Design of an explicit crack bridging constitutive model for engineered cementitious composites using polyvinyl alcohol (PVA) or polyethylene (PE) fiber

1. INTRODUCTION

Fibers with characters of high elastic modulus, high elongation rate, and high ultimate tensile strength are usually used to overcome the crack sensibility and intrinsic brittleness of concrete (11. Lu C, Li VC, Leung CKY. 2018. Flaw characterization and correlation with cracking strength in Engineered Cementitious Composites (ECC). Cem. Concr. Res. 107:64-74. https://doi.org/10.1016/j.cemconres.2018.02.024.
, 22. Liao Q, Su YR, Yu JT, Yu KQ. 2022. Torsional behavior of BFRP bars reinforced engineered cementitious composites beams without stirrup. Eng. Struct. 268:114748. https://doi.org/10.1016/j.engstruct.2022.114748.
). ECC composited by PE or PVA, cement, and fine aggregates exhibit an applausive tensile pseudo strain hardening property, meanwhile, multiple tight cracking with a mean crack width under 100 μm was usually accompanied in the tensile plane (3-63. Li VC, Mishra DK, Wu HC. 1995. Matrix design for pseudo-strain-hardening fibre reinforced cementitious composites. Mater. Struct. 28(184):586-595. https://doi.org/10.1007/BF02473191.
4. Li VC, Wang SX, Wu C. 2001. Tensile strain-hardening behaviorof polyvinyl alcohol engineered cementitious composite (PVA-ECC). ACI Mater. J. 98(6):483-492. https://doi.org/10.1089/apc.2006.20.829.
5. Li M, Li VC. 2011. High-early-strength engineered cementitious composites for fast, durable concrete repair-material properties. ACI Mater. J. 108(1): 3-12. https://doi.org/10.14359/51664210.
6. Liao Q, Yu JT, Xie X, Ye J, Jiang F. 2022. Experimental study of reinforced UHDC-UHPC panels under close-in blast loading. J. Build. Eng. 46:103498. https://doi.org/10.1016/j.jobe.2021.103498.
). In addition, the fiber-reinforced concrete containing polyolefin and steel fibers also could exhibit great toughness and crack resistance if the fibers achieve a suitable distribution (7-97. Dupont D, Vandewalle L. 2005. Distribution of steel fibres in rectangular sections. Cem. Concr. Compos. 27(3):391-398. https://doi.org/10.1016/j.cemconcomp.2004.03.005.
8. Barr BIG, Lee MK, Hansen EJDP, Dupont D, Erdem E, Schaerlaekens S, Schnütgen B, Stang H, Vandewalle L. 2003. Round-robin analysis of the RILEM TC 162-TDF beam-bending test: Part 3-Fibre distribution. Mater. Struct. 36(9):631-635. https://doi.org/10.1007/BF02483283.
9. Barr BIG, Lee MK, Hansen EJDP, Dupont D, Erdem E, Schaerlaekens S, Schnütgen B, Stang H, Vandewalle L. 2003. Round-robin analysis of the RILEM TC 162-TDF uni-axial tensile test: Part 2. Mater. Struct. 36(4):275-280. https://doi.org/10.1007/BF02479621.
), while it is difficult to enhance the ductility obtained by the strain hardening and multiple cracks expanding action compared to PE-ECC and PVA-ECC. Hence, the superior tensile properties of ECC contributed to its wide application prospects in infrastructural construction.

On basis of fracture mechanics and micromechanical principles, ECC was designed by linking from single fiber to single crack scale to achieve the tensile pseudo strain hardening properties (1010. Li VC, Wu HC. 1992. Conditions for pseudo strain-hardening infiber reinforced brittle matrix composites. J. Appl. Mech. Rev. 45(8):390–398. https://doi.org/10.1115/1.3119767.
, 1111. Zhu B, Pan JL, Zhang M, Leung CKY. 2022. Predicting the strain-hardening behaviour of polyethylene fibre reinforced engineered cementitious composites accounting for fibre-matrix interaction. Cem. Concr. Compos. 134:104770. https://doi.org/10.1016/j.cemconcomp.2022.104770.
). The crack bridging model of composites expresses the relation of fiber bridging stress versus crack opening distance, which was of primary importance to implement the multiple crack expanding and tensile pseudo strain hardening characters by energy-based criterion and strength-based criterion (1212. Kanda T, Lin Z, Li VC. 2000. Tensile stress-strain modeling of pseudostrain hardening cementitious composites. J. Mater. Civ. Eng. 12(2):147–156. https://doi.org/10.1061/(ASCE)0899-1561(2000)12:2(147).
). Hence, tailoring raw microstructure including fiber, matrix, and fiber/matrix by fiber bridging law can capture a superior tensile property of ECC (1313. Kanda T, Li VC. 2006. Practical design criteria for saturated pseudo strain hardening behavior in ECC. J. Adv. Concr. Technol. 4(1):59-72. https://doi.org/10.3151/jact.4.59.
, 1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
). Moreover, modeling the micromechanical fiber bridging stress vs crack opening width relation also could evolve the macroscopic tensile properties of ECC (1515. Curosu I, Muja E, Ismailov M, Ahmed AH, Liebscher M, Mechtcherine V. 2021. An experimental-analytical scale-linking study on the crack-bridging mechanisms in different types of SHCC in dependence on fiber orientation. Cem. Concr. Res. 152:106650. https://doi.org/10.1016/j.cemconres.2021.106650.
, 1616. Yao J, Leung CKY. 2020. Scaling up modeling of strain-hardening cementitious composites based on beam theory: From single fiber to composite. Cem. Concr. Compos. 108:103534. https://doi.org/10.1016/j.cemconcomp.2020.103534.
).

