The components utilized in the matrix included P୰O 42.5 cement, fly ash, and fine quartz sand with a particle size ranging from 100 to 210 μm and an average size of 150 μm. Figure 1 displays the particle size distribution curves for fly ash, sand and cement. The particle size of fly ash is finer compared to cement. It can be used to make the particles of the raw material pack more closely, meanwhile, it has higher hydration activity than low grade fly ash, thus improving the strength of the composite material to a certain extent. Referred to the previous research results (Ma et al., 2017; Zhang et ak., 2020a), the mass ratio of fly ash and cement was set as 4:1 to guarantee a substantial ductility for the designed composite material.

Previous literatures suggested that the optimal proportion of cement to fine sand should be between 0.6 and 1.8 (Liu and Tan, 2017a; Liu et al., 2017; Ramasamy and Shanmughasundaram, 2018; Tinoco and Silva, 2021; Arain et al., 2023). This study set the mass ratio of sand and cement as 1.8:1 in order to increase the density of the particles, bringing them closer to the theoretical curve, as depicted in Figure 2. More details about the modified A&A model can be obtained in previous literatures (Andreasen and Andersen, 1930; Funk and Dinger, 1994; Hunger, 2010; Karim et al., 2019).

The water to binder ratio was set as 0.34. The fresh MsHySHCC’s workability was modified using superplasticizer at 0.5 wt% of binder. Three kinds of reinforcing fibrous materials were employed in this study, as shown in Figure 3. Table 1 displays the composition of various fibers in each designed group. Meanwhile, the 28 days compressive strength (70.7 mm³ cube) of each group is also give in Table 1. It can be seen that the designed composites had a relative high compressive strength.

The process of preparing the specimen is illustrated in Figure 4. Dog-bone-shaped specimen was prepared. The tension specimen size is shown in Figures 4, 5. To ensure the precision of the experimental findings, three samples were prepared for every design mixture.

Tensile test set-ups are shown in Figure 5. A 15 mm range extensometer was used to monitor the tensile deformation of the specimen. The tensile loading rate is 0.1 mm/min.

Figure 6 shows the tensile stress-strain relationships of control groups with or without CaCO₃ whiskers. Table 2 summarizes the average tensile values of strength, ultimate strain and toughness of the three groups of materials. This paper quantifies the tensile toughness by calculating the enclosed region beneath the tensile stress-strain graph, which is employed to assess the specimen’s ability to absorb energy under tensile force. By observing Figure 6; Table 2, it becomes evident that CaCO₃ whiskers can enhance both the tensile strength and tensile strain capacity of the mortar matrix. And the tensile strength and ultimate strain are further improved with increasing the content of CaCO₃ whiskers from 1% to 2%. The enhancements can be explained by the micro-mechanisms of CaCO₃ whiskers as illustrated in Figure 7. However, it is worth mentioning that despite the enhancement of tensile properties in the mortar matrix due to the utilization of CaCO₃ whisker, the failure process still exhibited evident brittleness and lacked post peak toughness (Cao et al., 2013; Cao et al., 2019). Because the particle size of CaCO₃ whisker is small, it can only strengthen and toughen the mortar matrix at the microscopic level, thus can’t improve the post peak toughness of the matrix like other macro fibers.

The relationship between tensile stress and strain is depicted in Figure 8. Experimental tensile results of strength, ultimate strain and toughness for each group are provided in Figure 9. Experimental data of tensile parameters in previous available literatures are summarized in Table 3. Figure 10 illustrates the comparisons of the experimental findings between previous literature and the current study. From Figures 8–10; Table 3, the following findings can be addressed.

The experimental curves exhibited a slight improvement in robustness when PVA fibers were partially replaced with CaCO₃ whiskers and micro hooked steel fibers, as depicted in Figures 8A–G. The microscopic mechanisms of CaCO₃ whisker shown in Figure 7 have the potential to enhance the quantity and evenness of micro-cracks within the mortar matrix (Cao et al., 2015; Ma et al., 2017; Pan et al., 2018). Ma et al. (2017) discovered that additional crack sources could potentially contribute to the multiple cracking performance of SHCC (Pan et al., 2018). As a result, it is possible to improve the robustness of the tensile stress-strain curves.

