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

Authors: Ding Cong; Guo Liping; Ren Jinming; Wang Yongming; Li Xinyu; Gao Yuan; Liu Wanpeng; Li Ruize
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

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 (frictionτ0, chemical bonding force Gd, 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.


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## Body

## 1. Introduction

High ductility cementitious composites (HDCCs) exhibit an excellent tensile ductility ability accompanied by closely spaced multiple cracks appearing before the final failure [1, 2]. The 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]. These extraordinary characteristics can be designed by the “bridging law,” which give the fiber bridging stress versus crack opening relation for short randomly distributed fibers [5–7]. With a proper design, the fibers bridging stress can be transferred efficiently back to the HDCC matrix after the first cracking event, which enables the composite to undergo multiple cracking and have a high ductility behavior [8, 9]. The bridging model has been extensively investigated in the past as a key theory for HDCC [10]. Nevertheless, this existing fiber bridging model is usually limited to specific application conditions that do not account for chemical bonding or fiber slipping rupture analysis during crack propagation.In the fiber bridging model design, handling ways of the fiber/matrix interfacial parameters (friction stressτ, chemical bonding force Gd, and slip-hardening coefficient β) determine the accuracy and the applicable conditions. In simplified bridging model [10], the Gd and β are ignored. Besides, the friction τ was assumed to be constant and equal to an τ0. Based on those assumptions, only hydrophobic high-strength fibers, such as PE fiber and carbon fiber, are qualified. However, some hydrophilic fibers, such as PVA fiber, can form a high chemical bond and strong slip-hardening effect with the surrounding matrix [11–13], and the simplified 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 Gd and β to improve the bridging model [5]. Therefore, 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 Gd, 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. The 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 δ0 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. The complex slipping rupture phenomena of the entire rupture process of the fiber in the matrix were accurately analyzed. This work presents the current model for the evolution of the fiber bridging stress with crack opening based on fiber rupture analysis, which included the frictionτ0, the chemical bonding force Gd, 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.


## 2. Modified Micromechanical Bridging Model

### 2.1. Single-Fiber Bridging Model

#### 2.1.1. Single-Fiber Pullout Behavior of HDCC

During crack propagation, the uniform randomly distributed fibers in HDCC will have five states, namely, debonding state, slipping state, debonding rupture state, slipping rupture state, and complete pullout state. The specific state depends on the fiber’s spatial location and the interfacial bonding. Fibers in HDCC are assumed to be randomly distributed in all 3 dimensions, and the spatial locations of the fibers are expressed asfz and fθ [19]. Detailed descriptions are given in section 2.1. The 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.Figure 1
Typical pullout behavior curves of PVA fiber.As can be seen in Figure1, the slip-hardening phenomenon occurs after debonding stage. The slip-hardening effect describes the phenomenon that the pullout load increases continuously after the debonding stage of the fiber. The main reason is that the fiber surface is abraded during pullout process. The 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. The interfacial parameters (Gd,τ0,β) are calculated by equations (1) to (3) [20].(1)Gd=2Pa−Pb2π2Efdf3,(2)τ0=PbπdfLe,(3)β=dfLe1τ0πdfΔPΔS|S⟶S0+1,where Gd is the chemical debonding energy value (J/m2), τ0 is the frictional bond strength (Pa), β is the slip-hardening coefficient, Pa is the peak load of the pullout curve in the debonding stage (N), Pb is the load after the sudden drop following Pa (N), Ef is the fiber modulus of elasticity (Pa), df is the fiber diameter (m), Le is the fiber embedment length (m), ΔP/Δs|S⟶S0 is the slope of the pullout curve after full debonding, and S0 is the displacement corresponding to a full debonding (m).Figure 2
The abrasion and delamination phenomenon.


#### 2.1.2. Single-Fiber Bridging Stress

A theoretical single-fiber bridging pullout model was derived by Lin and Kanda [15]. In the single-fiber bridging model, there are several main assumptions: (1) the Poisson effect of fiber is negligible. (2) The elastic deformation of fiber in the slipping stage is negligible compared to the slipping magnitude. (3) The interfacial friction τ is considered to be linearly increasing instead of constant τ0. (4) The longer embedded side of the bridging fiber is always in the debonding stage because it 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 beδ 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 δ0 is calculated by equation (4) for θ = 0 and is given by equation (5) for θ = 0.(4)σbδ=Δσbdδ=4τ0δ+2Gd1+ηEfdf,0≤δ<δ0,Δσbpδ=4τ0df2Le−δ−2δ0df+βδ−2δ0,δ0≤δ≤Lf2,(5)δ0=2τ0Le21+ηEfdf+8GdLe21+ηEfdf,where σbd is the fiber bridging stress in the debonding stage and σbp is the fiber bridging stress at slipping stage (Pa). δ is the single main bridge crack opening, and δc is the full debonding crack opening (m). η=VfEf/VmEm, where Em is the elastic modulus of the matrix (Pa), and Vf and Vm represent the volume fraction of the fibers and the matrix, respectively.Figure 3
The sketch of fiber bridging crack [17]. (a) Single-side pullout model and (b) two-side pullout model.
(a)(b)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 inclined angle θ. This degradation effect can be represented by equation (7) [19, 22, 23].(6)σbθ=Δσbdθ=σbdefθ,Δσbpθ=σbpefθ,(7)σfuθ=σfune−f′θ,where f is the snubbing coefficient, f′ is the reduction of the apparent fiber strength, θ is the inclined angle, σfu is the apparent fiber tensile strength, and σfun is the nominal fiber tensile strength (Pa).


## 2.1. Single-Fiber Bridging Model

### 2.1.1. Single-Fiber Pullout Behavior of HDCC

During crack propagation, the uniform randomly distributed fibers in HDCC will have five states, namely, debonding state, slipping state, debonding rupture state, slipping rupture state, and complete pullout state. The specific state depends on the fiber’s spatial location and the interfacial bonding. Fibers in HDCC are assumed to be randomly distributed in all 3 dimensions, and the spatial locations of the fibers are expressed asfz and fθ [19]. Detailed descriptions are given in section 2.1. The 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.Figure 1
Typical pullout behavior curves of PVA fiber.As can be seen in Figure1, the slip-hardening phenomenon occurs after debonding stage. The slip-hardening effect describes the phenomenon that the pullout load increases continuously after the debonding stage of the fiber. The main reason is that the fiber surface is abraded during pullout process. The 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. The interfacial parameters (Gd,τ0,β) are calculated by equations (1) to (3) [20].(1)Gd=2Pa−Pb2π2Efdf3,(2)τ0=PbπdfLe,(3)β=dfLe1τ0πdfΔPΔS|S⟶S0+1,where Gd is the chemical debonding energy value (J/m2), τ0 is the frictional bond strength (Pa), β is the slip-hardening coefficient, Pa is the peak load of the pullout curve in the debonding stage (N), Pb is the load after the sudden drop following Pa (N), Ef is the fiber modulus of elasticity (Pa), df is the fiber diameter (m), Le is the fiber embedment length (m), ΔP/Δs|S⟶S0 is the slope of the pullout curve after full debonding, and S0 is the displacement corresponding to a full debonding (m).Figure 2
The abrasion and delamination phenomenon.