To date, continuous efforts have been made to establish the crack bridging relation based on the fiber failure models and interface properties. Li et al. (1717. Li VC, Leung CKY. 1992. Steady-state and multiple cracking of short random fiber composites. J. Eng. Mech. 118(11):2246-2264. https://doi.org/10.1061/(ASCE)0733-9399(1992)118:11(2246).
) first established the fiber pullout model (FPM) which considered the interfacial frictional stress as a constant during the fiber pullout process. Thereafter, Lin et al. (1818. Zhong L, Li VC. 1997. Crack bridging in fiber reinforced cementitious composites with slip-hardening interfaces. J. Mech. Phys. Solids. 45(5):763-787. https://doi.org/10.1016/S0022-5096(96)00095-6.
) introduce the interfacial slip-hardening coefficient to advance the precision of FPM where interfacial frictional strength is enhanced by the increasing fiber pullout distance. However, fiber fracture was neglected in above two models when fiber bridging stress is greater than the fiber apparent tensile strength. Based on FPM, Maalej et al. (1919. Maalej M, Li VC, Hashida T. 1995. Effect of fiber rupture on tensile properties of short fiber composites. J. Eng. Mech. 121(8):903-913. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:8(903).
) first considered fiber rupture in debonding process and proposed the fiber pullout rupture model (FPRM). Afterward, Kanda et al. (2020. Kanda T, Li VC. 1999. Effect of fiber strength and fiber-matrix interface on crack bridging in cement composites. J. Eng. Mech. 125(3):290–299. https://doi.org/10.1061/(ASCE)0733-9399(1999)125:3(290).
) included a chemical bond stress to advance FPRM where the chemical bond stress and interfacial frictional stress control the fiber-matrix interface properties. Thereafter, Lin et al. (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
) further introduced the interfacial slip-hardening coefficient to the above advanced FPRM. Noticeably, fiber bridging stress during pullout stage could exceed the fiber tensile strength due to the slip-hardening character of single fiber (2222. Li VC, Wu C, Wang S, Ogawa A, Saito T. 2002. Interfacetailoring for strain-hardening polyvinyl alcohol-engineered cementi tious composite (PVA-ECC). ACI Mater. J. 99(5):463–472. Retrieved from https://acemrl.engin.umich.edu/wp-content/uploads/sites/412/2018/10/Interface-Tailoring-for-Strain-hardening-PVA-ECC.pdf.
). However, the situation of fiber rupture in slipping stage was no considered in FPRM and two advanced FPRMs.

In recent years, fiber rupture in slipping stage was gradually introduced to advance the fiber bridging model. Based on the advanced FPRM (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
),Yang et al. (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
) considered fiber rupture in the debonding and slipping process, matrix micro-spalling, and Cook-Gordon effect to improve the precision accuracy of the fiber bridging model. However, no explicit expression was presented in the model. Yu et al. (2424. Yu J, Chen YX, Leung CKY. 2018. Micromechanical modeling of crack-bridging relations of hybrid-fiber strain-hardening cementitious composites considering interaction between different fibers. Constr. Build. Mater. 182:629-636. https://doi.org/10.1016/j.conbuildmat.2018.06.115.
) created the implicit micromechanical modeling of crack-bridging relations of hybrid-fiber strain-hardening cementitious composites considering the interaction between different fibers based on Yang’s model. In addition, based on the advanced FPRM, Huang et al. (1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
) developed an explicit crack bridging model that included fiber rupture in debonding and slipping stage (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), while the fiber two way pullout was explicated in this model. However, the crack opening width at the fiber complete debonding point was neglected in the fiber limiting embedment and bridging stress, while Cook-Gordon effect was also excluded. Cook-Gordon effect presents a premature fiber/matrix interface debonding normal to the fiber axis caused by a tensile stress located ahead of a blunt matrix crack implementing towards a fiber under remote tensile load, which makes fiber debonding take place ahead of the matrix crack, resulting in stretching of a free fiber segment and additional crack opening (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
). Thus, excluding Cook-Gordon effect would reduce the crack width and mitigate the prediction accuracy of fiber bridging model. Based on the reviewed fiber bridging model, it is more reasonable to establish an advanced crack bridging model including the overall properties of the fibers, slip-hardening interfacial coefficient, fiber rupture in debonding and slipping stage, Cook-Gordon effect, and fiber two-way pullout.

The main objective of this paper was to establish an explicit fiber bridging model of ECC that could be easily adopted for all ECC researchers. On basis of Lin et al.’s FPRM model (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), the fiber rupture zone in this study in debonding and slipping stage was separately calculated. Then, the fiber bridging model was derived when complete fiber bridging stress with no fracture subtracted the bridging stress in the fiber debonding and slipping zone. In addition, Cook-Gordon effect and fiber two-way pullout were included to modify the crack bridging model, thus evolving the ultimate $\text{σ}\left(\text{δ}\right)$ relation of ECC. Finally, the current $\text{σ}\left(\text{δ}\right)$ relation was verified against experimental fiber bridging behavior in a series of ECCs. This analytical model was expected to favor ECC material design in terms of pseudo strain hardening and multiple crack opening behavior.

