Polyvinyl Alcohol Fiber Length Optimization for High Ductility Cementitious Composites with Different Compressive Strength Grades

+e fiber length has a significant impact on the fiber bridging capacity and the mechanical properties of high ductility cementitious composites (HDCCs), which is related to fiber/matrix interfacial bonding. However, this fundamental knowledge of HDCCs design has rarely been investigated systematically. To this end, this study deeply investigates the effect of the fiber length on the bridging stress and the complementary energy with various fiber/matrix interfacial bonds in theory. +en, the mechanical performances of HDCCs with various fiber lengths and compressive strengths were evaluated experimentally. In micromechanical design, longer fibers can achieve stronger bridging stress and more sufficient complementary energy regardless of the fiber/matrix interfacial bonding properties. However, it should be noted that the increase in bridging capacity was quite slow for the overlong fibers and excessive interfacial bonding. +e experiments indicated that overlong fibers (18mm and 24mm) easily twined on the mixer blade and were hard to disperse evenly. +e HDCCs with shorter fibers displayed better workability. +e compressive strength was less affected by the fiber length, and most striking differences were less than 5.0%, while the flexural properties and the tensile properties first increased and then decreased when the fiber length ranged from 6mm to 24mm. Consequently, the fibers with lengths of 9mm and the fibers with lengths of 12mm were better candidates for the HDCCs with compressive strengths of 30MPa to 80MPa, and fibers with lengths of 9mm caused the HDCCs to exhibit higher ductility properties in general.


Introduction
High ductility cementitious composites (HDCCs) are a special kind of civil engineering materials that have attracted increasing interest in recent years due to their high tensile strain and robust strain hardening behaviour. Moreover, the excellent crack control ability of HDCCs features the characteristics of multiple cracks rather than a localized crack, and the typical crack width is generally less than 100 μm [1][2][3][4][5]. e unique mechanical properties of HDCCs materials overcome the weaknesses in the inherent brittleness and cracking sensitivity of traditional concrete. Hence, broad application prospects can be expected in civil infrastructures where the structures are resilient, durable, and sustainable [6,7]. In typical HDCCs, high strength and high modulus polyvinyl alcohol (PVA) fibers have been widely employed because of their reasonable price and strong crack-bridging capacity [8]. Currently, PVA-HDCCs with compressive strengths ranging from 30 MPa to 80 MPa have been widely developed to meet various engineering requirements [9][10][11]. To improve the mechanical performances of HDCCs, the study of tailored micromechanical parameters has always been a key research focus. Since the micromechanical bridging theory was introduced by Li et al., it has been a significant tool to guide the material design of HDCCs, which covers the fibers, matrix, and fiber/matrix interface [12,13]. Currently, many researchers focus on tailoring the HDCCs matrix and fiber/matrix interface bond to obtain high tensile ductility or tensile strength. However, few researchers have focused on the tailoring of fiber parameters (tensile strength, elastic modulus, length, and diameter). e tensile strength and elastic modulus of fibers are inherent properties that hardly change and depend on the material characteristics and production technology. In contrast, the fiber length can be easily customized and controlled [14], and it is advisable to tailor the fiber length to develop the full potential of fiber performance and maximize the fiber bridging effect.
e fiber length has a profound impact on the mode of failure and resulting strain capacity of a composite. In general, fibers with a longer length can provide a greater embedment length, and a stronger stress would be needed to achieve fiber debonding and slipping from the matrix [12]. If the fiber length is shorter, the fiber stress will be extremely weak, where the stress is far less than the maximum stress that the fiber can be afforded, and a large number of performance margins will not be utilized. In contrast, if the fiber length is long, the fiber stress will be larger than the fiber strength, the fiber will be more prone to rupture, and the ruptured fiber will no longer bridge cracks during propagation [12]. In addition, it is difficult to easily disperse long fibers. In either case, the inferior ductility and the absence of strain hardening behaviour will occur due to incorrect selection of fiber length [15]. erefore, an optimization of fiber length has great significance for HDCCs design. Figure 1 summarizes the PVA fiber length and diameter adopted in 83 related studies, and typical references can be found in [16][17][18][19][20][21][22][23][24][25][26][27][28]. PVA fibers with moderate lengths of 8 mm and 12 mm were universally employed in HDCCs, even though a length of 12 mm has become a universally accepted length. However, it is unclear which of these two fiber lengths should be selected in HDCCs design or whether shorter or longer fibers are a better candidate to achieve superior characteristics of HDCCs. In addition, whether the appropriate fiber length is related to the matrix properties has not been explored deeply.