### 2.1.2. Single-Fiber Bridging Stress

A theoretical single-fiber bridging pullout model was derived by Lin and Kanda [15]. In the single-fiber bridging model, there are several main assumptions: (1) the Poisson effect of fiber is negligible. (2) The elastic deformation of fiber in the slipping stage is negligible compared to the slipping magnitude. (3) The interfacial friction τ is considered to be linearly increasing instead of constant τ0. (4) The longer embedded side of the bridging fiber is always in the debonding stage because it 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 beδ 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 δ0 is calculated by equation (4) for θ = 0 and is given by equation (5) for θ = 0.(4)σbδ=Δσbdδ=4τ0δ+2Gd1+ηEfdf,0≤δ<δ0,Δσbpδ=4τ0df2Le−δ−2δ0df+βδ−2δ0,δ0≤δ≤Lf2,(5)δ0=2τ0Le21+ηEfdf+8GdLe21+ηEfdf,where σbd is the fiber bridging stress in the debonding stage and σbp is the fiber bridging stress at slipping stage (Pa). δ is the single main bridge crack opening, and δc is the full debonding crack opening (m). η=VfEf/VmEm, where Em is the elastic modulus of the matrix (Pa), and Vf and Vm represent the volume fraction of the fibers and the matrix, respectively.Figure 3
The sketch of fiber bridging crack [17]. (a) Single-side pullout model and (b) two-side pullout model.
(a)(b)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 inclined angle θ. This degradation effect can be represented by equation (7) [19, 22, 23].(6)σbθ=Δσbdθ=σbdefθ,Δσbpθ=σbpefθ,(7)σfuθ=σfune−f′θ,where f is the snubbing coefficient, f′ is the reduction of the apparent fiber strength, θ is the inclined angle, σfu is the apparent fiber tensile strength, and σfun is the nominal fiber tensile strength (Pa).


## 2.1.1. Single-Fiber Pullout Behavior of HDCC

During crack propagation, the uniform randomly distributed fibers in HDCC will have five states, namely, debonding state, slipping state, debonding rupture state, slipping rupture state, and complete pullout state. The specific state depends on the fiber’s spatial location and the interfacial bonding. Fibers in HDCC are assumed to be randomly distributed in all 3 dimensions, and the spatial locations of the fibers are expressed asfz and fθ [19]. Detailed descriptions are given in section 2.1. The 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.Figure 1
Typical pullout behavior curves of PVA fiber.As can be seen in Figure1, the slip-hardening phenomenon occurs after debonding stage. The slip-hardening effect describes the phenomenon that the pullout load increases continuously after the debonding stage of the fiber. The main reason is that the fiber surface is abraded during pullout process. The 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. The interfacial parameters (Gd,τ0,β) are calculated by equations (1) to (3) [20].(1)Gd=2Pa−Pb2π2Efdf3,(2)τ0=PbπdfLe,(3)β=dfLe1τ0πdfΔPΔS|S⟶S0+1,where Gd is the chemical debonding energy value (J/m2), τ0 is the frictional bond strength (Pa), β is the slip-hardening coefficient, Pa is the peak load of the pullout curve in the debonding stage (N), Pb is the load after the sudden drop following Pa (N), Ef is the fiber modulus of elasticity (Pa), df is the fiber diameter (m), Le is the fiber embedment length (m), ΔP/Δs|S⟶S0 is the slope of the pullout curve after full debonding, and S0 is the displacement corresponding to a full debonding (m).Figure 2
The abrasion and delamination phenomenon.


## 2.1.2. Single-Fiber Bridging Stress

A theoretical single-fiber bridging pullout model was derived by Lin and Kanda [15]. In the single-fiber bridging model, there are several main assumptions: (1) the Poisson effect of fiber is negligible. (2) The elastic deformation of fiber in the slipping stage is negligible compared to the slipping magnitude. (3) The interfacial friction τ is considered to be linearly increasing instead of constant τ0. (4) The longer embedded side of the bridging fiber is always in the debonding stage because it 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 beδ 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 δ0 is calculated by equation (4) for θ = 0 and is given by equation (5) for θ = 0.(4)σbδ=Δσbdδ=4τ0δ+2Gd1+ηEfdf,0≤δ<δ0,Δσbpδ=4τ0df2Le−δ−2δ0df+βδ−2δ0,δ0≤δ≤Lf2,(5)δ0=2τ0Le21+ηEfdf+8GdLe21+ηEfdf,where σbd is the fiber bridging stress in the debonding stage and σbp is the fiber bridging stress at slipping stage (Pa). δ is the single main bridge crack opening, and δc is the full debonding crack opening (m). η=VfEf/VmEm, where Em is the elastic modulus of the matrix (Pa), and Vf and Vm represent the volume fraction of the fibers and the matrix, respectively.Figure 3
The sketch of fiber bridging crack [17]. (a) Single-side pullout model and (b) two-side pullout model.
(a)(b)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 inclined angle θ. This degradation effect can be represented by equation (7) [19, 22, 23].(6)σbθ=Δσbdθ=σbdefθ,Δσbpθ=σbpefθ,(7)σfuθ=σfune−f′θ,where f is the snubbing coefficient, f′ is the reduction of the apparent fiber strength, θ is the inclined angle, σfu is the apparent fiber tensile strength, and σfun is the nominal fiber tensile strength (Pa).


