioned assumptions, the crack opening on one side should be δ for a given general crack opening 2δ in the debonding stage, as illustrated in Figure 3(b). Consequently, the stress of a single bridging fiber can be calculated by equation (4) [17]. When full debonding is completed, the crack opening δ 0 is calculated by equation (4) for θ � 0 and is given by equation (5) for where σ b d is the fiber bridging stress in the debonding stage and σ bp is the fiber bridging stress at slipping stage (Pa). δ is the single main bridge crack opening, and δ c is the full debonding crack opening (m). η where E m is the elastic modulus of the matrix (Pa), and V f and V m represent the volume fraction of the fibers and the matrix, respectively.
Equation (4) applies to the fibers embedded in a direction perpendicular to the crack surface. However, in the more general case, randomly distributed fibers will intersect the crack plane with different inclined angles θ. For inclined fibers, the fiber bridging stress is magnified by the snubbing effect and given by equation (6) [21]. Moreover, the apparent fiber tensile strength will be decreased because of the    Advances in Materials Science and Engineering 3 inclined angle θ. is degradation effect can be represented by equation (7) [19,22,23].
where f is the snubbing coefficient, f ′ is the reduction of the apparent fiber strength, θ is the inclined angle, σ fu is the apparent fiber tensile strength, and σ n fu is the nominal fiber tensile strength (Pa).

Without Fiber Rupture
Analysis. e composite bridging stress versus crack opening relation is used to link the properties of the matrix, the fiber, and the fiber/matrix interface. e spatial location and interfacial parameters of the randomly distributed fibers are contained in the relationship. Li et al. [15] used a double integral method to add up the contributions of every single fiber in the crack plane: where V f is the volume fraction of the fibers, and f(z) and f(θ) are the probability density functions at the inclined angle θ and centroid distance z of fibers from the crack plane, respectively. For a 3D random distribution, ). e geometric relations between L e , L f /2, and z are revealed in Figure 3(a). e embedment length L e is converted using rough an integral conversion, an extended expression is given by the following equation [8,11]: Ideally, without considering fiber rupture during crack propagation, the general bridging stress σ c (δ) is calculated by equation (10). e fibers with a shorter embedment length go through the debonding stage and then the slipping stage, but the fibers with a longer embedment length are still in the debonding stage until the crack opening expands to 2δ 0 (L e ). Figure 4 shows the specific state of randomly distributed fibers in the composites for a given crack opening δ r . e full debonding stage is only completed for every single fiber in the case of δ r � 2δ 0 (L e � L f /2).
One case of the relationship between the bridging stress and the crack opening without fiber rupture analysis is shown in Figure 5. e physical properties of the PVA fibers are listed in Table 1 and the volume fraction of the fibers is 2%. e predicted peak bridge stress reaches 51 MPa, which is significantly higher than the actual value. erefore, the fiber bridging capacity is seriously overestimated. When the crack expands to 0.18 mm, the debonding stage is over. Due   Advances in Materials Science and Engineering to the slip-hardening effect, the bridging stress still rapidly increases in the slipping stage. However, the fiber stress will be much larger than the fiber tensile strength and the fibers that have been ruptured cannot bridge the crack plane. Consequently, fiber rupture analysis is indispensable to obtain an accurate bridging model, especially for the slipping rupture.