The addition of steel fibers and CaCO₃ whiskers to replace some PVA fibers increased the tensile strength of PVA-SHCC. Compared to PVA-SHCC, when the whisker content maintained at 1%, the tensile strength exhibited an increase from 5.48 MPa to 6.32 MPa with the steel fiber content ranging from 0.25% to 0.75%. Increasing the amount of whiskers further enhanced the tensile strength to a small extent, despite the further reduction in PVA fiber content. The enhancements could be credited to the higher peak stress for crack bridging offered by steel fibers with hooks and the reinforcing effect at a microscopic level provided by CaCO₃ whiskers. The whiskers and steel fibers have greater modulus of elasticity and stiffness compared to PVA fibers. By limiting the development of cracks, these rigid fibers can effectively improve the principal tensile stress of the material, thus increasing its tensile strength (Rawat et al., 2022; Qasim et al., 2023). On the contrary, PVA fiber is a kind of flexible synthetic fiber that has a lesser impact on the tensile strength compared to steel fiber and CaCO₃ whisker.

As depicted in Figure 8; Table 3, MsHySHCC-1 exhibited greater tensile strength and ultimate tensile strain in comparison to PVA-SHCC, with 1% CaCO₃ whiskers hybrid 0.25% steel fibers partially replacing 0.25% PVA fibers. Nevertheless, a gradual deterioration in strain hardening behavior of SHCCs was observed when the PVA fiber content was reduced while the steel fiber and CaCO₃ whisker amounts were increased. It implies the strain hardening behavior dominates by the PVA fibers. The discrepancy in total quantity between PVA fiber and hooked steel fiber arises due to the constant volume content condition. The main determinant for enhancing hardening performance primarily relies on the quantity of fibers rather than just the fiber content. A decrease in the number of fiber pieces results in a reduction in the effective stress caused by fibers, consequently resulting in an unsaturated cracking performance and an unstable hardening process. Meanwhile, although the number of whiskers is very high, the size of the whiskers is tens of times smaller than that of the PVA fibers, so the whiskers can only have a positive effect on microscopic cracks in the matrix. Consequently, the SHCCs don’t exhibit significant ductility due to the excessive replacement of PVA fibers with whiskers and hooked steel fibers.

As shown in Table 3; Figure 10, based on the findings, it can be inferred that incorporating PVA fiber, steel fiber, and CaCO₃ whisker in the design process of multi-scale fibers is a viable approach to simultaneously enhance the strength and ductility in SHCC.

Available literatures have confirmed that double-line model can be used to describe the tensile constitutive relationship of PVA-SHCC, as illustrated in Figure 11. To the SHCC reinforced by steel and PVA hybrid fibers, it may have obvious post-peak softening stage due to the introduction of hooked steel fibers. Generally, the parameter of fiber reinforcing factor R I v dominates the softening behavior. Previous literatures (Ramasamy and Shanmughasundaram, 2018; Tinoco and Silva, 2021; Deshpande et al., 2019.) show that the softening stage should be considered when R I v ≥ 1 , which is presented as a triple-line constitutive model, as illustrated in Figure 11. However, through the experimental results in this paper, it is found that when CaCO₃ whiskers are added, a relatively obvious softening behavior begins to appear when R I v ≥ 0.876 . This is because the microscopic reinforcing and toughening effects of CaCO₃ whiskers improves the matrix properties and then enhances the pullout behavior of steel fibers and PVA fibers (Cao et al., 2015; Pan et al., 2018). Therefore, a higher residual bearing capacity for post-peak stage can be achieved, i.e., a more noticeable softening stage for tensile stress-strain curves.

