*2.1. Experimental Process*

According to the "Specification for Concrete Mix Design" (JGJ55-2011) [19], high-strength concrete with strength grades of C60 and C70 was prepared. Cement used was ordinary Portland cement P II52.5 R, and the coarse aggregate used was basalt gravel. The size of the crushed stone was 5–20 mm and the bulk density was 1455 kg/m<sup>3</sup> . The fine aggregate adopts medium-coarse sand with a fineness modulus of 2.73 and a mud content of 1.45%. The admixture was an NF–F compound high-efficiency admixture, and the proportions of slag and silica fume in C60 and C70 were the same, 73% and 20%, respectively. The mix proportion of C60 and C70 high-strength concrete was determined by an orthogonal experimental design (see Table 1).


**Table 1.** Mixture proportions of C60 and C70 high-strength concrete.

The conventional triaxial loading equipment for high-strength concrete specimens was the ZTCR-2000 low-temperature rock triaxial system (Figure 1). During the test, the concrete specimens were first preloaded to 0.5 MPa, and then confining pressure was applied to the predetermined value at the loading speed of 50 N/s according to the "load" control mode. The test confining pressure was set at 0, 5, 10, 15, and 20 MPa. After the confining pressure reached the predetermined value, the pressure was stabilized for 30 s, and finally, the axial pressure was applied at the loading speed of 0.05 mm/min according to the "displacement" control mode until the concrete specimens was destroyed. The conventional triaxial loading equipment for high-strength concrete specimens was the ZTCR-2000 low-temperature rock triaxial system (Figure 1). During the test, the concrete specimens were first preloaded to 0.5 MPa, and then confining pressure was applied to the predetermined value at the loading speed of 50 N/s according to the "load" control mode. The test confining pressure was set at 0, 5, 10, 15, and 20 MPa. After the confining pressure reached the predetermined value, the pressure was stabilized for 30 s, and finally, the axial pressure was applied at the loading speed of 0.05 mm/min according to the "displacement" control mode until the concrete specimens was destroyed.

*Crystals* **2020**, *10*, x FOR PEER REVIEW 3 of 19

**Figure 1.** Rock mechanics test system. **Figure 1.** Rock mechanics test system.

### *2.2. Analysis of Test Results 2.2. Analysis of Test Results*

*σ* ( ) **MPa** <sup>3</sup>

from high to low. This shows that there are obvious differences in the brittleness characteristics of concrete under different confining pressures. Figure 2 shows the stress–strain curves of C60 and C70 high-strength concrete specimens under different confining pressures. It can be seen from Figure 2 that, when the confining pressure is 0 MPa, there is no obvious yield point in the pre-peak curve of high-strength concrete; after reaching the peak strength, the stress drops rapidly, and the post-peak curve is steep, indicating that the brittleness of the concrete is relatively strong at this time and close to an ideal brittle material. With the increase of confining pressure, obvious yield steps appear in the pre-peak curve, and the post-peak curve and the stress drop trends gradually tend to be gentle. The post-peak phase gradually transforms from the strain softening state to the ideal plastic state, and its brittleness level gradually reduces, showing that, with the increase of confining pressure, the brittleness of high-strength concrete develops from high to low. This shows that there are obvious differences in the brittleness characteristics of concrete under different confining pressures. *Crystals* **2020**, *10*, x FOR PEER REVIEW 4 of 19

**Figure 2.** Full stress–strain curves of high-strength concrete under different confining pressures. (**a**) C60; (**b**) C70. **Figure 2.** Full stress–strain curves of high-strength concrete under different confining pressures. (**a**) C60; (**b**) C70.

According to the full stress–strain curve of high-strength concrete, the basic mechanical

0 65.38 84.41 14.66 14.35 0.343 0.337 0.523 0.611 28 31 0.31 0.29 5 81.96 100.22 38.93 45.68 0.540 0.419 1.388 1.373 29 32 0.28 0.26 10 98.26 115.98 71.16 76.70 0.631 0.721 1.515 1.640 31 33 0.29 0.26 15 114.01 128.22 89.97 102.27 0.812 0.969 1.953 2.167 33 35 0.28 0.28 20 129.12 141.67 111.13 114.47 1.10 1.081 2.240 2.864 34 36 0.26 0.26

Figure 3 shows the variation of the concrete peak strain with confining pressure. It can be seen from Figure 3 that with the increase of confining pressure, the corresponding strain when the concrete stress reaches the peak strength gradually increases, indicating that it is more difficult for the concrete to enter the brittle failure state, and more energy is needed from the outside to assist the cracks inside the concrete to connect and penetrate to form a macroscopic fracture surface. Therefore, the peak strain reflects the difficulty of achieving brittle failure for high-strength concrete. The greater the peak strain, the higher the threshold of concrete brittleness, which indicates that the

**C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 C60 C70** 

μ

parameters of C60 and C70 high-strength concrete are shown in Table 2.

confining pressure can limit the brittleness of concrete.