## 3. Relationship between Bridging Stress and Crack Opening in the Composites

### 3.1. Without Fiber Rupture Analysis

The composite bridging stress versus crack opening relation is used to link the properties of the matrix, the fiber, and the fiber/matrix interface. The 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:(8)σcδ=Vf∫0π/2∫0Lf/2cosθσbδ,z,θfzfθdzdθ,where Vf is the volume fraction of the fibers, and fz and fθ 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, fz=2/Lf0≤z≤Lf/2 and fθ=sinθ0≤θ≤π/2. The geometric relations between Le, Lf/2, and z are revealed in Figure 3(a). The embedment length Le is converted using Le=Lf/2−z/cosθ. Through an integral conversion, an extended expression is given by the following equation [8, 11]:(9)σcδ=Vflf∫0π/2∫0Lf/2σbδ,Le,θ=0efθsin2θdLedθ.Ideally, without considering fiber rupture during crack propagation, the general bridging stressσcδ is calculated by equation (10). The 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 2δ0Le. Figure 4 shows the specific state of randomly distributed fibers in the composites for a given crack opening δr. The full debonding stage is only completed for every single fiber in the case of δr=2δ0Le=Lf/2.(10)σcδ=ΔVfLf∫0π/2∫δrldδrσbpδ,Leefθsin2θdLedθ+Vflf∫0π/2∫ldδrLf/2σbdδ,Leefθsin2θdLedθ,0≤δ<2δ0Le=Lf2ΔVfLf∫0π/2∫δrLf/2σbpδ,Leefθsin2θdLedθ,2δ0Le=Lf2≤δ≤Lf2.Figure 4
Fiber states without fiber rupture analysis.One case of the relationship between the bridging stress and the crack opening without fiber rupture analysis is shown in Figure5. The physical properties of the PVA fibers are listed in Table 1 and the volume fraction of the fibers is 2%. The predicted peak bridge stress reaches 51 MPa, which is significantly higher than the actual value. Therefore, the fiber bridging capacity is seriously overestimated. When the crack expands to 0.18 mm, the debonding stage is over. Due 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.Figure 5
σ-δ curve without fiber rupture analysis.Table 1
Physical properties of PVA fibers.
Fiber typeNominal tensile strengthσfun MPaModulusEf GPaLengthLf mmDiameterdfμ mff′PVA fiber1060a22a12a39a0.2a0.33aNote: a is obtained by Yang et al. (2015) and Yang et al. (2008) [8, 17].


### 3.2. With Fiber Rupture Analysis

Fiber rupture will occur once the fiber stress at the crack plane reaches the apparent fiber strength. According to section2.1, debonding rupture and slipping rupture can be used to analyze fiber rupture phenomenon. The boundary between potential debonding rupture and potential slipping rupture can be derived by the potential critical embedment length Ldθ and Lpθ.


#### 3.2.1. Fiber Debonding Rupture

The potential critical debonding embedment lengthLdθ is calculated for σbdθmax=σfuθ, as shown in equation (11). For selected composites, Ldθ is determined by the angle θ. The larger the angle θ, the smaller Ldθ will be. The maximum value Ld0 and the minimum value Ldπ/2 can be obtained from equation (11). If the embedded length Leθ exceeds Ldθ, the fibers will risk debonding rupture.(11)Ldθ=σfundf4τ0e−f+f′θ−2GdEfdf1+η2τ0.For different fiber/matrix interfacial conditions, theLdθ curves have three shapes, as shown in Figure 6. If Ldπ/2>Lf/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 Ldθ curve intersects with δ, the angle θdδ can be calculated from(12)θdδ=Δπ20≤δ≤Ldπ2Δ−1f+f′ln4τ0σfundf2GdEfdf1+η2τ0+δLdπ2<δ≤Lf2.Figure 6
Ld curves with diverse fiber/matrix interface parameters. (a) Ldπ/2≥Lf/2. (b) Ldπ/2<Lf/2andLd0>Lf/2. (c) Ld0≤Lf/2.Figure6 shows the potential debonding rupture zone determined by Ldθ. The fibers located in the potential debonding rupture zone are not simultaneously broken, but they are gradually destroyed during crack propagation. The 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 δr, the corresponding debonding rupture zone can be depicted by ldδr, which is calculated by σbdθ=σfuθ. The debonding rupture length ldδr is given by(13)ldδr=τ0δr+2GdEfdf4τ021+η−GdEfdf2τ021+η.Figure7 shows the developing process of ldδr when the crack expands from 5 μm to 250 μm. The debonding rupture zone is determined by the intersection of ldδr and Ldθ. When ldδr=Ldπ/2, the fibers initiate debonding rupture. Here, the crack opening δcd can be calculated from(14)δcd=dfσfun24Efτ01+ηe−2f+f′θ−2Gdτ0.δcdπ/2 is the minimum debonding rupture crack opening. When 0≤δ≤δcdπ/2, there is no debonding rupture. The growth of the fiber debonding rupture zone is illustrated in Figure 7. When ldδr=Lf/2, the debonding stage is over and the crack opening δdt is given by(15)δdt=Lf2τ01+ηEfdf+8Lf2Gd1+ηEfdf.Figure 7
The growth of the debonding rupture region during crack propagation.When the curve ofldδr intersects with Ld, the inclined angle θcδr can be calculated from equation(16)θcδr=π2,0≤δ≤δcπ2,−12f−f′In4Ef1+η/dfσfun2τ0δ+2Gdδcπ2<δ≤δdt,0,δ>δdt,


#### 3.2.2. 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. The 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. The potential critical embedment lengthLpθ can be calculated for σbpθmax=σfuθ. The peak bridging stress σbpθmax in the slipping stage is calculated from δ=Le/2+2δ0+df/2β.(17)Lpθ=dfσfune−f+f′θβτ0−1β.The boundary value ofLpθ can be obtained by Lp0 and Ldπ/2. If Lpθ≥Ldθ, the fibers with an embedded length Le>Lpθ will be debonding rupture first and will not undergo slipping rupture. Generally, the Lpθ curve is lower than the Ldθ curve for PVA fibers. When the curve of Lpθ intersects with δ, the inclined angle θpδ can be calculated from(18)θpδ=1−f+f′lnβτ0σfunδdf+1β2.The potential slipping rupture fibers are gradually destroyed during crack propagation. For a given crack openingδr, the current slipping rupture zone can be depicted by lpδr, which is calculated for σbpθ=σfuθ. The current slipping rupture length lpδr is given by(19)lpδr−δr+2δ0df+βδr−2δ0=df2σfune−f+f′θ4τ0.Equation (19) can be simplified for lpδr=df2σfune−f+f′θ/4τ0df+βδr−2δ0+δr−2δ0. In Huang’s model [8], the crack opening 2δ0 is omitted, which will lead to a larger current slipping rupture zone for a given crack opening δr. In this study, 2δ0 was considered in lpδr.A case of fiber slipping rupture analysis is shown in Figure8. It presents a 3D shape of lpδr with δ from 50 μm to 300 μm and θ from 0 to π/2. lpδr is not a flat but a coiled surface. The area enclosed by the coiled surface represents the slipping rupture space. The rupture analysis for δr (100, 150, 200, 250, and 500 μm) is shown in Figure 9. The slipping rupture zone forms a parabolic progression. The slipping rupture zone cannot be neglected compared to the debonding rupture zone.Figure 8
The 3D slipping ruptured space.Figure 9
Growth of the slipping rupture space during crack propagation.An interesting case was found for fiber rupture analysis. At the debonding stage, fibers rupture more easily for a longer embedmentLe. However, this may be invalid in the slipping stage. The bridging stress of fibers with a longer embedment may not reach the fiber tensile strength in the slipping stage. In equation (19), (δ−2δ0) represents the crack opening generated by the slipping stage. The longer Le, the larger 2δ0 and the smaller (δ−2δ0) will be. The smaller crack opening (δ−2δ0) in the slipping stage may not be enough to break the fiber. Therefore, a shorter embedment Le may rupture earlier.