With
Fiber Rupture Analysis. Fiber rupture will occur once the fiber stress at the crack plane reaches the apparent fiber strength. According to section 2.1, debonding rupture and slipping rupture can be used to analyze fiber rupture phenomenon. e boundary between potential debonding rupture and potential slipping rupture can be derived by the potential critical embedment length L d (θ) and L p (θ).
, as shown in equation (11). For selected composites, L d (θ) is determined by the angle θ. e larger the angle θ, the smaller L d (θ) will be. e maximum value L d (0) and the minimum value L d (π/2) can be obtained from equation (11). If the embedded length L e (θ) exceeds L d (θ), the fibers will risk debonding rupture.
For different fiber/matrix interfacial conditions, the L d (θ) curves have three shapes, as shown in Figure 6. If L d (π/2) > L f /2, debonding rupture will never occur no matter the value of θ (Figure 6(a)). Otherwise, debonding rupture will happen during crack propagation ( Figure 6(b), 6(c)). If the L d (θ) curve intersects with δ, the angle θ dδ can be calculated from Figure 6 shows the potential debonding rupture zone determined by L d (θ). e fibers located in the potential debonding rupture zone are not simultaneously broken, but they are gradually destroyed during crack propagation. e fibers with a longer embedment length and a bigger angle rupture first. Afterward, the fibers with a shorter embedment length and a smaller angle only start to rupture. For a given crack opening δ r , the corresponding debonding rupture Advances in Materials Science and Engineering zone can be depicted by l d (δ r ), which is calculated by . e debonding rupture length l d (δ r ) is given by .
(13) Figure 7 shows the developing process of l d (δ r ) when the crack expands from 5 μm to 250 μm. e debonding rupture zone is determined by the intersection of l d (δ r ) and L d (θ). When l d (δ r ) � L d (π/2), the fibers initiate debonding rupture. Here, the crack opening δ c d can be calculated from  [8,17].
Advances in Materials Science and Engineering δ c d (π/2) is the minimum debonding rupture crack opening. When 0 ≤ δ ≤ δ c d (π/2), there is no debonding rupture. e growth of the fiber debonding rupture zone is illustrated in Figure 7. When l d (δ r ) � L f /2, the debonding stage is over and the crack opening δ dt is given by When the curve of l d (δ r ) intersects with L d , the inclined angle θ c (δ r ) can be calculated from equation

Fiber Slipping Rupture.
Due to the slip-hardening effect, the stress of the fibers in the slipping stage can be higher than in the debonding stage. e fibers, which are not ruptured in the debonding stage, may rupture in the slipping stage. Slipping rupture should also be considered. Similar to the debonding rupture analysis, the potential slipping rupture zone and the current slipping rupture zone need to be determined. e potential critical embedment length L p (θ) can be calculated for σ bp (θ) max � σ fu (θ). e peak bridging stress σ bp (θ) max in the slipping stage is calculated from δ � L e /2 + 2δ 0 + d f /2β.
e boundary value of L p (θ) can be obtained by L p (0) and L d (π/2). If L p (θ) ≥ L d (θ), the fibers with an embedded length L e > L p (θ) will be debonding rupture first and will not undergo slipping rupture. Generally, the L p (θ) curve is lower than the L d (θ) curve for PVA fibers. When the curve of L p (θ) intersects with δ, the inclined angle θ pδ can be calculated from e potential slipping rupture fibers are gradually destroyed during crack propagation. For a given crack opening δ r , the current slipping rupture zone can be depicted by l p (δ r ), which is calculated for σ bp (θ) � σ fu (θ). e current slipping rupture length l p (δ r ) is given by Equation (19) can be simplified for l p (δ r ) � d 2 f σ n fu e − (f+f′)θ /4τ 0 (d f + β(δ r − 2δ 0 )) + (δ r − 2δ 0 ). In Huang's model [8], the crack opening 2δ 0 is omitted, which will lead to a larger current slipping rupture zone for a given crack opening δ r . In this study, 2δ 0 was considered in l p (δ r ).
A case of fiber slipping rupture analysis is shown in Figure 8. It presents a 3D shape of l p (δ r ) with δ from 50 μm to 300 μm and θ from 0 to π/2. l p (δ r ) is not a flat but a coiled surface. e area enclosed by the coiled surface represents the slipping rupture space. e rupture analysis for δ r (100,   Figure 9. e slipping rupture zone forms a parabolic progression. e slipping rupture zone cannot be neglected compared to the debonding rupture zone. An interesting case was found for fiber rupture analysis. At the debonding stage, fibers rupture more easily for a longer embedment L e . However, this may be invalid in the slipping stage. e bridging stress of fibers with a longer embedment may not reach the fiber tensile strength in the slipping stage. In equation (19), (δ− 2δ 0 ) represents the crack opening generated by the slipping stage. e longer L e , the larger 2δ 0 and the smaller (δ− 2δ 0 ) will be. e smaller crack opening (δ− 2δ 0 ) in the slipping stage may not be enough to break the fiber. erefore, a shorter embedment L e may rupture earlier.