According to the full stress–strain curve of high-strength concrete, the basic mechanical parameters of C60 and C70 high-strength concrete are shown in Table 2.


**Table 2.** Mechanical parameters of C60 and C70 high-strength concrete.

Figure 3 shows the variation of the concrete peak strain with confining pressure. It can be seen from Figure 3 that with the increase of confining pressure, the corresponding strain when the concrete stress reaches the peak strength gradually increases, indicating that it is more difficult for the concrete to enter the brittle failure state, and more energy is needed from the outside to assist the cracks inside the concrete to connect and penetrate to form a macroscopic fracture surface. Therefore, the peak strain reflects the difficulty of achieving brittle failure for high-strength concrete. The greater the peak strain, the higher the threshold of concrete brittleness, which indicates that the confining pressure can limit the brittleness of concrete. *Crystals* **2020**, *10*, x FOR PEER REVIEW 5 of 19

**Figure 3.** Variation of peak strain with confining pressure. **Figure 3.** Variation of peak strain with confining pressure.

### **3. Energy Evolution Law of High-Strength Concrete during Compression 3. Energy Evolution Law of High-Strength Concrete during Compression**

### *3.1. Theoretical Analysis 3.1. Theoretical Analysis*

where

σ σ

Assuming that there is no heat exchange between the concrete system and the external environment, according to the first law of thermodynamics, the energy *W*F input to the system by the external force during the concrete deformation and failure process is equal to the elastic strain energy *W*E plus the energy *W*D dissipated by the concrete specimen during the test [20–22]. The energy input by the external force to the system mainly includes the work done by the axial force when the concrete specimen undergoes axial deformation and the work done by the confining pressure when radial deformation occurs. The dissipative energy is mainly used for the Assuming that there is no heat exchange between the concrete system and the external environment, according to the first law of thermodynamics, the energy *W*<sup>F</sup> input to the system by the external force during the concrete deformation and failure process is equal to the elastic strain energy *W*<sup>E</sup> plus the energy *W*<sup>D</sup> dissipated by the concrete specimen during the test [20–22]. The energy input by the external force to the system mainly includes the work done by the axial force when the concrete specimen undergoes axial deformation and the work done by the confining pressure when radial deformation occurs. The dissipative energy is mainly used for the development of internal damage or plastic deformation in the concrete, including the surface energy consumed during the generation,

( ) = += 1 3 <sup>2</sup> F 11 33 F 0 0 d2 d

radial strain, respectively; *D H* , are the diameter and height of the concrete specimen,

2 E EE

*<sup>π</sup> W D HU VU*

<sup>=</sup> <sup>4</sup>

<sup>=</sup> <sup>4</sup>

2 D DD

*<sup>π</sup> W D HU VU*

*WWW* FED = + (1)

ε ε

, are the axial and

(3)

*W DH σε σε VU* (2)

environment. The above-mentioned energy relationship is shown in Equation (1):

4

The work done by the axial force and confining pressure during the test is [24]:

*π ε ε*

1 3 , are the axial pressure and confining pressure, respectively; 1 3

respectively; *V* is the volume of the concrete sample; and *U*F is the input energy density. In the same way, the elastic strain energy and dissipative energy are as follows:

> <sup>=</sup> <sup>=</sup>

development of internal damage or plastic deformation in the concrete, including the surface energy consumed during the generation, the expansion and penetration of the cracks, the plastic strain the expansion and penetration of the cracks, the plastic strain energy of the irreversible plastic deformation of concrete specimens, the heat energy generated by friction and slipping between cracks, and various radiation energies, etc. [23]; the magnitude of the dissipative energy is mainly related to the structural properties of the concrete itself and the stress environment. The above-mentioned energy relationship is shown in Equation (1):

$$\mathcal{W}\_{\rm F} = \mathcal{W}\_{\rm E} + \mathcal{W}\_{\rm D} \tag{1}$$

The work done by the axial force and confining pressure during the test is [24]:

