Tensile behavior of strain-hardening cementitious composites after self-healing based on a novel fiber-bridging model considering preloading and reloading

. The self-healing of strain-hardening cementitious composites (SHCCs) causes the recovery of the debonded fiber-to-matrix interface by the products of autogenous healing (mainly calcium carbonates). The recovery of chemical bond G d has been detected in the reactive magnesia cement (RMC)-based SHCC (SHMC), and the recovery of frictional bond τ 0 has been detected in both SHMCs and normal SHCCs. While these phenomena can significantly alter the fiber-bridging σ-w relationship in SHCCs, they have not been quantified in any existing analytical models. In this work, we present a new fiber-bridging model that captures the effect of self-healing of RMC-based SHCC. On the single-fiber level, the debonding and slip-hardening of the fiber-to-matrix interface induced by a tensile preloading as well as the recovery of the interface properties by self-healing are coherently quantified in a clear kinetic process. On the fiber-bridging level, the tensile stress vs. crack width curve is formed by summing individual fibers’ tensile load vs. displacement relationship. The modeling results can well capture the fiber-bridging behavior of the self-healed SHCC specimens. Further, a parametric study is conducted to investigate the tensile behavior of SHCC after self-healing. The effects of preloading levels, recovered τ 0 , and fiber strength are discussed.