The triple-line model needs to determine three key parameters, namely, stable cracking stress σ s s , ultimate tensile stress σ c u and ultimate tensile strain ε c u , as illustrated in Figure 11. For the convenience of calculation, the first-peak stress σ 0 is numerically equal to the stable cracking stress σ s s approximately, that is, the stress at the beginning of the strain hardening behavior. Therefore, the relationship of tensile stress σ ε and tensile strain ε for MsHySHCCs can be expressed as Eq. 1. σ ε = E c ε 0 ≤ ε < σ s s E c σ s s + k 1 ε − σ s s E c σ s s E c ≤ ε < ε c u k 2 σ c u ε c u ≤ ε (1)

Where k 1 is the hardening coefficient, k 1 = σ c u − σ s s ε c u − σ s s / E c ; E c is the elastic modulus of designed MsHySHCCs, which can be empirically obtained by fitting experimental data in this study and E c = − 10.3 R I v 2 + 27.8 R I v + 1.8 ; k 2 is the softening coefficient, k 2 = ε c u ε 4 R I v 2 ; R I v is the fiber reinforcing parameter (Ou et al., 2011; Ning et al., 2015), and R I v can be obtained by Eq. 2. R I v = ∑ i = 1 n β V i l i d i E i E s f p (2)

The interfacial bonding parameter β can be considered as 1.0 for PVA fiber and 1.2 for steel fiber with hooked shape. Additionally, a stiffness factor p can be set as 1.3 for PVA fiber. The fiber volume fraction, fiber geometric length, and diameter are denoted as V i , l i , and d i , respectively. Furthermore, E i represents the fiber elastic modulus, while E s f signifies the elastic modulus of hooked steel fiber.

The ultimate tensile stress σ t u can be calculated by using the equation provided by Li and Leung in 1992, as shown in Eq. 3. The first-peak stress σ 0 can be calculated by Eq. 4. The hardening coefficient g can be determined by Eq. 5. The interfacial bonding strength τ i can be determined according to Table 4. σ t u = g σ 0 (3) σ 0 = V i , p v a τ i , p v a 2 l i , p v a d i , p v a + V i , s f τ i , s f 2 l i , s f d i , s f × F b e (4) g = 2 4 + f 2 1 + e π f / 2 (5)

The hooked steel fiber characteristic parameter F b e was set as 1.7. Additionally, the hardening effect was taken into consideration using the factor f (Ahmed et al., 2007). In this study, the f values assigned to the matrix with 0%, 1%, and 2% CaCO₃ whiskers were 0.085, 0.105, and 0.125 respectively.

To characterize the ductility of SHCC, it is necessary to theoretically calculate the ultimate tensile strain ε t u (Lin and Li, 1997), as provided in Eq. 6. ε t u = δ t u x d (6)

Based on earlier research (Li and Stang, 1997; Lin and Li, 1997; Kanda and Li, 2000), the ultimate crack opening displacement δ t u can be calculated by Eq. 7. δ t u = l ∼ f 2 c − 2 3 c (7)

Where c = γ l ∼ f 2 d ∼ f ; γ is a dimensionless factor (Wang et al., 1988; Shao et al., 1993), set as 0.5 in this study (Wu and Li, 1999). The parameter l ∼ f and d ∼ f signifies the generalized mean value of fiber length and fiber diameter, respectively.

According to previous literature (Wu and Li, 1992; Alwan, 1994; Kanda and Li, 1998; Wu and Li, 1995), the crack spacing value x d can be determined by Eq. 8. x d = l ∼ f − l ∼ f 2 − 2 π ψ l ∼ f x 2 − 2 F c ¯ m c (8)

Where ψ = 4 π g , x = V m σ m u d ∼ f 4 V f τ ∼ f , F c ¯ m c = ⁡ exp − 1 λ c ¯ 0 c ¯ m c m and c ¯ 0 = π 2 K m σ m u 2 . The tensile strength of mortar σ m u for 0%, 1% and 2% whiskers is taken as 3.0, 3.5, and 3.75 MPa, respectively; V m is the volume fraction of mortar; the Weibull parameter m is taken as 2 and the normalized flaw size is set as 10 μm (Wu and Li, 1992); the fracture toughness of mortar is set as 0.37 MPa·m^1/2 (Ahmed et al., 2007).

Figures 12, 13 showcase the theoretical findings related to the tensile parameters and stress-strain curves. It can be seen that the proposed model adequately characterizes the tensile stress-strain correlation of MsHySHCC. Nevertheless, to a certain degree, the proposed model overestimates the cracking saturation degree of MsHySHCC and traditional PVA-SHCC during the experiment. Consequently, the calculated ultimate tensile strain values surpass the corresponding experimental measurements.