Based on the analysis of micromechanical models, Maalej and Li [12,14] proposed that the optimum fiber length depends on the fiber diameter, fiber strength, and fiber/matrix interfacial bonding strength, beyond which an increase in the fiber length slightly increases the tensile strength but significantly reduces the fracture energy. Additionally, the optimum fiber length decreases with increasing fiber/matrix bonding strength. Lu et al. [29] proposed that the PVA fiber with 12 mm can make the HDCCs exhibit the best ductility by means of theoretical calculation. Although the longer fibers were more likely to rupture, the ultimate tensile strength of HDCCs tended to be higher. Georgiou et al. [30] presented that the use of longer fibers can improve the tensile strength and strain capacity. Furthermore, the mechanical performance of the composites with 12 mm fibers was noticeably better than that of the composites with 8 mm fibers. Sasmal et al. [31] investigated the flexural properties of HDCCs with 8 mm and 12 mm fibers, while they concluded that the variation in the fiber length from 8 mm to 12 mm did not induce any significant improvement in the flexural strength and even showed a slight decrease of 2.0% in the volume fraction, which was distinctly different from the conclusions obtained by Dai et al. [32] who used polyethylene (PE) fiber (6 mm, 12 mm and 18 mm) and seawater sea-sand to prepare HDCCs; they found that the fiber length had no obvious effect on the compressive strength. Longer fiber length had no obvious effect on the tensile strength but led to a significant increase in the tensile strain capacity, from 2.5% for 6 mm fiber to over 7% for 18 mm fiber. Yilmaz et al. [33] investigated the effect of polypropylene (PP) fiber length (6,12, and 18 mm) on strength behaviour of fiber-reinforced cementitious composites. Results showed that the longer fiber length does not necessarily lead to the higher unconfined compressive strength of samples, but the integrity of the sample under the same external load is better. In view of the above conflicting conclusions, it is still necessary to further investigate the effect of the fiber length on the fiber bridging capacity and mechanical performance of HDCCs. In addition, the effect of fiber length is still unclear for the HDCCs with different strength grade.
Accordingly, this paper made an in-depth study on the influence of fiber length on fiber bridging capacity and HDCCs properties by combining experimental and theoretical method. e purpose was conducted to tailor the PVA fiber length to achieve better mechanical performance of HDCCs. To this end, PVA fibers with lengths of 6 mm, 9 mm, 12 mm, 18 mm, and 24 mm were used, and HDCCs with diverse compressive strengths ranging from 30 MPa to 80 MPa were prepared. On the one hand, a systematic result regarding the effect of the fiber length can be achieved by employing shorter/moderate/longer fiber lengths. On the other hand, the strength grades for the PVA-HDCCs are typical, as discussed before. In the present work, the compressive strength, four-point flexural properties, and uniaxial tensile properties of HDCCs were investigated, and then an appropriate fiber length was determined by this series of experiments. In addition, the micromechanical mechanism was elaborated based on the fiber bridging theory. Curves of the fiber bridging stress versus the crack opening length for different fiber lengths and interfacial bonding strengths were drawn, and the effects on the fiber bridging stress σ 0 and the complement energy J b ′ were summarized to guide the selection of fiber parameters.

Micromechanical Theoretical Mechanism of the Affected Bridging Capacity
e fiber bridging micromechanical theory describes the relationship between the bridging stress σ and the crack opening width δ (Figure 2), which links the micromechanical properties (matrix, fiber, matrix/fiber interface property, etc.) to the composite properties (tensile strength, tensile ductility, crack width, etc.) [34]. e fiber bridging capacity can be characterized by the fiber bridging stress σ 0 and the complementary energy J b ′ , as shown in Figure 2 [34]. A detailed introduction to the micromechanical theoretical can be seen in [34].