### 3.3. The Influences of Interfacial Parameters on Rupture Analysis

According to the aforementioned analysis, the interfacial parameters determine the fiber rupture space. The influence ofτ0, Gd, and β on Ldθ and Lpθ is shown in Figure 10. When τ0 changes from 0.8 to 8 MPa, the Ldθ and Lpθ curves significantly shift down, as shown in Figure 10(a), which means the potential rupture space rapidly increases. Expectedly, the Ld curve is more affected by the change in τ0. In the debonding stage, Ldθ will be decreased because of Gd, as shown in Figure 10(b). Although the effect of Gd was less than τ0 on Ld, 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 Lpθ 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 τ0 and β, respectively. This can be used to tailor fibers/matrix interfaces and treat fiber surfaces.Figure 10
Interfacial parameter impact analysis to fiber rupture: (a) the effect ofτ0, (b) the effect of Gd, (c) the effect of β.
(a)(b)(c)The influence of different interfacial parameters on fiber rupture analysis is illustrated in this study through some specific cases, as shown in Figure11. The potential fiber rupture zone and the current rupture zone were analyzed when the crack opening was δr=100μm. In cases with a weaker slip-hardening effect, the Ldθ curves are close to Ldθ and slipping rupture can almost be ignored since debonding rupture dominates. If the Ldθ curve is above Ldθ, there will be no significant slipping rupture. The interfacial parameters play a decisive role in determining which type of fiber rupture dominates.Figure 11
Fiber rupture analysis with different interfacial parameters whenδ=100μm.


### 3.4.σδ Relationship When considering Fiber Rupture

Through a comprehensive consideration of debonding rupture and slipping rupture, the effective fiber bridging stressσeffectiveδ is calculated from equation (20), where σRDδ and σRSδ represent the fiber debonding rupture stress and slipping rupture stress, respectively.(20)σeffectiveδ=σcδ−σRDδ−σRSδ.During crack propagation,σRDδ is calculated by integrating the zone for which the fiber debonding rupture occurs.(21)σRDδ=VfLf∫θcδrπ/2∫ldδrLf/2σbdδ,Leefθsin2θdθdLe+VfLf∫θcδrπ/2∫Ldldδrσbpδ,Leefθsin2θdθdLe,0≤δr<2δ0,VfLf∫θcδrπ/2∫LdLf/2σbpδ,Leefθsin2θdθdLe,2δ0≤δr≤Ldπ2,VfLf∫θdδπ/2δ∫δLf/2σbpδ,Leefθsin2θdθdLe+VfLf∫θcδrθdδ∫LdLf/2σbpδ,Leefθsin2θdθdLe,Ldπ2<δ≤Lf2.Due to the geometrical irregularity of the current slipping rupture zone, it is very complicated to calculateσRSδ. Figure 12 shows the growth process of the slipping rupture zone S1. The current slipping rupture zone S1 was represented by a parabolic enclosure subtract zone S2. σRSδ is calculated by integrating the current slipping rupture zone.Figure 12
Developing of the slipping rupture zoneS1. (a) 0<θv<π/2. (b) θv<0 and θs>0. (c) θv<0 and θs<0.
(a)(b)(c)lp1δr and lp2δr represent the upper part and lower part of the lpδr curve, respectively. As shown in Figure 12, lp1δr may be greater than Lf/2 during crack propagation. At this point, the value of lp1δr should be set to Lf/2. When δr=Lf/16, lp2δr is in close proximity of Lpθ, lp2δr is considered to be equal to Lpθ and σRSδ can be calculated by (5).(22)σRSδ=VfLf∫θvπ/2∫lp2δrlp1δrσbpδ,Leefθsin2θdθdLe−VfLf∫θsπ/2∫Ldlp1δrσbpδ,Leefθsin2θdθdLe,0≤δr<Lf16,VfLf∫θcδrπ/2∫LpLdσbpδ,Leefθsin2θdθdLe,Lf16≤δr≤Lpπ2,VfLf∫θpδπ/2∫δLdσbpδ,Leefθsin2θdθdLe+VfLf∫0θpδ∫LpLdσbpδ,Leefθsin2θdθdLe,Lpπ2<δ≤Lp0,VfLf∫0π/2∫δLdσbpδ,Leefθsin2θdθdLe,Lp0<δ≤Ldπ2,VfLf∫0θdδ∫δLdσbpδ,Leefθsin2θdθdLe,Ldπ2<δ≤Lf2,where θv is the inclined angle when lp1δr intersects with lp2δr and θs is the inclined angle when lpδr intersects with Ldθ.The process to determineσeffectiveδ 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. The influence of fiber debonding rupture and slipping rupture on the bridging stress is clearly shown in Figure 14. The current fiber bridging stress, debonding rupture stress, and slipping rupture stress are perfectly determined during crack propagation. The 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. The predicted critical embedment length and σδ relation will be compared with experimental result in the following section to verify further the accuracy of the current bridging model.Figure 13
A flowchart for the current bridging model.Figure 14
Fiber bridging stress versus crack opening relation.


