*4.1. Brittleness Evaluation Index*

The performance of a material's brittleness is affected by its own properties as well as the specimen shape, size effect, and loading conditions. In the evaluation of material brittleness, attention should be paid to its ability to resist plastic deformation before the peak, the slope of the post-peak stress–strain curve and residual strength, and other characteristic factors, and the two stages before and after the peak need to be considered comprehensively when characterizing the brittleness of materials. Therefore, an approach based on the full stress–strain characteristics of high-strength concrete is an effective brittleness evaluation method to characterize the difficulty of brittle failure and the degree of brittleness of high-strength concrete in the form of energy parameters.

In the energy evolution process of concrete, the more strain energy is absorbed before the peak, the faster the stress–strain curve will drop after the peak, and the less additional energy will be provided by the external work required for concrete fracture after the peak; the energy required to cause concrete fracture mainly comes from the elastic strain energy absorbed before the peak, while the dissipative energy before the peak and the magnitude of the external work after the peak directly affect the fracture degree of the concrete after the peak. Based on this, it is proposed that the pre-peak elastic strain energy accumulation rate and the pre-peak dissipative energy dissipation rate should be used to define the pre-peak brittleness of concrete based on the whole process of concrete failure. The pre-peak brittleness index can be expressed as:

$$B\_{\rm pre} = 1 - \frac{W\_{\rm D}}{W\_{\rm E(\rm B)} + W\_{\rm D}} = \frac{W\_{\rm E(\rm B)}}{W\_{\rm E(\rm B)} + W\_{\rm D}} = \frac{\sigma\_{1\rm p}^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu\sigma\_{1\rm P}\sigma\_3}{2\mathbb{E}\left(\int\_0^{\varepsilon\_{1\rm p}} \sigma\_1 \mathbf{d}\,\varepsilon\_1 + 2\sigma\_3\varepsilon\_{3\rm P}\right)}\tag{17}$$

where *W*E(B) is the elastic strain energy at the peak point and *W*<sup>D</sup> is the dissipative energy before the peak.

*B*pre represents the ability to store elastic strain energy before the peak, and the value range is (0,1). When the concrete is in the ideal elastic state, *B*pre = 1, whereas when the concrete is in the fully plastic state, *B*pre = 0. Therefore, the larger the value of *B*pre, the higher the brittleness level of concrete.

Regarding the energy required for concrete failure, the less additional energy is provided from the outside after the peak, the higher the proportion of elastic strain energy that is stored before the peak and the more energy is provided for the self-fracture of concrete after the peak, which indicates that the brittleness of concrete is increased. Therefore, the release rate of post-peak elastic energy is used to define the post-peak brittleness of concrete, and the post-peak brittleness evaluation index can be expressed as

$$\begin{split} B\_{\text{post}} &= 1 - \frac{W\_{\text{F}(\text{post})}}{W\_{\text{E}(\text{B})} + W\_{\text{F}(\text{post})} - W\_{\text{r}}} = \frac{W\_{\text{E}(\text{B})} - W\_{\text{r}}}{W\_{\text{E}(\text{B})} + W\_{\text{F}(\text{post})} - W\_{\text{r}}} = \frac{W\_{\text{E}(\text{B})} - W\_{\text{r}}}{W\_{\text{f}}} \\ &= \frac{\sigma\_{\text{1p}}^{2} - \sigma\_{\text{1r}}^{2} - 4\mu \left(\sigma\_{\text{1p}} - \sigma\_{\text{1r}}\right) \sigma\_{\text{3}}}{2\mathbb{E}\left[\int\_{\epsilon\_{\text{1p}}}^{\epsilon\_{\text{1p}}} \sigma\_{1} \mathrm{d}\epsilon\_{1} + 2\sigma\_{\text{3}} \left(\epsilon\_{\text{3r}} - \epsilon\_{\text{1r}}\right) \right] + \left[\sigma\_{\text{1p}}^{2} - \sigma\_{\text{1r}}^{2} - 4\mu \left(\sigma\_{\text{1p}} - \sigma\_{\text{1r}}\right) \sigma\_{\text{3}}\right]} \end{split} \tag{18}$$

where *W*F(post) is the additional energy provided by the external testing machine, *W*<sup>r</sup> is the residual elastic strain energy, and *W*<sup>f</sup> is the fracture energy.

*B*post characterizes the ability of concrete to maintain self-fracture and crack propagation at the post-peak stage, and the value range is (0, 1). When concrete is in an ideal plastic state, *B*post = 0; when concrete is in an ideal brittle state, *B*post = 1. Therefore, the larger the value of *B*post, the more obvious the brittleness of concrete.