Introduction
Straining-hardening cementitious composites (SHCCs) are a group of high-performance fiber-reinforced cementitious composites. As indicated by its name, this composite can behave in a pseudo-strain-hardening manner after matrix cracking, and thus achieve an ultrahigh tensile strain capacity (1-10% [1]). SHCCs can engage autogenous crack healings because their crack widths are well controlled by fiber-bridging [2][3][4][5]. It has been noticed that the healing can happen at the fiber-tomatrix interface, i.e., the fine crack resulting from the fiber debonding from the matrix [6]. The healinginduced tensile recovery is mainly attributed to the mechanical recovery of the interface [7]. Like most discrete fiber-reinforced composites, the post-cracking behaviors of SHCC are essentially governed by the fiber-bridging constitutive laws [8]. Predicting the tensile stress-crack width relationship of fiber-bridging after self-healing could be significant to the design and application of SHCCs.
The mix proportion design of SHCCs is guided by the micromechanics-based analytical model that predicts the fiber-bridging constitutive law [9]. For more accurately predicting the crack widths in SHCC, which is crucial for the matrix cracks to heal in the presence of water, Yang and Li [10] added the two-way fiber pullout and Cook-Gordon effect to the model. Ranadae et al. [11] incorporated the mechanism of inclination-dependent * Corresponding author: yangqingliu@shu.edu.cn hardening into the fiber-pullout analytical model for high-strength SHCC. For predicting SHCC's resistance to impact loading, Yang et al. [12][13] added the effect of strain rate to the fiber-to-matrix interface properties and in-situ fiber strength to the model. For predicting SHCC's fatigue resistance, Qiu et al. [14][15] added the effect of fatigue-induced fiber debonding, sliphardening, and fiber strength reduction to the model. However, there has been no analytical model to quantify the post-healing behavior of the fiber-to-matrix interface properties.
Therefore, this study modifies the classic two-way pullout fiber-bridging law [10] by adding the proloading effects. On the single-fiber level, the debonding and slip-hardening of the fiber-to-matrix interface induced by a tensile preloading as well as the recovery of the interface properties by self-healing are coherently quantified in a clear kinetic process. On the fiberbridging level, the tensile stress-crack width curve is formed by summing individual fibers' tensile load vs. displacement relationship. The modeling results can capture the fiber-bridging behavior of the self-healed SHCC specimens. Further, a parametric study based on the new model is conducted. The effects of preloading levels, recovered interface property coefficients, and fiber strength on the fiber-bridging behavior during the reloading are discussed. The methodology of the analytical fiber-bridging model for SHCC under static loading is briefly described here for two reasons: first, it will be used to determine the damage level of the fiber-to-matrix interfaces in a preloaded SHCC, i.e., the debonding length a and the fiber slippage u; second, the algebraic equations in this model will be modified to include new terms for quantifying the carbonation and interface healing in the new model.
Lin et al. [9] derived Eqs. 1 and 2 to calculate the tensile force P on the fiber based on the fiber pullout displacement δ.
The interfacial properties Gd (J/m 2 ) is the fracture energy for the interfacial crack to propagate unit area; τ0 (N/m 2 ) is the friction force exerted by unit debonded area; β is a dimensionless term that quantifies the enhancement of τ0 because of slip-hardening. In these equations, Ef and df are the elastic modulus and diameter of the fibers; η is defined as VfEf /VmEm, where Vf and Vm are the volume fractions of fibers and matrix, respectively; Le is the fiber embedment length. Particularly, δ0 is the fiber pullout displacement at the moment of fiber full debonding (i.e., a=Le), which can be calculated by Eq. 3.
( ) ( ) Based on the above equations, Yang et al. [10] calculated the fiber-bridging σ-w curve with two-way fiber pullout scenarios considered in three steps: (1) translate the P-δ relationship into P-w relationship by combining the fiber displacement at both the shortembedded (δs) and long-embedded side (δl) into w; (2) calculate the σ in SHCC at a given w by adding up the P(δ) in all bridging fibers; (3) repeating the second step for periodically increased w.
There are three different scenarios of two-way fiber pullout, i.e., both short-and long-embedded sides are debonding (Scenario 1), the short-embedded is slipping while the long-embedded is still debonding (Scenario 2), both sides are slipping (Scenario 3). In all three scenarios, tensile force in the fiber of both sides should be equal, i.e., P(δs) = P(δl); the tensile force will return to zero if the short-embedded side is completely pulled out, i.e., us>Les, or if the fiber is ruptured, i.e., P/Af>σf. In the next section, the three scenarios will be extended to 15 scenarios for the two-way pullout of preloadedand-healed fibers.
Eq. 4 describes the method of integrating forces of individual fibers into the SHCC tensile stress, where the variation of fiber orientation φ and the location z (which determines Le) are considered by the probabilistic function p(φ) (dimensionless) and p(z) (m -1 ). When applying the iteration to calculate the σ-w relation, the effect of fiber pullout and rupture mentioned above will be included by deducting the force P of fibers of certain φ and Le. Fig. 1 shows the interface of a partially debonded-andhealed fiber and a fully debonded-and-healed fiber, respectively. Under reloading, both interfaces will undergo debonding and slippage again; here we use the red triangle cursor to illustrate how far the interfacial crack has propagated in the debonding stage, and the blue triangle cursor to illustrate how far the fiber end has slipped. If the fiber was partially debonded, four scenarios would successively occur during the interfacial failure, i.e., D1 -the crack frontier is still in the healed segment; D2 -the crack frontier has entered into the intact segment; S3 -the fiber has been fully debonded and slipped, the fiber end being in the intact segment; S4 -the fiber end has entered into the healed segment. Therefore, the P-δ of such fibers needs to be described with four separate analytical equations. If the fiber was fully debonded, only two scenarios, i.e., D1 and S4 would occur. Correspondingly, the P-δ of such fibers needs to be described with two separate analytical equations.   2 illustrates the two cases of single fiber being pulled out during monotonic reloading after the interfacial self-healing. In the two cases, the fiber debonding respectively occurs in the recovered or pristine region, i.e., the current debonding length a is smaller or larger than a', the maximum debonding length in the preloading. At the recovered interface, the chemical bond, frictional bond, and slip-hardening coefficient are G'd, τ'0, and β', while they equal Gd, τ0, and β at the pristine interface.