## 3.1. Without Fiber Rupture Analysis

The composite bridging stress versus crack opening relation is used to link the properties of the matrix, the fiber, and the fiber/matrix interface. The 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:(8)σcδ=Vf∫0π/2∫0Lf/2cosθσbδ,z,θfzfθdzdθ,where Vf is the volume fraction of the fibers, and fz and fθ 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, fz=2/Lf0≤z≤Lf/2 and fθ=sinθ0≤θ≤π/2. The geometric relations between Le, Lf/2, and z are revealed in Figure 3(a). The embedment length Le is converted using Le=Lf/2−z/cosθ. Through an integral conversion, an extended expression is given by the following equation [8, 11]:(9)σcδ=Vflf∫0π/2∫0Lf/2σbδ,Le,θ=0efθsin2θdLedθ.Ideally, without considering fiber rupture during crack propagation, the general bridging stressσcδ is calculated by equation (10). The 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 2δ0Le. Figure 4 shows the specific state of randomly distributed fibers in the composites for a given crack opening δr. The full debonding stage is only completed for every single fiber in the case of δr=2δ0Le=Lf/2.(10)σcδ=ΔVfLf∫0π/2∫δrldδrσbpδ,Leefθsin2θdLedθ+Vflf∫0π/2∫ldδrLf/2σbdδ,Leefθsin2θdLedθ,0≤δ<2δ0Le=Lf2ΔVfLf∫0π/2∫δrLf/2σbpδ,Leefθsin2θdLedθ,2δ0Le=Lf2≤δ≤Lf2.Figure 4
Fiber states without fiber rupture analysis.One case of the relationship between the bridging stress and the crack opening without fiber rupture analysis is shown in Figure5. The physical properties of the PVA fibers are listed in Table 1 and the volume fraction of the fibers is 2%. The predicted peak bridge stress reaches 51 MPa, which is significantly higher than the actual value. Therefore, the fiber bridging capacity is seriously overestimated. When the crack expands to 0.18 mm, the debonding stage is over. Due 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.Figure 5
σ-δ curve without fiber rupture analysis.Table 1
Physical properties of PVA fibers.
Fiber typeNominal tensile strengthσfun MPaModulusEf GPaLengthLf mmDiameterdfμ mff′PVA fiber1060a22a12a39a0.2a0.33aNote: a is obtained by Yang et al. (2015) and Yang et al. (2008) [8, 17].


## 3.2. With Fiber Rupture Analysis

Fiber rupture will occur once the fiber stress at the crack plane reaches the apparent fiber strength. According to section2.1, debonding rupture and slipping rupture can be used to analyze fiber rupture phenomenon. The boundary between potential debonding rupture and potential slipping rupture can be derived by the potential critical embedment length Ldθ and Lpθ.


### 3.2.1. Fiber Debonding Rupture

The potential critical debonding embedment lengthLdθ is calculated for σbdθmax=σfuθ, as shown in equation (11). For selected composites, Ldθ is determined by the angle θ. The larger the angle θ, the smaller Ldθ will be. The maximum value Ld0 and the minimum value Ldπ/2 can be obtained from equation (11). If the embedded length Leθ exceeds Ldθ, the fibers will risk debonding rupture.(11)Ldθ=σfundf4τ0e−f+f′θ−2GdEfdf1+η2τ0.For different fiber/matrix interfacial conditions, theLdθ curves have three shapes, as shown in Figure 6. If Ldπ/2>Lf/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 Ldθ curve intersects with δ, the angle θdδ can be calculated from(12)θdδ=Δπ20≤δ≤Ldπ2Δ−1f+f′ln4τ0σfundf2GdEfdf1+η2τ0+δLdπ2<δ≤Lf2.Figure 6
Ld curves with diverse fiber/matrix interface parameters. (a) Ldπ/2≥Lf/2. (b) Ldπ/2<Lf/2andLd0>Lf/2. (c) Ld0≤Lf/2.Figure6 shows the potential debonding rupture zone determined by Ldθ. The fibers located in the potential debonding rupture zone are not simultaneously broken, but they are gradually destroyed during crack propagation. The 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 δr, the corresponding debonding rupture zone can be depicted by ldδr, which is calculated by σbdθ=σfuθ. The debonding rupture length ldδr is given by(13)ldδr=τ0δr+2GdEfdf4τ021+η−GdEfdf2τ021+η.Figure7 shows the developing process of ldδr when the crack expands from 5 μm to 250 μm. The debonding rupture zone is determined by the intersection of ldδr and Ldθ. When ldδr=Ldπ/2, the fibers initiate debonding rupture. Here, the crack opening δcd can be calculated from(14)δcd=dfσfun24Efτ01+ηe−2f+f′θ−2Gdτ0.δcdπ/2 is the minimum debonding rupture crack opening. When 0≤δ≤δcdπ/2, there is no debonding rupture. The growth of the fiber debonding rupture zone is illustrated in Figure 7. When ldδr=Lf/2, the debonding stage is over and the crack opening δdt is given by(15)δdt=Lf2τ01+ηEfdf+8Lf2Gd1+ηEfdf.Figure 7
The growth of the debonding rupture region during crack propagation.When the curve ofldδr intersects with Ld, the inclined angle θcδr can be calculated from equation(16)θcδr=π2,0≤δ≤δcπ2,−12f−f′In4Ef1+η/dfσfun2τ0δ+2Gdδcπ2<δ≤δdt,0,δ>δdt,


### 3.2.2. 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. The 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. The potential critical embedment lengthLpθ can be calculated for σbpθmax=σfuθ. The peak bridging stress σbpθmax in the slipping stage is calculated from δ=Le/2+2δ0+df/2β.(17)Lpθ=dfσfune−f+f′θβτ0−1β.The boundary value ofLpθ can be obtained by Lp0 and Ldπ/2. If Lpθ≥Ldθ, the fibers with an embedded length Le>Lpθ will be debonding rupture first and will not undergo slipping rupture. Generally, the Lpθ curve is lower than the Ldθ curve for PVA fibers. When the curve of Lpθ intersects with δ, the inclined angle θpδ can be calculated from(18)θpδ=1−f+f′lnβτ0σfunδdf+1β2.The potential slipping rupture fibers are gradually destroyed during crack propagation. For a given crack openingδr, the current slipping rupture zone can be depicted by lpδr, which is calculated for σbpθ=σfuθ. The current slipping rupture length lpδr is given by(19)lpδr−δr+2δ0df+βδr−2δ0=df2σfune−f+f′θ4τ0.Equation (19) can be simplified for lpδr=df2σfune−f+f′θ/4τ0df+βδr−2δ0+δr−2δ0. In Huang’s model [8], the crack opening 2δ0 is omitted, which will lead to a larger current slipping rupture zone for a given crack opening δr. In this study, 2δ0 was considered in lpδr.A case of fiber slipping rupture analysis is shown in Figure8. It presents a 3D shape of lpδr with δ from 50 μm to 300 μm and θ from 0 to π/2. lpδr is not a flat but a coiled surface. The area enclosed by the coiled surface represents the slipping rupture space. The rupture analysis for δr (100, 150, 200, 250, and 500 μm) is shown in Figure 9. The slipping rupture zone forms a parabolic progression. The slipping rupture zone cannot be neglected compared to the debonding rupture zone.Figure 8
The 3D slipping ruptured space.Figure 9
Growth of the slipping rupture space during crack propagation.An interesting case was found for fiber rupture analysis. At the debonding stage, fibers rupture more easily for a longer embedmentLe. However, this may be invalid in the slipping stage. The bridging stress of fibers with a longer embedment may not reach the fiber tensile strength in the slipping stage. In equation (19), (δ−2δ0) represents the crack opening generated by the slipping stage. The longer Le, the larger 2δ0 and the smaller (δ−2δ0) will be. The smaller crack opening (δ−2δ0) in the slipping stage may not be enough to break the fiber. Therefore, a shorter embedment Le may rupture earlier.