2. FIBER BRIDGING THEORY

To obtain pseudo strain hardening and multiple crack expanding behavior of ECC under tensile load, the fiber bridging theory was fundamentally framed based on the energy-based criterion and strength-based criterion (2020. Kanda T, Li VC. 1999. Effect of fiber strength and fiber-matrix interface on crack bridging in cement composites. J. Eng. Mech. 125(3):290–299. https://doi.org/10.1061/(ASCE)0733-9399(1999)125:3(290).
, 2525. Kanda T. 1999. New micromechanics design theory for pseudo strain hardening cementitious composite. J. Eng. Mech. 125:373–381. https://doi.org/10.1061/(ASCE)0733-9399(1999)125:4(373).
). The typical $\text{σ}\left(\text{δ}\right)$ relation is shown in Figure 1.

Where ${\sigma }_c$ and ${\sigma }_0$ are the matrix tensile cracking stress and peak fiber bridging stress in $\text{σ}\left(\text{δ}\right)$ curve, respectively. Strength-based criterion is a prerequisite for multiple cracking development, indicating that fiber bridging stress could be effectively transferred from initial crack to the next crack only when the maximum peak fiber bridging stress exceeds the matrix tensile cracking stress depended on the fracture toughness and flaw size of the matrix (2626. Li VC, Wang SX. 2006. Microstructure variability and macroscopic composite properties of high performance fiber reinforced cementitious composites. Probab. Eng. Eng. Mech. 21(3):201-206. https://doi.org/10.1016/j.probengmech.2005.10.008.
), which control the start-up of cracks.

Where $J_{tip}$ and $J_b{}^{\text{'}}$ are the crack tip toughness and complementary energy, respectively. The specific formula of $J_{tip}$ and $J_b{}^{\text{'}}$ are shown in Figure 1. Energy-based criterion was derived by the reference (2727. Marshall DBAC, COX BN. 1988. A J-integral method for calculating steady-state matrix cracking stresses in composites. Mech. Mater. 7(2):127-133. https://doi.org/10.1016/0167-6636(88)90011-7.
) using Jintegral analysis, which guides saturated multiple cracking development only if the crack tip toughness is lower than the complementary energy. Moreover, crack tip toughness is approximately equal to the formula $K_m^2/E_m$ , where $E_m$ is tensile elastic modulus. Moreover, matrix cracking can absorb energy in the cracking process, which can affect the stress field intensity factor (KI). With KI increases, the stress at the crack tip is large enough to reach the fracture strength of the material, and the crack will expand and lead to fracture. So, the Km used in this work is the matrix fracture toughness, which is the critical value of KI. The matrix fracture toughness could be obtained by single crack three-point bend test per ASTM E 1820 -05a (2828. ASTM E1820. 2008. Standard test method for measurement of fracture toughness. ASTM International, West Conshohocken, PA, USA.
).

Simultaneous requirement of Equation [1] and [2] is indispensable to achieve the characteristics of pseudo strain hardening and multiple cracking of ECC. Both criterions are well employed to guide the materials tailoring to achieve high ductility performances by strain-hardening strength indices ${PSH}_{strength}={\sigma }_0/{\sigma }_c$ and strain-hardening energy indices ( ${PSH}_{energy}={J_b}^{\text{'}}/J_{tip}$ ) (2929. Li VC, Kanda T. 1998. Multiple cracking sequence and saturation in fiber reinforced cementitious composites. Concr. Res. Technol. 9:19–33. https://doi.org/10.3151/crt1990.9.2_19.
). For PVA-ECC, the $PS{H}_{strength}$ and $PS{H}_{energy}$ should exceed 1.5 and 3, respectively, while that of PE-ECC should greater than 1.2 and 3, respectively (1313. Kanda T, Li VC. 2006. Practical design criteria for saturated pseudo strain hardening behavior in ECC. J. Adv. Concr. Technol. 4(1):59-72. https://doi.org/10.3151/jact.4.59.
).

3. SINGLE FIBER PULLOUT BEHAVIOR

Based on the crack plane, the fiber pullout model was designed during debonding and slipping stage. Figure 2 shows the fiber inclined angle and embedment length scheme, where $\text{θ}$ is the inclined angle $\text{θ}\in \left(\mathrm{0,}\frac{\text{π}}{2}\right)$ , ${\text{L}}_{\text{L}}and{\text{L}}_{\text{S}}$ are the embedment length of the long end and short end, respectively. ${\delta }_{0L}and{\delta }_{0S}$ are the crack opening distance corresponding to the complete debonding point of the long end and short end, respectively. The fiber starts to slip when $\text{δ}$ exceeds 2 ${\text{δ}}_{0\text{S}}$ . The composite bridging stress can be computed by summing all single fiber stressing fibers crossing the crack plane (1818. Zhong L, Li VC. 1997. Crack bridging in fiber reinforced cementitious composites with slip-hardening interfaces. J. Mech. Phys. Solids. 45(5):763-787. https://doi.org/10.1016/S0022-5096(96)00095-6.
). The relation between fiber bridging stress and crack opening distance was established, which mainly included fiber, matrix, and fiber-matrix properties, as shown in Equation [3]. Meanwhile, the crack displacement ${\delta }_0$ at full-debonding point was defined in Equation [4] (1818. Zhong L, Li VC. 1997. Crack bridging in fiber reinforced cementitious composites with slip-hardening interfaces. J. Mech. Phys. Solids. 45(5):763-787. https://doi.org/10.1016/S0022-5096(96)00095-6.
, 2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
).

