# Investigation into the Flexural Toughness and Methods of Evaluating Ductile Concrete
## 1 Introduction  
Concrete is one of the most widely used construction and building materials due to its low cost, high strength, and versatility [1; 2]. However, the poor tensile behavior, brittle failure mode, and low deformability of concrete limit its application in civil engineering [3; 4]. One way to overcome these drawbacks is to add different fibers, such as steel, polyvinyl alcohol, and polypropylene fibers, to obtain higher tensile strength, better flexural toughness, and excellent seismic performance [5; 6; 7; 8]. Nowadays, FRC is used in a variety of applications such as new civil structures, the repair and retrofit of building structures, and underground structures to improve fatigue resistance and seismic performance [9; 10; 11]. Flexural toughness fundamentally reflects the toughening effects of fibers and the inner structural performance of matrices; reinforced concrete structures are designed through their application. Many researchers have focused on the flexural behavior and toughness properties of FRC. [PERSON] found that the flexural strength of steel fiber-reinforced concrete increased with concrete age and fiber volume fraction, but the first crack development significantly decreased by increasing the fiber volume fraction [12]. [PERSON] et al. investigated the influence of fiber volume content and concrete strength on the flexural behavior of steel fiber-reinforced concrete and observed that the fiber volume content significantly affected the initial and residual flexural toughness ratio of the specimens [13]. [PERSON] et al. presented that a high-strength matrix with a high-strength fiber gave the best performance from the viewpoint of flexural and toughness performance according to a three-point flexural loading test [14]. The test results showed that using straight steel and micropolyvinylalcohol fibers produced composites demonstrating stable deflection-hardening with a multiple cracking phenomenon [15]. [PERSON] et al. reported that a hybrid fiber-reinforced concrete showed significant improvements in the flexural toughness of the concrete and correlated the fractured surface roughness with the flexural toughness of the concrete by topographical analysis [16]. Previous research has shown that the addition of fibers improves post-cracking behavior, reduces the opening of cracks, and counteracts their propagation, as well as increases the toughness of concrete owing to their deboning and pull-out failure mechanisms [15; 17]. The specimen response in a bending test was also consistent with the tensile properties of a strain hardening response material, and an inverse analysis was discussed by [PERSON] et al. [18; 19; 20; 21]. However, there is no consistent and recognized flexural toughness evaluation method for FRC, especially for special FRC that exhibits strain hardening behavior under tension, high toughness under compression, and is flexural, which are termed engineered cementitious composites (ECCs) [22; 23; 24], high-performance fiber-reinforced cementitious composites (HPFRCCs) [25; 26], strain hardening cementitious composites (SHCCs) [27; 28], and high ductile fiber-reinforced concrete (HDC) [29; 30; 31; 32]. [PERSON] et al. pointed out that the ASTMC-1018 method is not appropriate for evaluating the toughness of ultra-high toughness cementitious composites, and modified JSCE-SF4 by expanding its range of deflection limit [33]. [PERSON] et al. recommended additional toughness parameters for the evaluation of toughness results obtained by ASTMC-1609 [34]. [PERSON] et al. evaluated the flexural toughness of steel fiber-reinforced lightweight aggregate concrete by using ASTMC-1018, ASTMC-1609, JSCE-SF4, and JG/T 472-2015 methods. They considered that the JG/T 472-2015 method could reveal the influence of the fiber in terms of the pre-peak and post-peak behavior [35]. In other cases, the evaluation methods mentioned above were directly used or slightly modified to estimate the toughness of FRC; therefore, the shortcomings of the existing methods have not been conquered.