## 3.2.1. Fiber Debonding Rupture

The potential critical debonding embedment lengthLdθ is calculated for σbdθmax=σfuθ, as shown in equation (11). For selected composites, Ldθ is determined by the angle θ. The larger the angle θ, the smaller Ldθ will be. The maximum value Ld0 and the minimum value Ldπ/2 can be obtained from equation (11). If the embedded length Leθ exceeds Ldθ, the fibers will risk debonding rupture.(11)Ldθ=σfundf4τ0e−f+f′θ−2GdEfdf1+η2τ0.For different fiber/matrix interfacial conditions, theLdθ curves have three shapes, as shown in Figure 6. If Ldπ/2>Lf/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 Ldθ curve intersects with δ, the angle θdδ can be calculated from(12)θdδ=Δπ20≤δ≤Ldπ2Δ−1f+f′ln4τ0σfundf2GdEfdf1+η2τ0+δLdπ2<δ≤Lf2.Figure 6
Ld curves with diverse fiber/matrix interface parameters. (a) Ldπ/2≥Lf/2. (b) Ldπ/2<Lf/2andLd0>Lf/2. (c) Ld0≤Lf/2.Figure6 shows the potential debonding rupture zone determined by Ldθ. The fibers located in the potential debonding rupture zone are not simultaneously broken, but they are gradually destroyed during crack propagation. The 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 δr, the corresponding debonding rupture zone can be depicted by ldδr, which is calculated by σbdθ=σfuθ. The debonding rupture length ldδr is given by(13)ldδr=τ0δr+2GdEfdf4τ021+η−GdEfdf2τ021+η.Figure7 shows the developing process of ldδr when the crack expands from 5 μm to 250 μm. The debonding rupture zone is determined by the intersection of ldδr and Ldθ. When ldδr=Ldπ/2, the fibers initiate debonding rupture. Here, the crack opening δcd can be calculated from(14)δcd=dfσfun24Efτ01+ηe−2f+f′θ−2Gdτ0.δcdπ/2 is the minimum debonding rupture crack opening. When 0≤δ≤δcdπ/2, there is no debonding rupture. The growth of the fiber debonding rupture zone is illustrated in Figure 7. When ldδr=Lf/2, the debonding stage is over and the crack opening δdt is given by(15)δdt=Lf2τ01+ηEfdf+8Lf2Gd1+ηEfdf.Figure 7
The growth of the debonding rupture region during crack propagation.When the curve ofldδr intersects with Ld, the inclined angle θcδr can be calculated from equation(16)θcδr=π2,0≤δ≤δcπ2,−12f−f′In4Ef1+η/dfσfun2τ0δ+2Gdδcπ2<δ≤δdt,0,δ>δdt,


## 3.2.2. 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. The 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. The potential critical embedment lengthLpθ can be calculated for σbpθmax=σfuθ. The peak bridging stress σbpθmax in the slipping stage is calculated from δ=Le/2+2δ0+df/2β.(17)Lpθ=dfσfune−f+f′θβτ0−1β.The boundary value ofLpθ can be obtained by Lp0 and Ldπ/2. If Lpθ≥Ldθ, the fibers with an embedded length Le>Lpθ will be debonding rupture first and will not undergo slipping rupture. Generally, the Lpθ curve is lower than the Ldθ curve for PVA fibers. When the curve of Lpθ intersects with δ, the inclined angle θpδ can be calculated from(18)θpδ=1−f+f′lnβτ0σfunδdf+1β2.The potential slipping rupture fibers are gradually destroyed during crack propagation. For a given crack openingδr, the current slipping rupture zone can be depicted by lpδr, which is calculated for σbpθ=σfuθ. The current slipping rupture length lpδr is given by(19)lpδr−δr+2δ0df+βδr−2δ0=df2σfune−f+f′θ4τ0.Equation (19) can be simplified for lpδr=df2σfune−f+f′θ/4τ0df+βδr−2δ0+δr−2δ0. In Huang’s model [8], the crack opening 2δ0 is omitted, which will lead to a larger current slipping rupture zone for a given crack opening δr. In this study, 2δ0 was considered in lpδr.A case of fiber slipping rupture analysis is shown in Figure8. It presents a 3D shape of lpδr with δ from 50 μm to 300 μm and θ from 0 to π/2. lpδr is not a flat but a coiled surface. The area enclosed by the coiled surface represents the slipping rupture space. The rupture analysis for δr (100, 150, 200, 250, and 500 μm) is shown in Figure 9. The slipping rupture zone forms a parabolic progression. The slipping rupture zone cannot be neglected compared to the debonding rupture zone.Figure 8
The 3D slipping ruptured space.Figure 9
Growth of the slipping rupture space during crack propagation.An interesting case was found for fiber rupture analysis. At the debonding stage, fibers rupture more easily for a longer embedmentLe. However, this may be invalid in the slipping stage. The bridging stress of fibers with a longer embedment may not reach the fiber tensile strength in the slipping stage. In equation (19), (δ−2δ0) represents the crack opening generated by the slipping stage. The longer Le, the larger 2δ0 and the smaller (δ−2δ0) will be. The smaller crack opening (δ−2δ0) in the slipping stage may not be enough to break the fiber. Therefore, a shorter embedment Le may rupture earlier.