Where ${\sigma }_{bb}$ and ${\sigma }_{bp}$ presents the fiber bridging stress during debonding and slipping periods, respectively. ${\tau }_0$ is the matrix/fiber frictional stress, $G_d$ is the chemical bond stress, $E_f$ is the modulus of elasticity of fiber, $\beta$ is the slip hardening coefficient, $d_f$ presents fiber diameter. $L_e$ is the fiber embedment length, which is less than $L_f/2$ Moreover, $f$ is the snubbing coefficient (3030. Li VC, Wang Y, Backer S. 1990. Effect of inclining angle, bundling and surface treatment on synthetic fibre pull-out from a cement matrix. Composites. 21(2):132-140. https://doi.org/10.1016/0010-4361(90)90005-H.
). $\eta =V_f{E}_f/(1-V_f)E_m$ , where $V_f$ is the fiber volume fraction and $E_m$ is the matrix tensile elastic modulus.

Additionally, the fiber apparent tensile strength exhibits a degradation effect, which decreases with the increasing inclined angle (3131. Tetsushi K, Li VC. 1992. Multiple cracking sequence and saturation in fiber reinforced cementitious composites. J. Adv. Concr. Technol. 9(2):19-33. https://doi.org/10.3151/crt1990.9.2_19.
). For PVA fiber, the strength reduction effect could be accounted by Equation [5].

Where ${\sigma }_{fu}$ is fiber tensile fracture strength, $f^{\prime }$ is the fiber tensile strength reduction coefficient.

4. CRACK BRIDGING MODEL OF ECC

4.2. No Fiber fractures

 ⌅

Firstly, assuming that all fiber in crack plane would not fracture under axial tensile force, the embedment length $l_d\left(\delta \right)$ at full debonding point can be derived when crack opening width $\delta$ reaches 2 ${\delta }_0$ (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), as shown in Equation [11].

$l_d\left(\delta \right)=\sqrt{\frac{({\tau }_0\delta +2G_d)}{4(1+\eta ){{\tau }_0}^2}}- \sqrt{\frac{E_f{d}_f{G}_d}{2\left(1+\eta \right){{\tau }_0}^2}}$  [11]

Figure 3 illustrates the fiber embedment at full debonding point with varying crack opening width. Fibers between $\delta$ to $l_d\left(\delta \right)$ are in slipping stage, in which fibers between $l_d\left(\delta \right)$ to $L_f/2$ are in debonding behavior. Moreover, the crack opening width ${\delta }^{*}$ can be obtained when fibers having a peak embedment length of $L_f/2$ are in complete debonding point, which is given in Equation [12].

FIGURE 3.  The fiber embedment at full debonding point with a varying crack opening width.

${\delta }^{*}=2{\delta }_0\left(L_e=\frac{L_f}{2}\right)=\frac{{\tau }_0(1+\eta ){L_f}^2}{d_f{E}_f}+\sqrt{\frac{8G_d(1+\eta ){L_f}^2}{d_f{E}_f}}$  [12]

Hence, Fiber states without fiber rupture could be divided into two kinds, as plotted in Figure 4. The crack opening width could be distinguished into two types according to the value of $\delta$ , ${\delta }^{*}$ , and $L_f/2$ . Then, the crack bridging stress with no fibers rupture ${\sigma }_N\left(\delta \right)$ for ECC can be calculated by integrating all contributory fibers bridging stress, which is given in Equation [13].

FIGURE 4.  Fiber states without fiber rupture. (a) $0<\delta \le {\delta }^{*}$ ; (b) ${\delta }^{*}<\delta \le \frac{L_f}{2}$ .

${\sigma }_N\left(\delta \right)=\left\{\begin{array}{c}\frac{V_f}{L_f}\left\{\begin{array}{c}{\int }_0^{\frac{\pi }{2}}{\int }_{\delta }^{l_d\left(\delta \right)}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta +\\ {\int }_0^{\frac{\pi }{2}}{\int }_{l_d\left(\delta \right)}^{\frac{L_f}{2}}{\sigma }_{bd}\left(\delta ,\theta \right)sin2\theta d{L}_{e}d\theta \end{array}\right. 0 < \delta \le {\delta }^{*} \\ \frac{V_f}{L_f}{\int }_0^{\frac{\pi }{2}}{\int }_{\delta }^{\frac{L_f}{2}}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta {\delta }^{*}<\delta \le \frac{L_f}{2}\end{array}\right.$  [13]

4.3. Fiber rupture in debonding stage

 ⌅

Fibers will be fractured when the maximum fiber bridging stress is higher than its apparent tensile strength ( ${{\sigma }_{bd}\left(\delta ,\theta \right)}_{max}\ge {\sigma }_{fu}\left(\theta \right))$ during debonding process. Specially, the fiber critical embedment length $L_d$ is derived by making the fiber bridging stress in complete debonding point equal the fiber tensile fracture strength ${\sigma }_{fu}\left(\theta \right)$ (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
)，as given in Equation [14].

