*3.2. Three-Point Bending Test Results and Fracture Energy of the Cement Matrix*

Figure 13 shows the failure mode of the three-point bending test piece. Because PE fiber was not added to the cement matrix, each group of the test pieces was brittle when they were damaged, and they all broke from the notch. Equations (3)–(6), which are recommended by Tada [42], show that, based on the peak load of the specimen failure, the mass of the specimen and the elastic modulus measured under uniaxial tension, the fracture energy of the five groups of matrixes can be obtained, as listed in Table 4 and Figure 14.

For Group A and Group B, the fracture energies are 72.5 J/m2 and 67 J/m2, respectively, which are small and conducive to achieving a high toughness. For ordinary sand, the matrix fracture energies of Groups B to E are 67.0 J/m2, 90.6 J/m2, 109.6 J/m2 and 96.5 J/m2, respectively. The matrix fracture energy of Group B was the smallest, and that of Group C, Group D and Group E was 35.2%, 63.6% and 44.0% higher than that of Group B. It can be seen that the larger the particle size is, the higher the fracture energy is.

**Figure 13.** Failure mode of the three-point bending test specimen.


**Table 4.** Fracture energy of the ECC matrix with different sand sizes.

**Figure 14.** Fracture energy of the cement matrix with different sand particle sizes.

#### *3.3. Single-Seam Tensile Test Results and Complementary Energy*

Figure 15 shows the failure mode of the single-seam tensile specimen. The failure occurs at the notch, and no cracks appear in other places. The stress–displacement curve is shown in Figure 16. The peak stress and the opening width at the peak stress can be obtained. The complementary energy *J* <sup>b</sup> can be obtained by integrating the axis of the stress, as listed in Table 5.

**Figure 15.** Failure mode of the single-seam tensile specimen.

**Figure 16.** Tensile stress–displacement curve of the single seam with different sand particle sizes.


**Table 5.** Complementary energy of the ECC single-seam cracking test.

## *3.4. Analysis and Discussion of Fracture Toughness*

The multiple cracking performance (MCP) and the pseudostrain hardening (PSH) could be obtained, respectively, according to the matrix's fracture energy *J*tip, the complementary energy *J* <sup>b</sup> and the initial crack strength, which was measured by the three-point bending test, the single-seam tensile test and the uniaxial tensile test, respectively, as shown in Table 6 and Figure 17.

**Table 6.** MCP and PSH values of the desert sand and ordinary sand with different particle sizes.


**Figure 17.** Comparative analysis of MCP and PSH values for the desert sand and ordinary sand.

The MCP of Group A with desert sand and Group B with ordinary sand is 2.88 and 2.33, respectively. The desert sand group's MCP is slightly larger than that of the ordinary sand group, and both are greater than 1.3, which meets the requirements of the strength criteria in Equation (1). Both groups of specimens have obvious characteristics of multiple cracking, which have been verified via the uniaxial tensile stress–strain curve and the specimen failure phenomenon. The PSH of the two groups of specimens is 8.76 and 8.17, respectively, which are both much larger than 2.7. The desert sand group's PSH is slightly larger than that of the ordinary sand group, and both meet the requirements of the energy criteria in Equation (2). The higher PSH of the desert sand group is conducive to achieving a high toughness.

For ordinary sand, the MCP of Groups B to E is 2.33, 2.44, 2.45 and 2.40, respectively, which are all greater than 1.3 and meet the requirements of the strength criteria in Equation (1). The four groups of specimens have the characteristic of multiple cracking. The PSH of the four groups of ordinary sand specimens are 8.17, 5.39, 5.05 and 6.00, which are all greater than 2.7 and meet the requirements of the energy criteria in Equation (2). It can be seen that the smaller the grain size of the ordinary sand is, the easier it is to achieve stable multiple cracking and strain hardening. Similar conclusions were reflected in M. Sahmaran's research [33]. In that research, dolomite limestone sand and gravel sand with a maximum particle size of 1.19 mm and 2.38 mm were used to replace micro-silica sand with maximum particle sizes of 0.2 mm when preparing the ECCs, and the production cost of the ECCs was reduced. The tensile strength and deformation of the ECC materials prepared by using larger grains of sand were reduced to varying degrees.