To attain sufficient strain hardening behaviour and prevent the formation of multiple cracks in HDCCs, the strength criterion (σ 0 ≥ σ cr ) and energy criterion (J b ′ ≥ J tip ) need to be satisfied synchronously [34]. e strength criterion requires that the fiber bridging stress σ 0 exceeds the initial cracking strength σ cr , which is the prerequisite for steady-state cracking. e energy criterion requires that the complementary energy J b ′ exceeds the crack tip toughness J tip , which provides an adequate energy supply for multiple saturated cracks [35,36]. e failure to meet these two key criteria will result in the Griffith cracking pattern being valid for an unrestricted crack width when the crack is extended [37,38]. For the micromechanical design of HDCCs, pseudo-strain hardening (PSH) indexes (PSH S � σ 0 /σ cr and PSH E � J b ′ /J tip ) were proposed to ensure multiple saturated cracks and robust tensile strain hardening [39]. erefore, an appropriate fiber length should make the bridging capacity achieve the fundamental requirements of PSH indexes.
In essence, the excellent performances of HDCCs are derived from the sufficient bridging capacity of fibers. Ideally, longer fibers have greater fiber bridging capacity, whereas the fibers will rupture during the bridging process once the fiber stress exceeds the fiber tensile strength. e fiber rupture zone is illustrated in Figure 3, and detailed descriptions about the critical rupture length can be found in the literature [13]. e fiber rupture effect is caused by a large fiber embedment length or excessive fiber/matrix interfacial bonding. Generally, a longer fiber can provide a larger embedment length, and a higher-strength matrix will result in stronger fiber/matrix interfacial bonding; hence, the fiber rupture effect will be significant. Ruptured fibers can no longer bridge cracks, and then the bridging stress σ 0 and the complementary energy J b ′ will experience losses [40], as shown in Figure 4. e fiber ruptured effect weakens the fiber bridging capacity (σ 0 , J b ′ ). Different fiber length and fiber/matrix interfacial bonding can cause different degrees of fiber rupture, and its effect on fiber bridging capacity will also be significantly different. e detailed analysis will be given in the following.
To intuitively demonstrate the effect of the fiber length combined with fiber/matrix interfacial bonding, Figure 5 shows the affected fiber rupture zone, and Figure 6 draws the fiber bridging stress σ versus crack opening δ curves for fiber lengths L f ranging from 3 mm to 24 mm and fiber/matrix interfacial bonding strengths τ 0 ranging from 1.0 MPa to 8.0 MPa. e selected ranges for the fiber length L f and fiber/matrix interfacial bonding strength τ 0 cover the main values in related research.  23% 62% Fiber length Fiber diameter Figure 1: Adopted PVA fiber lengths and diameters in related research [16][17][18][19][20][21][22][23][24][25][26][27][28].

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As shown in Figure 5, the fibers have no rupture or a lower rupture proportion when the fiber/matrix interfacial bonding is weak. For example, the minimal critical rupture length reaches 5.3 mm when τ 0 � 1.0 MPa, which means that fibers with lengths less than 10.6 mm will survive. With the increase in interfacial bonding τ 0 , the critical rupture length decreases. e fiber rupture effect will occur as long as the fiber length exceeds 1.34 mm when τ 0 � 8.0 MPa, and the rupture proportion of fibers is considerable. As a result, Figure 6(a) shows that the σ-δ curves are gentle, and the fiber bridging stress σ 0 rapidly increases with the fiber length. With the increase in fiber/matrix interfacial bonding τ 0 , the fiber rupture effect is more critical, and the fiber bridging stress drops rapidly and is accompanied by a tighter cracking opening δ 0 after reaching the ultimate point; hence, the σ-δ curves become sharp. Furthermore, there are no significant differences in the fiber bridging capacity for the longer fibers, as illustrated in Figure 6(h). e contour profiles of the fiber bridging stress σ 0 and the complementary energy J b ′ are presented in Figure 7. e detailed bridging model can be found in [40] and will not be described here. Figure 7 demonstrates the effects of the fiber length L f and the interfacial bonding τ 0 on the fiber bridging stress σ 0 and the complementary energy J b ′ .
As shown in Figure 7(a), longer fibers can achieve a stronger bridging stress σ 0 even if fiber rupture occurs, but it should be noted that the increase in the bridging stress σ 0 is actually very slow when the fiber length exceeds 12 mm. Figure 7(b) exhibits a similar effect on the complementary energy J b ′ . Moreover, for weaker interfacial bonding τ 0 , the effects of the fiber length on the bridging stress σ 0 and the complementary energy J b ′ are more significant than that of higher interfacial bonding τ 0 . With the increase in interfacial bonding τ 0 , the bridging stresses σ 0 are enhanced for fibers with lengths of 6 mm to 24 mm, while the complementary energy values J b ′ increase first and then decrease for the fibers with lengths of 6 mm, 9 mm, and 12 mm, and the values continue to decrease for fibers with lengths of 18 mm and 24 mm.