3.3.
e Influences of Interfacial Parameters on Rupture Analysis. According to the aforementioned analysis, the interfacial parameters determine the fiber rupture space. e influence of τ 0 , G d , and β on L d (θ) and L p (θ) is shown in Figure 10. When τ 0 changes from 0.8 to 8 MPa, the L d (θ) and L p (θ) curves significantly shift down, as shown in Figure 10(a), which means the potential rupture space rapidly increases. Expectedly, the L d curve is more affected by the change in τ 0 . In the debonding stage, L d (θ) will be decreased because of G d , as shown in Figure 10(b). Although the effect of G d was less than τ 0 on L d , it needs to be considered for the accuracy of the model. In the slipping stage, the slip-hardening phenomenon cannot be ignored. When β changes from 0.01 to 1.0, the L p (θ) curves evolve, as shown in Figure 10(c). We can conclude that the main factor for the critical embedment length in the debonding stage and the slipping stage was τ 0 and β, respectively. is can be used to tailor fibers/matrix interfaces and treat fiber surfaces. e influence of different interfacial parameters on fiber rupture analysis is illustrated in this study through some specific cases, as shown in Figure 11. e potential fiber rupture zone and the current rupture zone were analyzed when the crack opening was δ r � 100μm. In cases with a weaker slip-hardening effect, the L d (θ) curves are close to L d (θ) and slipping rupture can almost be ignored since debonding rupture dominates. If the L d (θ) curve is above L d (θ), there will be no significant slipping rupture. e interfacial parameters play a decisive role in determining which type of fiber rupture dominates.

σ(δ) Relationship When considering Fiber Rupture.
rough a comprehensive consideration of debonding rupture and slipping rupture, the effective fiber bridging stress σ effective (δ) is calculated from equation (20), where σ R D (δ) and σ RS (δ) represent the fiber debonding rupture stress and slipping rupture stress, respectively.
During crack propagation, σ R D (δ) is calculated by integrating the zone for which the fiber debonding rupture occurs.
Due to the geometrical irregularity of the current slipping rupture zone, it is very complicated to calculate σ RS (δ). Figure 12 shows the growth process of the slipping rupture zone S1. e current slipping rupture zone S1 was represented by a parabolic enclosure subtract zone S2. σ RS (δ) is calculated by integrating the current slipping rupture zone.   Advances in Materials Science and Engineering l p1 (δ r ) and l p2 (δ r ) represent the upper part and lower part of the l p (δ r ) curve, respectively. As shown in Figure 12, l p1 (δ r ) may be greater than L f /2 during crack propagation. At this point, the value of l p1 (δ r ) should be set to L f /2.
When δ r � L f /16, l p2 (δ r ) is in close proximity of L p (θ), l p2 (δ r ) is considered to be equal to L p (θ) and σ RS (δ) can be calculated by (5). where θ v is the inclined angle when l p1 (δ r ) intersects with l p2 (δ r ) and θ s is the inclined angle when l p (δ r ) intersects with L d (θ). e process to determine σ effective (δ) when considering fiber rupture is illustrated by the flowchart in Figure 13. Finally, the current fiber bridging model can be built. Figure 14 shows the calculation result based on the whole flowchart. e influence of fiber debonding rupture and slipping rupture on the bridging stress is clearly shown in Figure 14.
e current fiber bridging stress, debonding rupture stress, and slipping rupture stress are perfectly determined during crack propagation. e peak bridging stress is 5.0 MPa corresponding to a peak crack opening of 104 μm. Compared to result without considering fiber rupture, the predicted value is more reliable. e predicted critical embedment length and σ(δ) relation will be compared with experimental result in the following section to verify further the accuracy of the current bridging model. Portland cement conform to the Chinese standard GB175-2007 and Class F fly ash conform to the ASTM C618 standard were used. e river sand used in the experiments had a maximum size and fineness modulus of 0.60 mm and 1.40, respectively. Water reducer and hydroxypropyl methyl cellulose (HPMC) were used to adjust plastic viscosity of the paste. e mixing steps of HDCC paste are as follows: all cementitious materials, fine sand, water reducer, and HPMC were weighed accurately and mixed for 1 min at a speed of 140 rpm. en, water was added and mixed for 5 min at a speed of 280 rpm.