## 3.3. The Influences of Interfacial Parameters on Rupture Analysis

According to the aforementioned analysis, the interfacial parameters determine the fiber rupture space. The influence ofτ0, Gd, and β on Ldθ and Lpθ is shown in Figure 10. When τ0 changes from 0.8 to 8 MPa, the Ldθ and Lpθ curves significantly shift down, as shown in Figure 10(a), which means the potential rupture space rapidly increases. Expectedly, the Ld curve is more affected by the change in τ0. In the debonding stage, Ldθ will be decreased because of Gd, as shown in Figure 10(b). Although the effect of Gd was less than τ0 on Ld, 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 Lpθ 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 τ0 and β, respectively. This can be used to tailor fibers/matrix interfaces and treat fiber surfaces.Figure 10
Interfacial parameter impact analysis to fiber rupture: (a) the effect ofτ0, (b) the effect of Gd, (c) the effect of β.
(a)(b)(c)The influence of different interfacial parameters on fiber rupture analysis is illustrated in this study through some specific cases, as shown in Figure11. The potential fiber rupture zone and the current rupture zone were analyzed when the crack opening was δr=100μm. In cases with a weaker slip-hardening effect, the Ldθ curves are close to Ldθ and slipping rupture can almost be ignored since debonding rupture dominates. If the Ldθ curve is above Ldθ, there will be no significant slipping rupture. The interfacial parameters play a decisive role in determining which type of fiber rupture dominates.Figure 11
Fiber rupture analysis with different interfacial parameters whenδ=100μm.


## 3.4.σδ Relationship When considering Fiber Rupture

Through a comprehensive consideration of debonding rupture and slipping rupture, the effective fiber bridging stressσeffectiveδ is calculated from equation (20), where σRDδ and σRSδ represent the fiber debonding rupture stress and slipping rupture stress, respectively.(20)σeffectiveδ=σcδ−σRDδ−σRSδ.During crack propagation,σRDδ is calculated by integrating the zone for which the fiber debonding rupture occurs.(21)σRDδ=VfLf∫θcδrπ/2∫ldδrLf/2σbdδ,Leefθsin2θdθdLe+VfLf∫θcδrπ/2∫Ldldδrσbpδ,Leefθsin2θdθdLe,0≤δr<2δ0,VfLf∫θcδrπ/2∫LdLf/2σbpδ,Leefθsin2θdθdLe,2δ0≤δr≤Ldπ2,VfLf∫θdδπ/2δ∫δLf/2σbpδ,Leefθsin2θdθdLe+VfLf∫θcδrθdδ∫LdLf/2σbpδ,Leefθsin2θdθdLe,Ldπ2<δ≤Lf2.Due to the geometrical irregularity of the current slipping rupture zone, it is very complicated to calculateσRSδ. Figure 12 shows the growth process of the slipping rupture zone S1. The current slipping rupture zone S1 was represented by a parabolic enclosure subtract zone S2. σRSδ is calculated by integrating the current slipping rupture zone.Figure 12
Developing of the slipping rupture zoneS1. (a) 0<θv<π/2. (b) θv<0 and θs>0. (c) θv<0 and θs<0.
(a)(b)(c)lp1δr and lp2δr represent the upper part and lower part of the lpδr curve, respectively. As shown in Figure 12, lp1δr may be greater than Lf/2 during crack propagation. At this point, the value of lp1δr should be set to Lf/2. When δr=Lf/16, lp2δr is in close proximity of Lpθ, lp2δr is considered to be equal to Lpθ and σRSδ can be calculated by (5).(22)σRSδ=VfLf∫θvπ/2∫lp2δrlp1δrσbpδ,Leefθsin2θdθdLe−VfLf∫θsπ/2∫Ldlp1δrσbpδ,Leefθsin2θdθdLe,0≤δr<Lf16,VfLf∫θcδrπ/2∫LpLdσbpδ,Leefθsin2θdθdLe,Lf16≤δr≤Lpπ2,VfLf∫θpδπ/2∫δLdσbpδ,Leefθsin2θdθdLe+VfLf∫0θpδ∫LpLdσbpδ,Leefθsin2θdθdLe,Lpπ2<δ≤Lp0,VfLf∫0π/2∫δLdσbpδ,Leefθsin2θdθdLe,Lp0<δ≤Ldπ2,VfLf∫0θdδ∫δLdσbpδ,Leefθsin2θdθdLe,Ldπ2<δ≤Lf2,where θv is the inclined angle when lp1δr intersects with lp2δr and θs is the inclined angle when lpδr intersects with Ldθ.The process to determineσeffectiveδ 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. The influence of fiber debonding rupture and slipping rupture on the bridging stress is clearly shown in Figure 14. The current fiber bridging stress, debonding rupture stress, and slipping rupture stress are perfectly determined during crack propagation. The 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. The predicted critical embedment length and σδ relation will be compared with experimental result in the following section to verify further the accuracy of the current bridging model.Figure 13
A flowchart for the current bridging model.Figure 14
Fiber bridging stress versus crack opening relation.


## 4. Model Verification

### 4.1. Verification of Fiber Rupture Analysis

HDCC matrix composition consists of Type II Portland cement, fly ash, fine sand, and water in a proportion of 0.4 : 0.6 : 0.3 : 0.3. Type II Portland cement conform to the Chinese standard GB175-2007 and Class F fly ash conform to the ASTM C618 standard were used. The 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. The 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. Then, water was added and mixed for 5 min at a speed of 280 rpm.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 Table2. The 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. The fibers were vertically embedded in the matrix, and the embedment length Le was between 1 mm and 5 mm to determine the critical embedment lengths Ldθ and Lpθ. Single-fiber bridging stress versus pullout displacement curves are shown in Figure 15(b)–15(f). The single-fiber embedment length Le was set to 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm. Here, the interfacial bonding performance was recorded as L1, L2, L3, L4, and L5, respectively.Table 2
Physical properties of the PVA fibers used.
Fiber typeNominal tensile strengthσfun MPaModulusEf GPaLengthLf mmDiameterdfμ mff′PVA fiber12603012390.20.33Figure 15
Single-fiber bridging strength and pullout displacement curves for different values ofLe. (a) Fiber tensile strength. (b) Single-fiber pullout with Le  = 1 mm. (c) Single-fiber pullout with Le  = 2 mm . (d) Single-fiber pullout with Le  = 3 mm. (e) Single-fiber pullout with Le  = 4 mm. (f) Single-fiber pullout with Le  = 5 mm.
(a)(b)(c)(d)(e)(f)The fiber rupture phenomenon can be seen in Figure15 from all pullout curves with Le from 1 to 5 mm. The rupture strength of PVA fibers only reached 580 to 760 MPa, which was far less than the measured tensile strength. The 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. The fiber pullout zone was seriously abraded. The 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.Figure 16
SEM images of fiber rupture zone. (a) Direct tensile rupture manner. (b) Tensile rupture from matrix manner.
(a)(b)For fibers with anLe of 5 mm, the interface parameters could not be exactly obtained because of the fiber ruptured before complete debonding. The interface parameters (Gd, τ0, β) of PVA fiber/matrix were calculated using equations (1)–(4), as shown in Figure 17. The values of τ0 and β slowly decreased when Le increased from 1 mm to 4 mm. The average values of Gd, τ0, and β are given in Table 3.Figure 17
τ0 and β for different embedment lengths.Table 3
Fiber/matrix interface parameters for PVA fiber.
Seriesτ0 (MPa)Gd (J/m2)βL12.1 ± 0.120.8 ± 0.320.21 ± 0.01L21.2 ± 0.040.5 ± 0.100.17 ± 0.01L31.2 ± 0.040.2 ± 0.080.15 ± 0.02L41.3 ± 0.080.3 ± 0.110.17 ± 0.01Figure18 reveals the critical embedment length Ldθ and Lpθ for different embedment lengths. Due to data fluctuation with the interfacial parameters, Ldθ 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. The experimental results mostly followed the debonding rupture analysis. The 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, Lpθ changed from 1.2 to 1.9 mm when θ was 0, which was slightly higher than the experimental result. Figure 15(b) shows that fibers with an embedment length of 1 mm already underwent slipping rupture. The fibers actually undergo slipping rupture more easily, which also revealed how essential it is to include slipping rupture in the analysis to translate the real situation.Figure 18
Critical embedment length analysis.