Based on the requirements of the strength criterion and energy criterion, a higher bridging stress and complementary energy can achieve better mechanical performance for HDCCs. eoretical analyses indicate that longer fibers can perform better than shorter fibers. However, it should be   Crack opening δ (μm) Crack opening δ (μm) Crack opening δ (μm) Crack opening δ (μm) Advances in Materials Science and Engineering 5      Advances in Materials Science and Engineering pointed out that the theoretical analysis is based on the uniform dispersion of fibers. In practice, overlong fibers will inevitably encounter the problem of difficult dispersion, which will cause a decrease in the effective fiber content and will be explored in subsequent experiments.

Materials and Mixture Preparation.
e HDCCs binders were included of Portland cement (P II 42.5 R and P II 52.5 R) and fly ash (Class F), which conformed to Chinese standard GB175-2007 and ASTM C618, respectively. e chemical composition was provided in Table 1. River sand, as fine aggregates, had a maximum size and a fineness modulus of 0.60 mm and 1.40, respectively. A polycarboxylate-based high-performance water reducing admixture (HPWR) combined with hydroxypropyl methyl cellulose (HPMC) was utilized to control the fluidity of the paste. In this study, PVA fibers with 6 mm, 9 mm, 12 mm, 18 mm, and 24 mm were adopted to study the influence of the fiber length on the workability and mechanical properties of HDCCs, and the aspect ratio of PVA fiber ranges from 154 to 615. e physical/mechanical properties and geometry of the PVA fibers were provided in Table 2, and Figure 8 shows visual images of PVA fibers of various lengths.
To achieve a compressive strength that ranges from 30 MPa to 80 MPa for the HDCCs, fly ash/cement ratios of 0.80 and 0.60 and water/binder ratios of 0.30, 0.25, and 0.22 were selected. e mixture proportions of HDCCs are listed in Table 3. For the preparation of HDCCs, the mixture proportion of the matrix is fixed, and only the fiber length is variable. A typical fiber volume fraction of 2% (by total volume) was used. Specifically, P II 52.5 R cement was only used to obtain high-strength HDCCs for the mixture of 0.22-60%, whereas the other mixtures used P II 42.5 R cement. All the HDCCs mixtures were prepared using a mortar mixer with a capacity of 20 L. e preparation steps were as follows. First, the binders and river sand were dry mixed for 2 min; subsequently, the water were gradually  Advances in Materials Science and Engineering added to the dry mixture and mixed for another 5-10 min, and HPWR and HPMC were added to adjust the paste fluidity; finally, the PVA fibers were added and then mixed for another 5 min. e fresh HDCCs were cast into moulds and covered with plastic sheets to prevent moisture loss. e specimens were demoulded after 24 h and then cured at a temperature of 20 ± 1°C and relative humidity of 95% for 28 d. e workability of fresh HDCC was measured using a Hägermann flow table (cone dimensions: 70 mm top diameter, 100 mm base diameter, and 60 mm height) conforming to the requirements in ASTM C1437. e fresh HDCCs exhibited distinct fluidity states with the incorporation of PVA fibers of different lengths.
is is because longer fibers can anchor more cement paste, especially hydrophilic PVA fibers [41]. e anchoring effect restricts the flow of HDCCs paste and leads to worse composite fluidity, as illustrated in Figure 9. In addition, Figure 10(b) shows that fibers with lengths of 18 mm and 24 mm tended to twine on the mixer blade during the mixing procedure, even though this phenomenon was almost eliminated, which resulted in a decrease in the effective content of fibers. Consequently, the fiber bridging capacities will be greatly affected, which will be discussed in subsequent sections. For    Note. e symbol 'x' represents the fiber length, that is, 0.30-80%-6 is a mixture with a water/binder ratio of 0.30, a fly ash/cement ratio of 80%, and PVA fibers with a length of 6 mm. Fiber length (mm) Advances in Materials Science and Engineering fibers with lengths of 6 mm, 9 mm, and 12 mm, twining barely occurred, as shown in Figure 10(a).