$L_d\left(\theta \right)=\frac{d_f{\sigma }_{fu}}{4{\tau }_0}{e}^{-(f+f^{\text{'}})\theta }-\frac{\sqrt{2d_f{E}_f{G}_d\left(1+\eta \right)}}{2{\tau }_0}$  [14]

Figure 5 illustrates the fiber critical embedment and debonding rupture zone with varying crack opening width. The fiber embedment increases as the crack opening width increases, which could reach $L_f/2$ when crack opening width is up to ${\delta }^{*}$ . Moreover, the debonding rupture zone increases with the crack opening width increases up to ${\delta }^{*}$ . As the crack opening width $\delta$ increases, three scenarios could be divided to calculate the fiber bridging stress at debonding rupture zone, as shown in Figure 6. From Figure 6a, when $\delta$ lies between 0 and ${\delta }^{*}$ , three main zones could be classified for all fibers, which include slipping zone (SZ), debonding zone (DZ), and debonding rupture zone (DRZ). Fibers in debonding zone (DZ1) will be fractured under tensile load, while fibers in debonding zone (DZ2) suffer full debonding and slipping process without rupture. Moreover, fibers in debonding rupture zone have fractured, which have reached the embedment length $l_d\left(\delta \right)$ at full debonding point.

${\theta }_d$ produced by the intersection point between $L_d\left(\theta \right)$ and $L_f/2$ can be calculated by Equation [15]. Moreover, ${\theta }_c$ could be derived by making $L_d\left(\theta \right)$ be equal to $l_d\left(\delta \right)$ (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), as presented in Equation [16] and [17].

FIGURE 5.  The fiber critical embedment and debonding rupture zone with a varying crack opening width.

FIGURE 6.  Fiber debonding rupture zone. (a) $0\le \mathit{\delta }\le {\mathit{\delta }}^{\mathit{*}}$ ; (b) ${\mathit{\delta }}^{\mathit{*}}\le \mathit{\delta }\le {\mathit{L}}_{\mathit{d}}\left(\frac{\mathit{\pi }}{2}\right)$ ; (c) ${\mathit{L}}_{\mathit{d}}\left(\frac{\mathit{\pi }}{2}\right)\le \mathit{\delta }\le \frac{{\mathit{L}}_{\mathit{f}}}{2}$ .

${\theta }_d=\left\{\begin{array}{c}\frac{\pi }{2} \frac{L_f}{2}\le L_d\left(\frac{\pi }{2}\right)\\ -\frac{1}{f+f^{\text{'}}}\ln \frac{2L_f{\tau }_0+2\sqrt{2E_f{d}_f\left(1+\eta \right)}}{d_f{\sigma }_{fu}} L_d\left(\frac{\pi }{2}\right)\le \frac{L_f}{2}\le L_d\left(0\right) \\ 0 \frac{L_f}{2}>L_d\left(0\right)\end{array}\right.$  [15]

${{\theta }_c}^{\text{'}}= -\frac{1}{2\left(f+f^{\text{'}}\right)}\ln \frac{{4E}_f\left(1+\eta \right)}{{{\sigma }_{fu}}^2{d}_f}\left(\delta {\tau }_0+2G_d\right)$  [16]

${\theta }_c\left(\delta \right)=\left\{\begin{array}{c}\frac{\pi }{2} \frac{\pi }{2}< {{\theta }_c}^{\text{'}} \\ {{\theta }_c}^{\text{'}} {\theta }_d\le {{\theta }_c}^{\text{'}}\le \frac{\pi }{2}\\ {\theta }_d {{\theta }_c}^{\text{'}}<{\theta }_d\end{array}\right.$  [17]

When $\delta$ lies between ${\delta }^{*}$ and $L_d\left(\frac{\pi }{2}\right)$ , the debonding rupture zone (DRZ) and slipping zone (SZ) were presented in Figure 6b. Fibers in DRZ have fractured and reached the embedment length $l_d\left(\delta \right)$ at full debonding point. Additionally, Figure 6c shows that the debonding rupture zone of fibers reduces as $\delta$ increases from $L_d\left(\frac{\pi }{2}\right)$ to $L_f/2$ . The inclined angle ${\theta }_a\left(\delta \right)$ could be calculated by making $L_d\left(\theta \right)$ equal $\delta$ (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), as defined by Equation [18]. Hence, the fiber rupture bridging stress ${\sigma }_D$ in bebonding stage versus the increasing crack opening width $\text{δ}$ relation can be evolved by combining the contributory fractured fibers, as given in Equation [19].

4.4. Fiber fracture in slipping stage

 ⌅

Fiber bridging stress can be strengthened since the slip hardening feature of PVA fiber emerges in slipping stage. Hence, fiber slipping fracture also can be presented when the maximum fiber bridging stress ${{\sigma }_{bp}(\delta ,\theta )}_{max}$ grow higher than the fiber apparent tensile strength ${\sigma }_{fu}\left(\theta \right)$ . The maximum fiber bridging stress could be deduced by Equation [20], corresponding to ${\delta }_{max}=\frac{L_e}{2}+2{\delta }_0+\frac{d_f}{2\beta }$ (3333. Wu C. 2001. Micromechanical tailoring of PVA-ECC for structural applications. University of Michigan.
). Furthermore, another fiber critical embedment length $L_p\left(\theta \right)$ could be derived by making ${\sigma }_{bp}\left({\delta }_{max},\theta \right)$ equal ${\sigma }_{fu}\left(\theta \right)$ , as given in Equation [21].