### 4.2. Verification of theσ-δ Relationship

Figure19 shows the relationship between the bridging stress and crack opening for the experimental results and the predicted model. The 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. The peak bridging stress was in the 1.5–1.2 MPa range corresponding to a peak crack opening in the 97–118 μm range. The predicted results were in the region of the experimental data. The predicted peak bridging stress was 1.4–1.2 MPa corresponding to a peak crack opening in the 93–125 μm range. The σ-δ relationship predicted by the bridging model is consistent with the experimental results.Figure 19
Comparison of predicted and experimentalσ−δ curve.


## 4.1. Verification of Fiber Rupture Analysis

HDCC matrix composition consists of Type II Portland cement, fly ash, fine sand, and water in a proportion of 0.4 : 0.6 : 0.3 : 0.3. Type II Portland cement conform to the Chinese standard GB175-2007 and Class F fly ash conform to the ASTM C618 standard were used. The 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. The 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. Then, water was added and mixed for 5 min at a speed of 280 rpm.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 Table2. The 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. The fibers were vertically embedded in the matrix, and the embedment length Le was between 1 mm and 5 mm to determine the critical embedment lengths Ldθ and Lpθ. Single-fiber bridging stress versus pullout displacement curves are shown in Figure 15(b)–15(f). The single-fiber embedment length Le was set to 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm. Here, the interfacial bonding performance was recorded as L1, L2, L3, L4, and L5, respectively.Table 2
Physical properties of the PVA fibers used.
Fiber typeNominal tensile strengthσfun MPaModulusEf GPaLengthLf mmDiameterdfμ mff′PVA fiber12603012390.20.33Figure 15
Single-fiber bridging strength and pullout displacement curves for different values ofLe. (a) Fiber tensile strength. (b) Single-fiber pullout with Le  = 1 mm. (c) Single-fiber pullout with Le  = 2 mm . (d) Single-fiber pullout with Le  = 3 mm. (e) Single-fiber pullout with Le  = 4 mm. (f) Single-fiber pullout with Le  = 5 mm.
(a)(b)(c)(d)(e)(f)The fiber rupture phenomenon can be seen in Figure15 from all pullout curves with Le from 1 to 5 mm. The rupture strength of PVA fibers only reached 580 to 760 MPa, which was far less than the measured tensile strength. The 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. The fiber pullout zone was seriously abraded. The 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.Figure 16
SEM images of fiber rupture zone. (a) Direct tensile rupture manner. (b) Tensile rupture from matrix manner.
(a)(b)For fibers with anLe of 5 mm, the interface parameters could not be exactly obtained because of the fiber ruptured before complete debonding. The interface parameters (Gd, τ0, β) of PVA fiber/matrix were calculated using equations (1)–(4), as shown in Figure 17. The values of τ0 and β slowly decreased when Le increased from 1 mm to 4 mm. The average values of Gd, τ0, and β are given in Table 3.Figure 17
τ0 and β for different embedment lengths.Table 3
Fiber/matrix interface parameters for PVA fiber.
Seriesτ0 (MPa)Gd (J/m2)βL12.1 ± 0.120.8 ± 0.320.21 ± 0.01L21.2 ± 0.040.5 ± 0.100.17 ± 0.01L31.2 ± 0.040.2 ± 0.080.15 ± 0.02L41.3 ± 0.080.3 ± 0.110.17 ± 0.01Figure18 reveals the critical embedment length Ldθ and Lpθ for different embedment lengths. Due to data fluctuation with the interfacial parameters, Ldθ 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. The experimental results mostly followed the debonding rupture analysis. The 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, Lpθ changed from 1.2 to 1.9 mm when θ was 0, which was slightly higher than the experimental result. Figure 15(b) shows that fibers with an embedment length of 1 mm already underwent slipping rupture. The fibers actually undergo slipping rupture more easily, which also revealed how essential it is to include slipping rupture in the analysis to translate the real situation.Figure 18
Critical embedment length analysis.


## 4.2. Verification of theσ-δ Relationship

Figure19 shows the relationship between the bridging stress and crack opening for the experimental results and the predicted model. The 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. The peak bridging stress was in the 1.5–1.2 MPa range corresponding to a peak crack opening in the 97–118 μm range. The predicted results were in the region of the experimental data. The predicted peak bridging stress was 1.4–1.2 MPa corresponding to a peak crack opening in the 93–125 μm range. The σ-δ relationship predicted by the bridging model is consistent with the experimental results.Figure 19
Comparison of predicted and experimentalσ−δ curve.


## 5. 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. The 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 stressτ0, the chemical bonding force Gd, 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. The modified bridging model synthetically takes into account fiber debonding rupture and slipping rupture. This 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. The abrasion effect was considered in the fiber rupture analysis.Finally, single-fiber pullout tests with different embedded lengths were carried out to validate the modified bridging model. The relationship between the bridging stress and crack opening predicted by the bridging model was consistent with the experimental results.



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*Source: 1461318-2022-05-24.xml*