Test
Programs. e compressive test was conducted in accordance with standard EN 1015-11. Six cubes of 40 mm × 40 mm × 40 mm were tested for each mixture, and the loading rate was controlled as 50 N/s.
Four-point bending test were conducted in accordance with standard test method for flexural performance of fiberreinforced concrete (ASTM C1609). ree plate specimen of 300 mm × 75 mm × 25 mm were tested for each mixture, a loading span was set as a length of 240 mm, and the midspan deflection was measured by a Linear Variable Differential Transformer (LVDT) placed at the center of each specimen. e loading was applied at a displacement rate of 0.5 mm/min. e device for four-point bending test was shown in Figure 11.
Uniaxial tension test was performed in accordance with the JSCE recommendation for design and construction of high-performance fiber-reinforced cement composites (HPFRCC). ree dogbone-shaped specimens with a cross section of 13 mm × 30 mm were tested for each mixture. e tensile strain was measured by an external LVDT attached diametrically to the specimens. Figure 12 showed the specimen geometry and test setup. e uniaxial tension test was conducted under quasistatic loading conditions at a displacement rate of 0.2 mm/s.

Compressive Properties.
e compressive strength of the matrix and the HDCCs with different fiber lengths are shown in Figure 13(a), and the increase/decrease in the compressive strength with respect to the matrix is shown in Figure 13(b).
As shown in Figure 13 [41][42][43]. On the one hand, the randomly distributed fibers can produce a strong lateral binding force and constrain lateral expansion of HDCCs, which restricts the rapid expansion of cracks and improves the composite compressive properties. Such behaviour can be likened to the lateral binding force generated by stirrups in the confined concrete. e distinguishing feature is that the binding force generated by the fibers is an internal binding force, that generated by the stirrups is an external binding force, and the schematic on two kinds of binding force is shown in Figure 14 [42]. On the other hand, the addition of fibers in cementitious composites can be regarded as flaws, and some artificial pores can be more easily introduced due to the worse workability, which introduces a negative effect on the compressive properties [42,44]. Consequently, the compressive strength of HDCCs was dependent on the combined actions of these opposite aspects. For the 0.30-80%, 0.30-60%, and 0.25-60% series, the positive effect was dominant compared with the negative  effect, and the compressive strength increased due to the incorporation of PVA fibers. Conversely, the negative effect was dominant for the 0.22-60% series because the lower water/binder ratio caused the matrix to be quite compact, and high sensitivity to flaws and artificial pores was exhibited. e literature also demonstrated that shorter fibers increase the compressive strength further compared with that of longer fibers due to better dispersion and consolidation effect, although long fibers can produce stronger bridging stress [41]. As shown in Figure 13(a), the HDCCs with 6 mm fibers achieved slightly higher compressive strength. Nevertheless, Figure 13(b) indicates that the most striking differences in compressive strength were less than 5.0%. e compressive properties were slightly affected by the fiber length, which mainly depended on the matrix properties. When PVA fibers with lengths from 6 mm to 24 mm and a volume content of 2% were added, the compressive strengths of the HDCCs were almost the same, although there was a slight difference.    As mentioned, the ranges of the compressive strength in this study were representative of the PVA-HDCCs. Based on the limitation of the mechanical properties of PVA fibers, most fibers are prone to rupture if a higher-strength matrix (larger than 80 MPa) is employed [40]; hence, the fiber bridging capacity will be insufficient to maintain steady-state propagation of cracks. erefore, the exploration of the effect of the fiber length on the mechanical performance is justified and worthy in this study. Figure 15 shows the flexural stress versus midspan deflection curves obtained from the four-point bending tests, and the flexural properties are summarized in Table 4. As shown in Figure 15, all the HDCCs specimens were characterized by deflectionhardening behaviour and multiple cracks. For identical mixtures with different fiber lengths, the flexural stress/ deformation capacities of the specimens were different due to the diverse bridging capacities, and the cracking patterns in the pure bending region are illustrated in Figure 16(a). For the different mixtures with identical fiber lengths, more significant differences in flexural properties can also be found, and the cracking pattern in the pure bending region is illustrated in Figure 16(b). e stress-deflection curves in the initial cracking stage are shown in Figure 17(a), and the initial cracking strength is shown in Figure 17(b). Significant deflection-hardening behaviour after the initial cracking point can be clearly observed (Figure 17(a)). e fibers across the crack plane hold the applied load and allow the increase in the flexural stress to be maintained. With the expansion of cracks, the sufficient bridging stress was beyond the initial cracking strength. In general, the initial cracking strength was significantly affected by the matrix properties [45]. e effect of the fiber bridging capacity on the flexural properties is mainly reflected in the postcracking stage. e bridging stress is transferred to the surrounding matrix by the fiber/matrix interface, and then cracks are formed again when the transferred stress exceeds the cracking strength.