$$\mathcal{W}\_{\rm F} = \frac{\pi}{4} D^2 H \left( \int\_0^{\varepsilon\_1} \sigma\_1 \mathbf{d} \varepsilon\_1 + 2 \int\_0^{\varepsilon\_3} \sigma\_3 \mathbf{d} \varepsilon\_3 \right) = V \mathcal{U}\_{\rm F} \tag{2}$$

where σ1, σ<sup>3</sup> are the axial pressure and confining pressure, respectively; ε1, ε<sup>3</sup> are the axial and radial strain, respectively; *D*, *H* are the diameter and height of the concrete specimen, respectively; *V* is the volume of the concrete sample; and *U*<sup>F</sup> is the input energy density.

In the same way, the elastic strain energy and dissipative energy are as follows:

$$\begin{cases} \begin{aligned} \mathcal{W}\_{\mathcal{E}} &= \frac{\pi}{4} D^2 H \mathcal{U}\_{\mathcal{E}} = V \mathcal{U}\_{\mathcal{E}} \\\ \mathcal{W}\_{\mathcal{D}} &= \frac{\pi}{4} D^2 H \mathcal{U}\_{\mathcal{D}} = V \mathcal{U}\_{\mathcal{D}} \end{aligned} \tag{3}$$

where *U*<sup>E</sup> is the elastic strain energy density and *U*<sup>D</sup> is the dissipative energy density.

According to the elastic theory [24], the elastic strain energy density is:

$$\mathcal{U}\_{\rm E} = \frac{1}{2} (\sigma\_1 \varepsilon\_1^{\rm e} + \sigma\_2 \varepsilon\_2^{\rm e} + \sigma\_3 \varepsilon\_3^{\rm e}) \tag{4}$$

The three-dimensional constitutive relationship of concrete is:

$$
\varepsilon\_{ij}^{\varepsilon} = \frac{1+\mu}{E\_{ij}} \sigma\_{ij} - \frac{\mu}{E\_{ij}} \sigma\_{kk} \delta\_{ij} \tag{5}
$$

where ε e *ij*(*i*, *j* = 1, 2, 3) is the elastic strain in the direction of the main stress, σ*ij*(*i*, *j* = 1, 2, 3) is the main stress, σ*kk* = σ<sup>1</sup> + σ<sup>2</sup> + σ3; δ*ij* is the Kronecker tensor and *Eij*(*i*, *j* = 1, 2, 3) is the unloading modulus of elasticity—for convenience of calculation, the initial elastic modulus *E* can be used instead [24]—and µ is the Poisson's ratio.

Equations (4) and (5) can be substituted into Equation (3) to obtain the elastic strain energy *W*E:

$$\mathcal{W}\_{\rm E} = \frac{1}{2E} (\sigma\_1^2 + \sigma\_2^2 + \sigma\_3^2 - 2\mu\sigma\_1\sigma\_2 - 2\mu\sigma\_1\sigma\_3 - 2\mu\sigma\_2\sigma\_3)V \tag{6}$$

In the conventional triaxial compression test, where σ<sup>2</sup> = σ3, Equation (6) can be simplified to:

$$\mathcal{W}\_{\rm E} = \frac{1}{2E} \Big[ \sigma\_1^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu\sigma\_1\sigma\_3 \Big] V \tag{7}$$

Substituting Equation (7) into Equation (1) and combining this with Equation (2) to obtain the dissipative energy results in the following:

$$\mathcal{W}\_{\rm D} = \left\{ \int\_0^{\varepsilon\_1} \sigma\_1 \mathrm{d}\varepsilon\_1 + 2 \int\_0^{\varepsilon\_3} \sigma\_3 \mathrm{d}\varepsilon\_3 - \frac{1}{2E} \left[ \sigma\_1^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu \sigma\_1 \sigma\_3 \right] \right\} V \tag{8}$$