${\theta }_a\left(\delta \right)=\left\{\begin{array}{c}\frac{\pi }{2} 0 \le \delta \le L_d\left(\frac{\pi }{2}\right)\\ -\frac{1}{\left(f+f^{\text{'}}\right)}\ln \frac{4{\tau }_0}{d_f{\sigma }_{fu}}\left(\delta +\frac{\sqrt{2d_f{E}_f{G}_d\left(1+\eta \right)}}{2{\tau }_0}\right) L_d\left(\frac{\pi }{2}\right)\le \delta \le \frac{L_f}{2} \end{array} \right.$  [18]

${\sigma }_D\left(\delta \right)=\left\{\begin{array}{c}\frac{V_f}{L_f}\left\{\begin{array}{c}{\int }_{{\theta }_c\left(\delta \right)}^{\frac{\pi }{2}}{\int }_{l_d\left(\delta \right)}^{\frac{L_f}{2}}{\sigma }_{bd}\left(\delta ,\theta \right)sin2\theta d{L}_{e}d\theta +\\ {\int }_{{\theta }_c\left(\delta \right)}^{\frac{\pi }{2}}{\int }_{L_d\left(\theta \right)}^{l_d\left(\delta \right)}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta \end{array} 0\le \delta \le {\delta }^{*}\right.\\ \frac{V_f}{L_f}{\int }_{{\theta }_c\left(\delta \right)}^{\frac{\pi }{2}}{\int }_{L_d\left(\theta \right)}^{\frac{L_f}{2}}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta {\delta }^{*}\le \delta \le L_d\left(\frac{\pi }{2}\right)\\ \frac{V_f}{L_f}\left\{\begin{array}{c}{\int }_{{\theta }_a\left(\delta \right)}^{\frac{\pi }{2}}{\int }_{\delta }^{\frac{L_f}{2}}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta \\ {\int }_{{\theta }_c\left(\delta \right)}^{{\theta }_a\left(\delta \right)}{\int }_{L_d\left(\theta \right)}^{\frac{L_f}{2}}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta \end{array}\right. L_d\left(\frac{\pi }{2}\right)\le \delta \le \frac{L_f}{2}\end{array}\right.$  [19]

${\sigma }_{bp}\left({\delta }_{max},\theta \right)=\frac{{\tau }_0{e}^{f\theta }}{d_f}\left(\frac{{L_e}^2\beta }{d_f}+2L_e+\frac{d_f}{\beta }\right)$  [20]

$L_p\left(\theta \right)=d_f\left(\sqrt{\frac{{\sigma }_{fu}{e}^{-\left(f+f^{\text{'}}\right)\theta }}{\beta {\tau }_0}}-\frac{1}{\beta }\right)$  [21]

As crack opening width expands, a new critical embedment length $l_p\left(\delta \right)$ resulting in fracturing for the slipping fibers can also be derived by letting ${\sigma }_{bp}\left(\delta ,\theta \right)$ equal ${\sigma }_{fu}(\theta$ , as shown in Equation [22], indicating that fibers with an embedment length higher than $l_p\left(\delta \right)$ has fractured in the current crack opening width (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
).

$l_p\left(\delta \right)=\frac{{d_f}^2{\sigma }_{fu}{e}^{-\left(f+f^{\text{'}}\right)\theta }}{4{\tau }_0(d_f+\beta \left(\delta -2{\delta }_0\right))}+\left(\delta -2{\delta }_0\right)$  [22]

Fiber slipping rupture zone is determined by the combined curve of $l_p\left(\delta \right)$ , $L_p\left(\theta \right)$ , and $L_d\left(\theta \right)$ . However, $l_p\left(\delta \right)$ is an inapprehensible curve since ${\delta }_0$ is the quadratic function of $L_e$ . Huang et al. (1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
) strictly omitted ${\delta }_0$ in Equation [22]. Actually, an essential ${\delta }_0$ should be derived to achieve the better prediction accuracy of σ(δ) relation. Hence, a new assuming in this study, embedment length $L_e$ of all fibers conservatively lie between $L_p\left(\frac{\pi }{2}\right)$ and $L_d\left(\frac{\pi }{2}\right)$ and presents a random uniform distribution in the slipping rupture zone, which was proposed to identify, and account for the slipping fracture section. Afterward, the probability density function of $L_e$ is proposed to get the average ${\delta }_0{}_{average}$ , as defined in Equation [23] and [24]. The revised current critical embedment length $l_p\left(\delta \right)$ can be shown as in Equation [25].