Four-Point Bending Properties.
e above process continues accompanied by multiple cracks in the pure bending region [46]. Finally, the bridging effect cannot sustain the tensile stresses caused by the external moment, and the flexural stress decreases gradually with unstable opening of the dominant crack. Accordingly, a stronger bridging capacity can result in a higher ultimate flexural strength and larger deflection. e effects of the fiber length on the ultimate flexural strength, midspan deflection, and fiber bridging energy dissipation are presented in Figure 18. e area of the curves for the flexural stress versus the midspan deflection from the initial cracking point σ i (y i ) to the ultimate cracking point σ u (y u ) provides the fiber bridging energy dissipation G B under four-point bending.
According to Figure 18, the HDCCs with 9 mm fibers or 12 mm fibers generally achieve a higher ultimate flexural strength and larger midspan deflection regardless of the matrix strength, and shorter or longer fibers cause the HDCCs to exhibit slightly weaker flexural properties. According to the theoretical analysis above, shorter fibers provide a weaker bridging capacity; thus, the results are    easy to understand. However, the longer fibers (18 mm and 24 mm) exhibited twining phenomena and poor dispersion, and the effective content of fibers were less than 2.0%, as can be found in Figure 10. In addition, previous results [45] also showed that the fiber dispersion decreased linearly with the increase of fiber length. us, HDCCs with 18 mm fibers or 24 mm fibers achieve weaker flexural properties than those of other lengths, although they experience a high fiber bridging stress σ 0 and complementary energy J b ′ in theory. In practice, the uneven distribution of longer fibers (18 mm and 24 mm) reduced the bridging capacity of fibers, and the actual bridging capacity was even lower than that of 9 mm and 12 mm fibers. erefore, the experimental results were predictable, and moderate length of PVA fiber (9 mm and 12 mm) brought about better flexural strength and deformation capacity than shorter (6 mm) and longer (18 mm and 24 mm) fibers. e cracking pattern intuitively shows the differences in the performance for fibers with different lengths, as shown in Figure 16. Regardless of the matrix properties, the 9 mm fibers and 12 mm fibers are all the better candidates, and the matrix properties for the two lengths depend on the quality control of preparation procedure.

Uniaxial Tensile Properties.
Based on the flexural properties, 9 mm and 12 mm PVA fibers were preferable candidates. Extremely longer fibers will tend to twine and are hard to disperse evenly, and inferior tensile properties will be expected. erefore, uniaxial tensile tests were conducted only for the HDCCs with fibers with lengths of 9 mm and 12 mm in this part.
e curves of the uniaxial tensile stress versus strain are shown in Figure 19. e ultimate tensile stress and tensile ductility are presented in Figure 20 and Table 4. As shown in Figure 19, all the HDCCs specimens exhibited tensile strain hardening characteristics, and the bridging stress of the 9 mm fibers and the 12 mm fibers all exceeded the matrix first cracking strength. For each mixture, the tensile properties affected by the fiber length were different. In general, the HDCCs with 9 mm fibers exhibited better tensile ductility, and the tensile strengths were slightly lower except for that of the 0.22-60% series.
For the 0.30-80% series, 9 mm fibers and 12 mm fibers made the HDCCs exhibit the same level of tensile properties. e ultimate tensile strength of the 0.30-80%-9 specimen  For the 0.30-60% and 0.25-60% series, the HDCCs with 9 mm fibers achieved tensile ductility values of 2.05% and 1.67%, respectively, which were 117.8% and 159.0% higher than that of the samples with 12 mm long fibers. However, HDCCs with 12 mm fibers can achieve higher ultimate tensile strengths of 4.91 MPa and 5.87 MPa, which were 4.7% and 23.8% higher than those of the 9 mm fibers.