G' d , τ'
Mechanical model of single fiber pullout after healing.
In the first case, randomly taking a small segment of the debonded fiber with a length of dx, the force equilibrium can be expressed as Eq. 5. Wherein σf2(x) represents the fiber cross-sectional tensile stress in the recovered region. According to the boundary condition at the fiber free end, the expression of σf2(x) can be solved. ( For the whole cross-section, the force equilibrium of the fiber and matrix is written as Eq. 8, and the equation of the matrix tensile stress σm2(x) is Eq. 9. Further, the relative displacement between the fiber and the matrix can be calculated by Eq. 10, where η=EfVf/Em(1-Vf).
From the perspective of energy balance, the energy criterion for debonding crack advancing is expressed as Eq. 11. Wherein dW, dWε, and dWf are the external work, strain energy, and friction-induced energy dissipation while the crack advancing by da; uf is the displacement of the fiber free end and can be calculated by Eq. 12, where Ec=VfEf+VmEm. Besides, the equation of Wf is given as Eq. 13. By substituting Eqs. 12 and 13 into Eq. 11, the solution to σ with respect to a can be expressed as Eq. 14.
According to Eqs. 10 and 14, the relative displacement of the fiber free end to matrix can be calculated by Eq. 15. Based on Eq. 15, the debonding length a can be expressed by Eq. 16. By combining Eqs. 14 and 16, the relation between the fiber force and displacement is established as Eq. 17.
With respect to the second case, the force equilibriums of the dx-long segments in the pristine and recovered region can be expressed as Eq. 18. Considering the boundary conditions at the fiber free end Eq. 6 and the bound between the two regions Eq. 19, the expression of σf1(x) and σf2(x) can be obtained.
( ) ( ) 4 , According to Eq. 8, the equations of the matrix tensile stress σm1(x) and σm2(x) are written as Eq. 21. In this case, the relative displacement between the fiber and the matrix is divided into two segments: , 0 1 Since the location of debonding crack advancing is in the pristine region, the energy balance is expressed as Eq. 23. The equations of uf and Wf also consist of two parts, as given in Eqs. 24 and 25.
By substituting Eqs. 24 and 25 into Eq. 23, the solution to σ with respect to a can be expressed as: Combining Eqs. 22 and 26, the relative displacement of the fiber free end to matrix is expressed as: Moreover, the debonding length a can be written as Eq. 28. By substituting Eq. 28 into Eq. 26, the fiber force can be calculated by the displacement, as given in Eq. 4 Based on the analysis above, the four analytical P-δ equations for the partially debonded fibers are given in Eq. 30; the two analytical P-δ equations for the fully debonded fibers are given in Eq. 31. If the fiber was fully deboned and had slipped, the effective fiber embedment length will be subtracting the slippage from the original embedment length, i.e., L'e=Le-u. During the preloading, the pulled-out segment of fiber should be equal to the fiber slippage u; therefore, during reloading, the force will be zero if displacement δ is smaller than u, the necessary displacement to straighten the soft pulled-out fiber segment and induce loading. In Eq. 31, the term δ' stands for the fiber displacement induced during reloading; the total fiber displacement induced during preloading and reloading is δ=u+δ'.
To use Eqs. 30 and 31, the fiber debonding length a' or fiber slippage u during the preloading should first be calculated based on the preloading level using Eqs. 32 and 33, respectively.

( ) ( )
During the reloading, δ'0 and δ0 in Eqs. 30 and 31 are the fiber pullout displacements corresponding to fully debonding of the healed and pristine segments and can be calculated using Eqs. 34 and 35. The terms δ'0 and δ0 will also be used for determining the location of the cursors.
( ) ( ) Fig. 3 is the flow chart of calculating the fiberbridging σ-w relationship with the effect of preloading and interface healing considered. It reflects the methodology described above. The steps of taking account of the preloading dependency are highlighted in red.
Start: input parameters, including crack depth and interface coefficients.
Check whether fiber has been ruptured or pulled out. If so, P = 0.
Accumulate forces of fibers and calculate fiber-bridging tensile stress.
For a given δ, determine stages of short and long segments; calculate fiber force.

Preloading
For a given δ, determine locations of left and right cursors; Calculate fiber force considering effect of residual slips.
Check whether fiber has been ruptured or pulled out. If so, P = 0.
Calculate two-way fiber-bridging stress during reloading.

