2.2.3. Single-Seam Tensile Test

To determine the parameters of the strength criteria and energy criteria, the maximum peak stress and maximum complementary energy should be determined based on the relationship between the bridging-fiber stress and crack width, in addition to the fracture energy of the cement matrix that is determined with the three-point bending test. To determine the relationship between the ECC stress and crack opening width, the multiple cracking needed to be artificially limited to ensure that single-seam cracking occurred in the case of failure. The size of the test piece was the same as that of the dumbbell-type test piece mentioned earlier. Each group contained three test pieces, for a total of five groups of fifteen test pieces. After 28 days of curing under standard conditions, an annular notch with a width of less than 1 mm was cut into the middle of the test piece. The depth of the notch is shown in Figure 7. The notch is cut using diamond-cutting pieces. During the notch-making process, care is taken to avoid damage to the other parts of the test piece to ensure that the ECC only experiences single-seam cracking when under uniaxial tension. The tensile test device is shown in Figure 8.

**Figure 7.** Cutting depth of the single-seam cracking specimen.

**Figure 8.** Single-seam cracking tensile test device.

#### *2.3. Evaluation Method of ECC Toughness*

On the basis of the design theory of ECC micromechanics [3], the ECCs must satisfy both the strength criteria and energy criteria to achieve the characteristics of multiple cracking and strain hardening; otherwise, stress softening occurs during the tensile process. (1) Strength Criteria

The strength criteria set the boundary condition of tensile stress for cracks starting from the initial defect, as they control the cracking process. Continuous multiple cracking must satisfy Equation (1).

$$
\sigma\_{\mathbb{C}} < \min \{ \sigma\_0 \} \tag{1}
$$

where *σ*<sup>c</sup> is the initial crack strength of the matrix and *σ*<sup>0</sup> is the peak stress.

(2) Energy Criteria

The crack-distribution phenomenon of the test conforms to the flat crack-propagation mode and satisfies the energy criteria. The stability of the crack width under a constant external load must satisfy Equation (2).

$$
\sigma\_0 \delta\_0^- \int\_0^{\delta\_0} \sigma(\delta) \mathbf{d} \delta = f'\_{\mathbf{b}} \ge f\_{\text{tip}} \tag{2}
$$

where *δ*<sup>0</sup> is the displacement corresponding to the bridging-fiber peak stress, *σ*(*δ*) denotes the relationship between the bridging-fiber stress and crack opening width, *J* <sup>b</sup> is the maximum complementary energy and *J*tip is the fracture energy of the matrix material.

Theoretically, *σ*0/*σ*<sup>c</sup> ≥ 1 and *J* b/*J*tip ≥ 1 can achieve stable tensile strain-hardening characteristics. However, in practice, because of the existence of uncertainty factors such as material fluctuation, an uneven manufacturing process, and test errors, satisfying the requirements for strain-hardening characteristics is difficult. Research shows that only PCM = *σ*0/*σ*<sup>c</sup> ≥ 1.3 and PSH = *J* b/*J*tip ≥ 2.7 can meet the characteristics of multiple cracking and strain hardening [43], where MCP is the multiple cracking performance and PSH is the pseudostrain hardening.

Based on the calculations of Equations (3)–(6) recommended by Tada [41], which are combined with the three-point bending test, the fracture energy of the cement matrix can be calculated.

$$J\_{\rm tip} = \frac{K\_{\rm m}^2}{E\_{\rm m}} \tag{3}$$

$$K\_{\rm m} = \frac{3(F + 10^{-3} \text{mg}/2)10^{-3} \text{L} \sqrt{a}}{2bh^2} f(a) \tag{4}$$

$$f(a) = \frac{1.99 - a(1 - a)\left(2.15 - 3.93a + 2.7a^2\right)}{(1 + 2a)(1 - a)^{3/2}}\tag{5}$$

$$
\alpha = \frac{a}{h} \tag{6}
$$

where *<sup>J</sup>*tip (J/m2) is the fracture energy, *<sup>K</sup>*<sup>m</sup> (MPa·m1/2) is the fracture toughness, *<sup>E</sup>*<sup>m</sup> (GPa) is the tensile modulus of elasticity, *F* (kN) is the three-point bending peak load, *m* (kg) is the test-piece quality, *g* (m/s2) is the gravitational acceleration, *L* (m) is the span of the three-point bending test piece, *a* (m) is the notch depth, *b* (m) is the width of the test piece, *h* (m) is the height of the test piece and *f*(*α*) is the test-piece shape parameters.

According to the single-seam tensile test, the relationship between the stress and crack opening width can be obtained, and the maximum complementary energy can be obtained by integrating it with the axis where the stress is located, as shown in Figure 9.

**Figure 9.** Relationship between the crack opening width and bridging-fiber stress.