Inferior tensile ductility can be found in the 0.22-60% series regardless of whether the contents of the fibers with lengths of 9 mm or 12 mm were all less than 0.5%. Nevertheless, tensile strain hardening behaviour still appeared. e tensile ductility values of the 0.22-60%-9 and 0.22-60%-12 samples were almost the same and were 0.40% and 0.39%, respectively. In addition, the former exhibited a higher ultimate tensile strength of 6.08 MPa, which was 7.2% higher than the latter. Although 12 mm fibers have a stronger bridging capacity based on micromechanical analysis, the HDCCs with 9 mm fibers achieved better or similar tensile ductility results. It can be concluded that the fiber distribution has a great influence on the tensile properties [47][48][49].
Although 12 mm fibers can achieve stronger bridging capacity, the tensile properties of HDCCs prepared by 12 mm fibers are not completely dominant. According to previous research results [45], shorter fibers can achieve better dispersion, while longer fibers are difficult to disperse evenly. From the results of uniaxial tensile properties, it can be inferred that the fiber dispersion has a significant effect on tensile properties and sometimes even exceeds the influence of fiber bridging capacity. For example, in Zhou's study [47], the tensile ductility of HDCCs reached 1.5% when the fiber dispersion coefficient is 0.72, while the tensile ductility of HDCCs reached 2.5% when the fiber dispersion coefficient is 0.72 increased to 0.78. e tensile ductility of HDCCs was significantly increased by 66.7% with the increase of fiber dispersion coefficient by 8.3%. Similar results can also be found in the studies of Torigoe [48] and Felekog�lu [49]. For HDCCs with different strength, the tensile properties showed similar effects. Due to the significant influence of fiber dispersion, thus it is recommended to select 9 mm fiber because it is easier to be evenly dispersed in the preparation of HDCCs.
Combined with the results of the tensile properties and flexural properties, for the HDCCs with compressive strength grades from 30 MPa to 80 MPa, the 9 mm fibers appear to be a better candidate. On the one hand, the 9 mm fibers can be easily dispersed; on the other hand, the HDCCs with 9 mm fibers achieved a balance in mechanical performance and workability.

Conclusions
e fiber length has a profound impact on the fiber bridging capacity and the mechanical performance of composites. To tailor the optimum fiber length for HDCCs, first, this study theoretically explored the effect of the fiber length on the bridging stress σ 0 and complementary energy J b ′ with various fiber/matrix interfacial bonds τ 0 . en, a series of experiments, including compressive tests, fourpoint flexural tests, and uniaxial tensile tests, were conducted, and five kinds of fiber lengths and four kinds of HDCCs mixtures were employed. e following conclusions can be drawn based on theoretical and experimental investigations: (1) eoretical analysis indicates that longer fibers can achieve a stronger bridging stress σ 0 and sufficient complementary energy J b ′ even if fiber rupture occurs. However, the increase in bridging stress σ 0 and complementary energy J b ′ are actually very slow when the fiber length exceeds 12 mm. With the increase in the interfacial bonding stress τ 0 , the bridging stress σ 0 was enhanced for the fibers with lengths that ranged from 6 mm to 24 mm, while the complementary energy J b ′ first increased and then decreased for the 6 mm, 9 mm, and 12 mm fibers and continued to decrease for the 18 mm and 24 mm fibers.
(2) For the HDCCs with compressive strength grades from 30 MPa to 80 MPa, the compressive properties are less affected by the fiber length. By incorporating PVA fibers, the compressive strengths were enhanced for the 0.30-80%, 0.30-60%, and 0.25-60% series regardless of the fiber length. However, a decreasing tendency was exhibited for the 0.22-60% series. (3) e HDCCs with different fiber lengths have similar initial cracking strengths, although there are fluctuations. e effect of the fiber bridging capacity on the flexural properties is mainly reflected in the postcracking stage. e HDCCs with 9 mm fibers or 12 mm fibers generally achieved higher ultimate flexural strength and larger midspan deflection regardless of matrix strength, and shorter or longer fibers made the HDCCs exhibit slightly weaker flexural properties. In addition, compared with 12 mm fibers, 9 mm fibers resulted in better tensile ductility, and the tensile strengths were slightly lower except for the 0.22-60% series. (4) Considering the fiber dispersion, HDCCs workability, and mechanical properties, the 9 mm fiber is considered to be the appropriate fiber length for HDCCs with compressive strength grades that range from 30 MPa to 80 MPa.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. 18 Advances in Materials Science and Engineering