$f\left(L_e\right)=\frac{1}{L_d\left(\frac{\pi }{2}\right)-L_p\left(\frac{\pi }{2}\right)}$  [23]

${{\delta }_0}_{Average}={\int }_{L_p\left(\frac{\pi }{2}\right)}^{L_d\left(\frac{\pi }{2}\right)}{\delta }_0(L_e)f\left(L_e\right)d{L}_e$  [24]

$l_p\left(\delta \right)=\frac{{d_f}^2{\sigma }_{fu}{e}^{-\left(f+f^{\text{'}}\right)\theta }}{4{\tau }_0(d_f+\beta \left(\delta -2{\delta }_{0\_average}\right))}+\left(\delta -2{\delta }_{0\_average}\right)$  [25]

Figure 7 illustrates the limiting embedment length at slipping state and slipping rupture zone with varying crack opening distance. The slipping rupture zone gradually increases as crack opening expands. Figure 8 shows the fiber slipping rupture zone. It is noteworthy to see that the slipping rupture zone starts to occur when crack plane expands a minimum width ${\delta }_{min}$ that can be approximately calculated by making $l_p\left({\delta }_{min},\frac{\pi }{2}\right)$ equal $L_d\left(\frac{\pi }{2}\right)$ using Equation [26]. Thus, the slipping rupture zone is zero when crack opening width is lower than ${\delta }_{min}$ , as shown in Figure 8a.

FIGURE 7.  The limiting embedment length in slipping stage and slipping rupture zone with varying crack opening distance.

$\frac{{d_f}^2{\sigma }_{fu}{e}^{-\left(f+f^{\text{'}}\right)\frac{\pi }{2}}}{4{\tau }_0(d_f+\beta \left({\delta }_{min}-2{\delta }_{0\_average}\right))}+\left({\delta }_{min}-2{\delta }_{0\_average}\right)= \frac{d_f{\sigma }_{fu}}{4{\tau }_0}-\frac{\sqrt{2d_f{E}_f{G}_d\left(1+\eta \right)}}{2{\tau }_0}$  [26]

Moreover, the slipping rupture zone increases as the crack plane increases from ${\delta }_{min}$ to ${\delta }^{*}$ (see Figure 8 b). The inclined angle ${\theta }_b$ could be calculated by letting $l_p\left(\delta \right)$ equal $L_f$ /2 (2121. Lin Z, Kanda T, Li VC. 1999. On interface property characterization and performance of fiber reinforced cementitious composites. J. Adv. Concr. Technol. 1(3):173–184. Retrieved from http://hdl.handle.net/2027.42/84718.
), as given in Equation [27] and [28]. The curve of $l_p\left(\delta \right)$ and $L_d\left(\theta \right)$ almost coincide when the crack opening width expand to $L_f$ /16 (see Figure 8 c). Generally, the crack opening width corresponding to the peak fiber bridging stress is lower than $L_f$ /16 (around 750 μm). Hence, the fiber rupture bridging stress ${\sigma }_S$ in slipping stage versus increasing crack opening width $\text{δ}$ (lower than $L_f$ /16) for composites can be deduced by integrating the contributory fractured fibers, as presented in Equation [29].

FIGURE 8.  Fiber slipping rupture zone. (a) ; (b); (c).

${{\theta }_b}^{\text{'}}= -\frac{1}{f+f^{\text{'}}}\ln \left[\frac{2{\tau }_0\left(\beta \left(\delta -2{\delta }_{0_{average}}\right)+d_f\right)\left(L_f-2\left(\delta -2{\delta }_{0_{average}}\right)\right)}{{\sigma }_{fu}{d}_f^2}\right]$  [27]

${\theta }_b=\left\{\begin{array}{c}0 {{\theta }_b}^{\text{'}}\le 0\\ {{\theta }_b}^{\text{'}} {{\theta }_b}^{\text{'}}>0\end{array}\right.$  [28]

${\sigma }_S\left(\delta \right)=\left\{\begin{array}{c}0 0\le \delta \le {\delta }_{min}\\ \frac{V_f}{L_f}\left\{\begin{array}{c}{\int }_{{\theta }_b}^{{\theta }_d}{\int }_{l_p\left(\delta ,\theta \right)}^{\frac{L_f}{2}}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta +\\ {\int }_{{\theta }_d}^{\frac{\pi }{2}}{\int }_{l_p\left(\delta ,\theta \right)}^{L_d\left(\theta \right)}{\sigma }_{bp}\left(\delta ,L_e,\theta \right)sin2\theta d{L}_{e}d\theta \end{array} {\delta }_{min}\le \delta \le \frac{L_f}{16}\right.\end{array} \right.$  [29]

5. FIBER TWO-WAY PULLOUT CONSIDERATION

Fiber two way pullout was first reported with respect to PVA fiber due to slip-hardening interface behavior (3434. Wang YJ, Li VC, Backer S. 1988. Modelling of fibre pull-out from a cement matrix. Int. J. Cement. Compos. Lightweight. Concr. 10(3):143-149. Retrieved from https://deepblue.lib.umich.edu/bitstream/handle/2027.42/84694/ywang_CCLC88.pdf?sequence=1.
). The ${\text{σ}}_{effective}\left(\text{δ}\right)$ exclude the fiber two-way pullout situation. In this study, the so-called nominal fiber was adopted to supply a reasonably approximate crack opening width contributed by fiber two-way pullout (1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
, 2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
).