The energy conversion diagram of the concrete failure process is shown in Figure 4. When the load reaches the yield stress σ1c, the elastic strain energy accumulated inside the concrete is *W*E(A) , showing that the concrete is in the linear elastic stage, without damage and energy dissipation; as the load increases, when the peak strength σ1p is reached, the total elastic strain energy accumulated in

the pre-peak stage is *W*E(B) . One part of the work *W*F(pre) done by the external testing machine is converted into dissipative energy *W*D, which is irreversible energy that is consumed in the plastic yield stage of concrete, and the other part is transformed into stored elastic strain energy, which is reversible. The energy relationship in the pre-peak stage is as follows:

$$\mathcal{W}\_{\rm F(pre)} = \mathcal{W}\_{\rm E(B)} + \mathcal{W}\_{\rm D} \tag{9}$$

$$\mathcal{W}\_{\rm F(pre)} = \left( \int\_0^{\varepsilon\_{1\rm p}} \sigma\_1 \mathbf{d} \, \varepsilon\_1 + 2 \int\_0^{\varepsilon\_{3\rm p}} \sigma\_3 \mathbf{d} \, \varepsilon\_3 \right) V = \left( \int\_0^{\varepsilon\_{1\rm p}} \sigma\_1 \mathbf{d} \, \varepsilon\_1 + 2 \sigma\_3 \varepsilon\_{3\rm p} \right) V \tag{10}$$

$$\mathcal{W}\_{\rm E(B)} = \frac{1}{2E} \left[ \sigma\_{\rm lp}^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu\sigma\_{\rm lp}\sigma\_3 \right] V \tag{11}$$

$$W\_{\rm D} = \left(\int\_0^{\varepsilon\_{\rm lp}} \sigma\_{\rm l} \mathrm{d}\varepsilon\_{\rm l} + 2\sigma\_3 \varepsilon\_{\rm lp}\right) V - \frac{1}{2\Xi} \left[\sigma\_{\rm lp}^2 + 2(1-\mu)\sigma\_3^2 - 4\mu\sigma\_{\rm lp}\sigma\_3\right] V \tag{12}$$

where *W*F(pre) is the area *S*<sup>1</sup> + *S*<sup>2</sup> enclosed by the stress–strain curve and the axial strain axis from the initial loading point to the peak strength, *W*E(B) is the area *S*<sup>2</sup> enclosed by the peak point unloading curve and the axial strain axis, and *W*<sup>D</sup> is the area *S*<sup>1</sup> between the loading curve and the unloading curve. unloading curve and the axial strain axis, and *W*D is the area *S*1 between the loading curve and the unloading curve.

**Figure 4.** Schematic diagram of energy conversion during concrete failure. **Figure 4.** Schematic diagram of energy conversion during concrete failure.

When the concrete stress reaches the peak strength, the specimen enters the failure state, and its internal storage of elastic strain energy *W*E(B) is not sufficient to support its complete failure: therefore, external work is required to provide additional energy *W*F(post) . When the concrete stress reaches the residual strength, the internal storage of elastic strain energy *W*E(B) and the external additional energy *W*F(post) are converted into the energy *W*<sup>f</sup> required for the failure of concrete, When the concrete stress reaches the peak strength, the specimen enters the failure state, and its internal storage of elastic strain energy *W*E(B) is not sufficient to support its complete failure: therefore, external work is required to provide additional energy *W*F(post) . When the concrete stress reaches the residual strength, the internal storage of elastic strain energy *W*E(B) and the external additional energy *W*F(post) are converted into the energy *W*<sup>f</sup> required for the failure of concrete, and at this time, a part of the elastic strain energy *W*<sup>r</sup> will remain inside the concrete specimen. The energy relationship in the post-peak stage is as follows:

$$\mathcal{W}\_{\rm f} = \mathcal{W}\_{\rm F(post)} + \mathcal{W}\_{\rm E(B)} - \mathcal{W}\_{\rm r} \tag{13}$$

$$\mathcal{W}\_{\rm F(post)} = \left( \int\_{\varepsilon\_{1\rm p}}^{\varepsilon\_{1\rm r}} \sigma\_1 \mathrm{d}\varepsilon\_1 + 2 \int\_{\varepsilon\_{3\rm p}}^{\varepsilon\_{3\rm r}} \sigma\_3 \mathrm{d}\varepsilon\_3 \right) V = \left[ \int\_{\varepsilon\_{1\rm p}}^{\varepsilon\_{1\rm r}} \sigma\_1 \mathrm{d}\varepsilon\_1 + 2 \sigma\_3 \left( \varepsilon\_{3\rm r} - \varepsilon\_{3\rm p} \right) \right] V \tag{14}$$

$$W\_{\rm r} = \frac{1}{2E} \left[ \sigma\_{\rm 1r}^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu\sigma\_{1r}\sigma\_3 \right] V \tag{15}$$

