*4.2. Brittleness Evaluation Method*

Since both *B*pre and *B*post are positively correlated with the brittleness of concrete, a multi-index multiplicative synthesis method is used to establish parameters that can comprehensively characterize the pre-peak and post-peak brittleness index. The evaluation results of this method are continuous and monotonous, i.e., the product of *B*pre and *B*post is used to characterize the brittleness index of the whole stress–strain process as follows:

$$B = B\_{\rm pre} B\_{\rm post} \tag{19}$$

The value range of the brittleness index *B* is (0, 1). For the ideal brittle state of concrete, *B* = *B*pre = *B*post = 1; for the ideal plastic state of concrete, *B* = *B*pre = *B*post = 0, so when the brittleness index increases from 0 to 1, the failure behavior of concrete changes from plastic to brittle. The method of comprehensively characterizing the brittleness evaluation index combines the pre-peak and post-peak brittleness evaluation indexes to establish a brittleness evaluation method that reflects the whole process of concrete failure.

### *4.3. Verification and Analysis*

### 4.3.1. Experimental Verification

According to the calculation method of the brittleness evaluation index *B*, the triaxial compression test data of C60 and C70 concrete under different confining pressures are analyzed. The calculation results are shown in Table 4. When the confining pressure is 0 MPa, the brittleness evaluation indexes (*B*) of C60 and C70 concrete are 0.329 and 0.402, respectively; since the brittleness evaluation index is positively correlated with the brittleness level, the brittleness of the concrete is also increased at this time. When the confining pressure is 20 MPa, the brittleness evaluation indexes (*B*) of C60 and C70 concrete are 0.005 and 0.006, respectively, indicating that the brittleness of concrete is low at this time. Under the same confining pressure, the brittleness evaluation index *B* of C70 concrete is greater than that of C60, indicating that the strength of concrete is positively correlated with its brittleness index. The higher the concrete strength, the higher the brittleness level. Figure 9 shows the variation of brittleness index with confining pressure; the brittleness indexes *B*pre, *B*post and *B* of C60 and C70 concrete decrease with the increase of confining pressure. When the confining pressure increases from 0 MPa to 20 MPa, the brittleness index of C60 and C70 concrete decreases by 98% and 99%, respectively, indicating that the confining pressure inhibits the development of the brittleness level of concrete, and the brittleness evaluation index of C70 concrete is more sensitive to the inhibition of the confining pressure. By fitting the calculated value of the brittleness evaluation index *B*, the relationship between the brittleness index and the confining pressure of C60 and C70 high-strength concrete is obtained, respectively, as shown below. Equations (20) and (21) reflect the brittleness characteristics of high-strength concrete under different confining pressures to a certain extent.

$$\text{C60}: \ B = 0.008 \ + \ 0.32 \cdot \text{e}^{-\frac{\sigma\_3}{2.56}}, \text{R}^2 = 0.999\tag{20}$$

$$\text{C70}: B = 0.008 + 0.39 \cdot \text{e}^{-\frac{\sigma\_3}{2.81}} \tag{21}$$


Notes: the standard deviation of brittleness evaluation indexes for C60 and C70 is 0.0029 and 0.0044, respectively.

*Crystals* **2020**, *10*, 1099

Comparing the brittleness evaluation indexes calculated by different evaluation methods in Figure 9, we can see that the brittleness index *B* in this paper has a high consistency with the trend of *B*<sup>1</sup> and *B*2, and the values of *B*, *B*1, and *B*<sup>2</sup> gradually decrease as the confining pressure increases, indicating that the brittleness of the concrete specimen gradually decreases, which is basically consistent with the experimental trend, indicating that the three brittleness indicators are negatively correlated with the confining pressure. However, *B*<sup>1</sup> and *B*<sup>2</sup> only consider the impact of recoverable elastic energy on the brittleness of concrete at the pre-peak stage, ignoring the effect of the additional energy provided by external work at the post-peak stage on the brittleness level of concrete; thus, they cannot well reflect the brittle behavior of the whole process of concrete failure, which indicates that these two brittleness characterization methods *B*<sup>1</sup> and *B*<sup>2</sup> have certain limitations. The brittleness level reflected by *B*<sup>3</sup> is contrary to reality; the value of *B*<sup>3</sup> gradually increases with the increase of confining pressure, indicating that *B*<sup>3</sup> is positively correlated with the confining pressure, and the value range of *B*<sup>3</sup> is larger, which is not conducive to measuring the brittleness level of the material, as the numerator (fracture energy) and denominator (nominal stress) in the expression do not belong to the same order of magnitude, so the physical meaning of *B*<sup>3</sup> is not sufficiently clear. The brittleness evaluation index *B* proposed in this paper is established based on the whole process of concrete energy evolution with a clear physical meaning: the value range is (0, 1), and the value of *B* is continuous and monotonous, so it has good adaptability. In summary, the brittleness level represented by the brittleness index *B* is more consistent with the physical test results, and it has a better evaluation effect on the brittleness of high-strength concrete materials under different confining pressures. *Crystals* **2020**, *10*, x FOR PEER REVIEW 15 of 19 increases with the increase of confining pressure, indicating that *B*3 is positively correlated with the confining pressure, and the value range of *B*3 is larger, which is not conducive to measuring the brittleness level of the material, as the numerator (fracture energy) and denominator (nominal stress) in the expression do not belong to the same order of magnitude, so the physical meaning of *B*3 is not sufficiently clear. The brittleness evaluation index *B* proposed in this paper is established based on the whole process of concrete energy evolution with a clear physical meaning: the value range is (0, 1), and the value of *B* is continuous and monotonous, so it has good adaptability. In summary, the brittleness level represented by the brittleness index *B* is more consistent with the physical test results, and it has a better evaluation effect on the brittleness of high-strength concrete materials under different confining pressures.