The single fiber bridging stress is the function of embedment length $L_e$ and inclined angle $\theta$ , thus Equation (10) can be rewritten as Equation [31] (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
) Where ${\xi }_b$ is the efficiency of fiber bridging, which represents the fiber contributing stress as fibers increasingly fracture across a crack plane. Thus, ${\xi }_b$ will gradually reduce to zero. It is noteworthy that the efficiency fiber contributing stress for fibers containing the rupture fiber in debonding and slipping stage is defined as ${\xi }_{effective}{}_b$ which is given in Appendix I. Hence, according to Equation [31], ${\text{σ}}_b$ is presented in Equation [33]. For fibers having a long embedment length $l_e(l_e\in (\frac{L_f}{2},L_f\left)\right)$ , under the effective bridging stress ${\text{σ}}_b$ , the displacement of fiber in debonding and pullout stage can be deduced (1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
), as given in Equation [34] and [35], respectively.

Where ${\text{σ}}_b$ is the nominal fiber bridging stress, ${\xi }_b$ can be derived in Equation [32].

Where ${\sigma }_0=\frac{{\tau }_0{V}_f{L}_f}{2d_f}$ , $\gamma =\frac{\sqrt{8G_d{E}_f{d}_f(1+\eta )}}{{\tau }_0{L}_f}$ , $u_0=\frac{{\tau }_0{L_f}^2}{E_f{d}_f(1+\eta )}$ , $\alpha =\frac{2d_f}{\beta L_f}$ . The displacement of ${\delta }_L$ is the function of maximum fiber bridging stress ${\text{σ}}_b{}_{max}$ and embedment length of the long side $l_e$ in the loading process, as shown in Equation [36].

Assuming $l_e$ is a uniform random distribution in the interval from $\frac{L_f}{2}$ to $L_f$ , thus the probability density function of $l_e$ is defined as $f\left(l_e\right)$ by Equation [37]. Afterwards, the remedial crack opening width ${\delta }_r$ is approximately averaging ${\delta }_L$ by the integral of $l_e$ , as presented in Equation [38].

Moreover, it is known from Equation [34], the maximum fiber bridging stress ${\text{σ}}_b{}_{max}$ in debonding stage has a critical value (see Equation [39]), which deduces the current fiber critical embedment length $l_{e0}$ , as given in Equation [40].

Where $\zeta =\frac{V_f{{\sigma }_b}_{max}}{4{\sigma }_0}$ . A long embedment length of fibers with higher than $l_{e0}$ is still at the debonding stage.

According to the above definition, two scenarios could be generated according to $l_{e0}$ .If $l_{e0}\le \frac{L_f}{2}$ , namely ${{\sigma }_b}_{max}\le \frac{4{\sigma }_0}{V_f}\left(1+\frac{\gamma }{2}\right)$ , implying that the long embedded side of fibers is in debonding process, ${\delta }_r$ is given by Huang et al. (1414. Huang T, Zhang YX, Su C, Lo SR. 2015. Effect of slip-hardening interface behavior on fiber rupture and crack bridging in fiber-reinforced cementitious composites. J. Eng. Mech. 141(10):04015035. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000932.
).

If $\frac{L_f}{2}<l_{e0}\le L_f$ , namely $\frac{4{\sigma }_0}{V_f}\left(1+\frac{\gamma }{2}\right)\le {{\sigma }_b}_{max}<\frac{4{\sigma }_0}{V_f}\left(2+\frac{\gamma }{2}\right)$ , which indicates that the long embedded side of fibers is in the debonding and pullout process, ${\delta }_r$ is given as follows [42]

6. COOK-GORDON EFFECT

Cook-Gordon effect will contribute the additional width to the ${\text{σ}}_{effective}\left(\text{δ}\right)$ , which describes a precocious fiber debonding normal process. Meanwhile, the fiber-matrix is separated under the fiber bridging stress related to the elastic crack tip field of the adjoining matrix crack in the horizontal plane, thus leading to an attached crack opening ${\delta }_{cg}$ generated from the fiber elastic stretching of the fiber segment (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
). Therefore, the fiber debonding process initiates ahead of the matrix cracking, leading to stretch a free fiber segment ${\delta }_{cg}$ , as given in Equation [43] (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
).

Where $\text{α}$ is the Cook-Gordon parameter. In general, α = 2 $d_f$ for PVA fiber was suggested, while α is set as 15 $d_f$ for polyethylene (PE), and polypropylene (PP) fibers (3535. Li VC. 1993. From micromechanics to structural engineering. The design of cementitous composites for civil engineering applications. Structural engineering/earthquake engineering. 1993(471):1-12. https://doi.org/10.2208/jscej.1993.471_1.
).

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

7. MODEL VALIDATION

Figure 9 shows the computation flow scheme of final $\text{σ}\left(\text{δ}\right)$ . The experimental data including fiber, matrix, and fiber/matrix parameters from Yang et al. (2323. Yang EH, Wang S, Yang Y, Li VC. 2008. Fiber-bridging constitutive law of engineered cementitious composites. J. Adv. Concr. Technol. 6(1):181-193. https://doi.org/10.3151/jact.6.181.
) were employed to verify the final $\text{σ}\left(\text{δ}\right)$ . The micro-parameters for PE-ECC and PVA-ECC are listed in Table 1. It is well known that PVA fiber has a strong chemical bond strength and slip hardening behavior in hardened pastes. Table 1 illustrates the micro-parameters for PVA-ECC and PE-ECC.