*<sup>E</sup>* (16)

$$\mathcal{W}\_{\rm f} = \left[ \int\_{\varepsilon\_{\rm 1p}}^{\varepsilon\_{\rm 1r}} \sigma\_1 \mathbf{d} \varepsilon\_1 + 2\sigma\_3 (\varepsilon\_{\rm 3r} - \varepsilon\_{\rm 3p}) \right] V + \frac{1}{2E} \left[ \sigma\_{1\rm p}^2 - \sigma\_{1\rm r}^2 - 4\mu \{ \sigma\_{1\rm p} - \sigma\_{1\rm r} \} \sigma\_3 \right] V \tag{16}$$

2 2

f 1 1 3 3r 3p 1p 1r 1p 1r 3

*<sup>ε</sup> W σε σε ε V σ σ μσ σ σ V*

2

1r 1p

obtained, as shown in Figures 5 and 6.

where *W*F(post) is the area *S*<sup>3</sup> enclosed by the stress–strain curve between the peak strength and the residual strength and the axial strain axis, *W*<sup>r</sup> is the area *S*<sup>4</sup> enclosed by the unloading curve of the residual strength point and the axial strain axis, and *W*<sup>f</sup> is the area shown by the orange area in the figure. residual strength and the axial strain axis, *W*<sup>r</sup> is the area *S*4 enclosed by the unloading curve of the residual strength point and the axial strain axis, and *W*<sup>f</sup> is the area shown by the orange area in the figure.

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( ) ( ) = + = +− 1r 3r 1r 1p 3p 1p F post 1 1 3 3 1 1 3 3r 3p d2 d d2 *εε ε*

*W σ μσ μσ σ V*

( ) ( ) = + − + −− −

2

f 1 1 3 3r 3p 1p 1r 1p 1r 3 <sup>1</sup> d2 4

*<sup>ε</sup> W σε σε ε V σ σ μσ σ σ V*

where *W*F post ( ) is the area *S*3 enclosed by the stress–strain curve between the peak strength and the

2

 +− − ( ) 2 2 r 1r 3 1r 3 <sup>1</sup> = 21 4

*εε ε W σε σε V σε σε ε V* (14)

2 2

*<sup>E</sup>* (15)

*<sup>E</sup>* (16)

### *3.2. Relationship between Energy Evolution and Axial Strain* By substituting the triaxial compression test results into Equations (2), (7), and (8), the

*3.2. Relationship between Energy Evolution and Axial Strain* 

1r 1p

*ε*

By substituting the triaxial compression test results into Equations (2), (7), and (8), the evolution curves of the input energy, elastic energy and dissipative energy with the axial strain of high-strength concrete under different confining pressures are obtained, as shown in Figures 5 and 6. evolution curves of the input energy, elastic energy and dissipative energy with the axial strain of high-strength concrete under different confining pressures are obtained, as shown in Figures 5 and 6.

**Figure 5.** Energy evolution curves of C60 high-strength concrete under different confining pressures. (**a**) 0 MPa; (**b**) 5 MPa; (**c**) 10 MPa; (**d**) 15 MPa; (**e**) 20 MPa. **Figure 5.** Energy evolution curves of C60 high-strength concrete under different confining pressures. (**a**) 0 MPa; (**b**) 5 MPa; (**c**) 10 MPa; (**d**) 15 MPa; (**e**) 20 MPa.

(**a**) (**b**)

0

0

140

280

Input Energy,Dissipative Energy (J)

(**c**) (**d**)

420

560

700

50

100

150

200

250

0.0 0.3 0.6 0.9 1.2 1.5 1.8

Axial Strain (%)

0.0 0.5 1.0 1.5 2.0 2.5

Axial Strain (%)

Input Energy Dissipative Energy

Elastic Energy

0

0

7

14

21

Elastic Energy (J)

28

35

42

5

10

15

Elastic Energy (J)

20

25

30

Elastic Energy

Input Energy Dissipative Energy

0

0

8

16

Elastic Energy (J)

24

32

40

4

8

12

Elastic Energy (J)

16

20

24

Input Energy Dissipative Energy

Elastic Energy

Input Energy Dissipative Energy

<sup>350</sup> Elastic Energy

0

Input Energy,Dissipative Energy (J)

9

18

27

Input Energy,Dissipative Energy (J)

36

45

54

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Axial Strain (%)

0.0 0.3 0.6 0.9 1.2 1.5 1.8

Axial Strain (%)

Input Energy Dissipative Energy

Elastic Energy

0

150

300

Input Energy,Dissipative Energy(J)