Model Verification
PVA fibers modified with a mass fraction of 1.2% oil agent were used in this study, and its mechanical and geometrical properties are given in Table 2. e measured average tensile strength of the PVA fibers reached 1260 MPa, as shown in Figure 15(a). To obtain the fiber/ matrix interfacial parameters, single-fiber pullout tests were carried out. e fibers were vertically embedded in the matrix, and the embedment length L e was between 1 mm and 5 mm to determine the critical embedment lengths L d (θ) and L p (θ). Single-fiber bridging stress versus pullout displacement curves are shown in Figure 15(b)-15(f ). e single-fiber embedment length L e was set to 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm. Here, the Advances in Materials Science and Engineering interfacial bonding performance was recorded as L1, L2, L3, L4, and L5, respectively. e fiber rupture phenomenon can be seen in Figure 15 from all pullout curves with L e from 1 to 5 mm. e rupture strength of PVA fibers only reached 580 to 760 MPa, which was far less than the measured tensile strength. e apparent strength of the PVA fibers dropped drastically even if the fibers were embedded vertically (θ � 0). Figure 16 shows an SEM image of the fiber rupture zone. e fiber pullout zone was seriously abraded. e abrasion effect causes a loss in fiber tensile strength and in the effective diameter. To take into account this effect, the apparent fiber strength was recalculated from equation (22), where k represents the strength reduction coefficient due to abrasion. In this study, k was set to 3.0. Finally, the fiber apparent strength in the matrix was recalculated at 670 MPa.
For fibers with an L e of 5 mm, the interface parameters could not be exactly obtained because of the fiber ruptured   before complete debonding. e interface parameters (G d , τ 0 , β) of PVA fiber/matrix were calculated using equations (1)-(4), as shown in Figure 17. e values of τ 0 and β slowly decreased when L e increased from 1 mm to 4 mm. e average values of G d , τ 0 , and β are given in Table 3. Figure 18 reveals the critical embedment length L d (θ) and L p (θ) for different embedment lengths. Due to data fluctuation with the interfacial parameters, L d (θ) changed from 3.2 to 5.8 mm when θ was 0, which meant that the fibers could rupture by debonding rupture when the embedment length is in this range. e experimental results mostly followed the debonding rupture analysis. e most of fibers with an embedment length of 5 mm underwent debonding rupture (Figure 15(f)), and a few fibers underwent debonding rupture when the embedment length was between 3 and 4 mm (Figures 15(d) and 15(e)). In addition, L p (θ) changed from 1.2 to 1.9 mm when θ was 0, which was slightly higher than the experimental result. Figure 15 slipping rupture more easily, which also revealed how essential it is to include slipping rupture in the analysis to translate the real situation. Figure 19 shows the relationship between the bridging stress and crack opening for the experimental results and the predicted   model. e fiber content in the composite is 0.5% by volume fraction so that a single crack opening easily occurs [17]. In Figure 19, the experimental results are marked in pale yellow. e peak bridging stress was in the 1.5-1.2 MPa range corresponding to a peak crack opening in the 97-118 μm range. e predicted results were in the region of the experimental data. e predicted peak bridging stress was 1.4-1.2 MPa corresponding to a peak crack opening in the 93-125 μm range. e σ-δ relationship predicted by the bridging model is consistent with the experimental results.

Conclusions
A new fiber bridging model for high ductility cementitious composites reinforced with PVA fiber was built based on the previous theories predicting the relationship between the bridging stress and crack opening. e main elements of the modified bridging model can be summarized as follows: (1) Due to the hydrophilic surface, the PVA fiber can form a strong chemical bond with the surrounding matrix. Furthermore, the strong slip-hardening effect cannot be ignored. Based on these elements, the main interfacial parameters, namely, the friction stress τ 0 , the chemical bonding force G d , and the slip-hardening coefficient β, were all included in the modified bridging model to better understand PVA-HDCC. (2) Fiber rupture will occur once the fiber stress in the crack plane reaches the apparent fiber strength during crack propagation. e modified bridging model synthetically takes into account fiber debonding rupture and slipping rupture.
is is crucial to build an accurate bridging model.
(3) An abrasion phenomenon on the fiber surface was observed during the pullout process, which could cause a significant loss of the fiber tensile strength and its effective diameter. e abrasion effect was considered in the fiber rupture analysis.

Advances in Materials Science and Engineering
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
ere are no conflicts of interest.