**Figure 9.** Relationship between brittleness evaluation index and confining pressure of high-strength concrete. (**a**) C60; (**b**) C70. **Figure 9.** Relationship between brittleness evaluation index and confining pressure of high-strength concrete. (**a**) C60; (**b**) C70.

### 4.3.2. Analysis of Influencing Factors 4.3.2. Analysis of Influencing Factors

its brittleness level.

The relationship between the characteristic strength of concrete under triaxial compression and the brittleness evaluation index *B* is shown in Figure 10. The figure shows that the brittleness evaluation index *B* is negatively correlated with the peak strength and residual strength, showing an exponential function relationship. With the increase of the peak strength and residual strength, the brittleness evaluation index *B* gradually decreased, and the brittleness level of concrete gradually decreased, indicating that the confining pressure can inhibit the initiation and propagation of micro-cracks in concrete and increase the threshold stress for micro-crack initiation The relationship between the characteristic strength of concrete under triaxial compression and the brittleness evaluation index *B* is shown in Figure 10. The figure shows that the brittleness evaluation index *B* is negatively correlated with the peak strength and residual strength, showing an exponential function relationship. With the increase of the peak strength and residual strength, the brittleness evaluation index *B* gradually decreased, and the brittleness level of concrete gradually decreased, indicating that the confining pressure can inhibit the initiation and propagation of micro-cracks in concrete and increase the threshold stress for micro-crack initiation and propagation in concrete, thereby improving the load-bearing capacity of concrete and reducing its brittleness level.

and propagation in concrete, thereby improving the load-bearing capacity of concrete and reducing

C70.

**Figure 10.** Relationship between brittleness evaluation index and strength parameters. (**a**) C60; (**b**) **Figure 10.** Relationship between brittleness evaluation index and strength parameters. (**a**) C60; (**b**) C70.

The relationship between the brittleness index *B* and the elastic energy stored before the peak, the total energy before the peak, the additional energy and the fracture energy is shown in Figure 11. It can be seen from Figure 11 that the brittleness evaluation *B* has an exponential function and negative correlation with the elastic energy stored before the peak, the total energy before the peak, the additional energy, and the fracture energy. In the pre-peak stage, with the increase of confining pressure, the load-bearing capacity of the concrete increases. At this time, the propagation and penetration of the micro-cracks inside the specimen needs to dissipate more energy, so *W*<sup>D</sup> gradually increases with the increase of confining pressure. The ratio of *W*D to *W*F(pre) increases gradually, while that of *W*E(B) to *W*F(pre) decreases gradually and the storage capacity of elastic energy decreases gradually, resulting in the gradual weakening of the brittleness level of concrete. In the post-peak stage, as the confining pressure increases, *W*r and *W*F(post) gradually increases and the ratio of *W*F(post) to *W*f gradually increases. This is because the confining pressure increases the threshold stress for crack initiation and improves the residual strength of the concrete specimen at the post-peak stage; furthermore, the self-sustaining fracture ability of the specimen weakens at the post-peak stage, thus weakening its brittleness. The relationship between the brittleness index *B* and the elastic energy stored before the peak, the total energy before the peak, the additional energy and the fracture energy is shown in Figure 11. It can be seen from Figure 11 that the brittleness evaluation *B* has an exponential function and negative correlation with the elastic energy stored before the peak, the total energy before the peak, the additional energy, and the fracture energy. In the pre-peak stage, with the increase of confining pressure, the load-bearing capacity of the concrete increases. At this time, the propagation and penetration of the micro-cracks inside the specimen needs to dissipate more energy, so *W*<sup>D</sup> gradually increases with the increase of confining pressure. The ratio of *W*<sup>D</sup> to *W*F(pre) increases gradually, while that of *W*E(B) to *W*F(pre) decreases gradually and the storage capacity of elastic energy decreases gradually, resulting in the gradual weakening of the brittleness level of concrete. In the post-peak stage, as the confining pressure increases, *W*<sup>r</sup> and *W*F(post) gradually increases and the ratio of *W*F(post) to *W*<sup>f</sup> gradually increases. This is because the confining pressure increases the threshold stress for crack initiation and improves the residual strength of the concrete specimen at the post-peak stage; furthermore, the self-sustaining fracture ability of the specimen weakens at the post-peak stage, thus weakening its brittleness. *Crystals* **2020**, *10*, x FOR PEER REVIEW 17 of 19

**Figure 11.** Relationship between brittleness evaluation index and energy parameters. (**a**) C60; (**b**) **Figure 11.** Relationship between brittleness evaluation index and energy parameters. (**a**) C60; (**b**) C70.

### C70. **5. Conclusions**

1. Under different confining pressures, the input energy and dissipative energy of C60 and C70 high-strength concrete specimens increase with the increase of axial strain, and the elastic strain energy shows a trend of first increasing and then decreasing. After the specimen reaches the peak strength, the elastic strain energy decreases gradually, and the dissipative energy increases gradually, reaching maximum and minimum values until the failure of the specimen. When the high-strength concrete specimen is damaged, the ratio of the additional energy *W*F(post) provided by the outside world to the fracture energy is proportional to the confining pressure.


**Author Contributions:** The author contributions of the paper are as follows: R.Z., H.C., and M.L. proposed the conceptualization and methodology; R.Z. and R.H. conducted the tests; L.Z. and M.L. analyzed the data; R.Z. wrote the paper; H.C. reviewed and edited the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation of China (NO.51474004; NO.51874005).

**Conflicts of Interest:** The authors declare no conflict of interest.