Reloading
For each fiber with different Les and φ, identify one of three scenarios.
End: Output tensile stress-crack width curves during preloading and reloading. Fig. 3. Flow chart for computing fiber-bridging constitutive law involving fiber-to-matrix interface healing. Table 1 shows the fiber properties used in the model, where Ef, df, Lf, and the in-situ fiber strength σfu are provided by the manufacturer; the strength reduction coefficient f′ and snubbing coefficient f refer to [16] and [17], respectively; Vf and Em are consistent with the testing scheme. Table 2 shows interfacial properties used in the model, which were obtained with a singlefiber preloading-healing-pullout test. The fibers were preloaded to reach full debonding, cured under waterdry cycles for interface healing, and reloaded to pullout or rupture. Each cycle consists of one day in water (W) and one day in the ambient air (A), e.g., P10W10A means preloading followed by ten healing cycles. A control group, i.e., not preloaded but cured under the identical water-dry cycles, was monotonically loaded to failure, was also included in the experiment. In the calculation, the crack width induced by preloading wpre is inversely calculated based on the tensile stress applied to fiber-bridging, based on the pristine interfacial properties.

Model validation
It should be noticed that the fiber-matrix interface close to the matrix crack mouth can experience full healing while the bond of fibers sitting inside a crack may have been less healed. The current study assumes that healing happens to all the fiber-matrix interfaces, which should be improved in future works. Table 1. Fiber-related properties as input of the model.  4 compares the tensile stress vs normalized crack width (σ vs. w/w0, where w0 is the crack width corresponding to the peak loading) curves calculated by the current model and the experimental results, which were obtained by conducting tensile test single-cracked FRCC that adopted the same PVA fibers and OPC matrix in the single-fiber test. The model can largely capture the fiber-bridging behaviors of both intact and self-healed FRCC. More importantly, it predicted the peculiar mechanical enhancement induced by the preloading and interface healing, i.e., the tensile strength of Fig. 4b is significantly higher than that of Fig. 4a.

Parametric study
A parametric study was conducted to evaluate the effects of preloading crack width wpre, recovered interface property τ'0, and fiber strength σfu. Other parameters were consistent with those in the validated model. Fig. 5a shows the reloading tensile stress-crack width curves of the cases with the wpre from 0 to 50 μm. When wpre=0, the reloading curve is exactly the same as the corresponding monotone preloading curve. The first peak of the curves rises with the preloading crack width, but the increment magnitude decreases with wpre. The reason is that more area of the fiber-to-matrix interfaces obtain a higher recovered interface strength as wpre increases. The ultimate strength first increases and then decreases with the growth of wpre, which indicates more fibers could be raptured under a higher preloading level. Fig. 5b shows the effect of τ'0 on the tensile stresscrack width curves. It is noted that enlarging the recovered frictional strength can significantly improve the first peak stress during reloading. The ultimate strength also rises with the increase of τ'0. However, the strain capacity and the strain-hardening behavior are weakened in the case with a high τ'0 because the fibers are more likely to break. Fig. 5c shows the tensile stresscrack width curves of the cases with varying fiber strength. When σfu is lower than 1000 MPa, the first peak strength and especially the ultimate strength are affected by the fiber strength. Due to the high recovered interface strength, fiber rapture frequently happens during reloading. By contrast, when σfu is no less than 1000 MPa, the difference in the tensile stress-crack width curves is inconsiderable.

Summary
To sum up, based on the classic two-way pullout fiberbridging law, this study developed a new analytical model that can quantify the effect of fiber-matrix interfacial healing. A kinetic process of single-fiber pullout considering the different scenarios of preloading was analyzed to evaluate the fiber-bridging behavior after interfacial self-healing. The analytical results matched well with the experimental results. Further, a parametric study with the new model was performed. The effects of preloading crack width, recovered interface property, and fiber strength on the fiberbridging behavior was discussed. More detailed analysis and discussion will be presented in our upcoming paper.