450

600

750

0.0 0.5 1.0 1.5 2.0 2.5

Axial Strain (%)

(**e**) **Figure 5.** Energy evolution curves of C60 high-strength concrete under different confining

0

10

20

Elastic Energy(J)

30

40

50

**Figure 6.** Energy evolution curves of C70 high-strength concrete under different confining pressures. (**a**) 0 MPa; (**b**) 5 MPa; (**c**) 10 MPa; (**d**) 15 MPa; (**e**) 20 MPa. **Figure 6.** Energy evolution curves of C70 high-strength concrete under different confining pressures. (**a**) 0 MPa; (**b**) 5 MPa; (**c**) 10 MPa; (**d**) 15 MPa; (**e**) 20 MPa.

It can be seen from Figures 5 and 6 that under different confining pressures, both the input energy and the dissipative energy increase with the increase of the axial strain, and the elastic strain energy first increases and then decreases. In the initial stage of loading, the initial pores inside the concrete gradually close under the action of external load, and most of the work done by external load is converted into elastic energy, which is stored inside the specimen. The elastic strain energy gradually increases with the axial strain, the dissipative energy is very low at this stage and the evolution curves of input energy, elastic energy and dissipative energy all show an upward concave It can be seen from Figures 5 and 6 that under different confining pressures, both the input energy and the dissipative energy increase with the increase of the axial strain, and the elastic strain energy first increases and then decreases. In the initial stage of loading, the initial pores inside the concrete gradually close under the action of external load, and most of the work done by external load is converted into elastic energy, which is stored inside the specimen. The elastic strain energy gradually increases with the axial strain, the dissipative energy is very low at this stage and the evolution curves of input energy, elastic energy and dissipative energy all show an upward concave trend. As the

limit of concrete, new cracks will appear inside the specimen, and the initiation and diffusion of cracks need to dissipate part of energy; thus, the slope of the elastic strain energy curve gradually decreases at this stage. When the external load reaches the peak strength, the elastic strain energy reaches the maximum value, and then the concrete specimen is destroyed and the stored elastic strain energy in the pre-peak stage is released rapidly; thus, the post-peak elastic strain energy gradually decreases with the axial strain, and most of the external input energy is dissipated by the cracks intersecting each other to form a macroscopic fracture surface. When the specimen reaches the peak strength, the elastic strain energy and dissipative energy gradually decrease and increase,

Based on the triaxial compression test data of high-strength concrete under different confining pressures, the characteristic energy of high-strength concrete is calculated by substituting Equations (9)–(16), and the results are shown in Table 3. It can be seen from Table 3 that when the confining pressure is 0 MPa, the additional energy *W*F post ( ) provided from the outside for the post-peak fracture of C60 and C70 concrete specimens is 14.65 J and 21.83 J, respectively, accounting for 52% and 49% of the fracture energy, respectively, which indicates that the elastic energy stored before the peak is the source power of the specimen failure. When the confining pressure is 20 MPa, the additional energy *W*F post ( ) required for the post-peak fracture of C60 and C70 concrete specimens is 400.11 J and 593.85 J, respectively, accounting for 97% of the fracture energy, indicating that the energy required for the failure of C60 and C70 concrete specimens at this time is mainly provided by external work, that the self-sustaining fracture ability of the specimens is poor and that the

respectively, until the specimen failure reaches the maximum value and minimum value.

*3.3. Relationship between Energy Evolution and Confining Pressure* 

brittleness level is low.

trend. As the external load increases, the concrete specimen enters the linear elastic deformation

external load increases, the concrete specimen enters the linear elastic deformation stage, and the external work is basically transformed into elastic strain energy; furthermore, the slopes of the three energy evolution curves reach the maximum value. This stage is the main stage of energy storage in the whole process of concrete failure. When the external load reaches the yield limit of concrete, new cracks will appear inside the specimen, and the initiation and diffusion of cracks need to dissipate part of energy; thus, the slope of the elastic strain energy curve gradually decreases at this stage. When the external load reaches the peak strength, the elastic strain energy reaches the maximum value, and then the concrete specimen is destroyed and the stored elastic strain energy in the pre-peak stage is released rapidly; thus, the post-peak elastic strain energy gradually decreases with the axial strain, and most of the external input energy is dissipated by the cracks intersecting each other to form a macroscopic fracture surface. When the specimen reaches the peak strength, the elastic strain energy and dissipative energy gradually decrease and increase, respectively, until the specimen failure reaches the maximum value and minimum value.

## *3.3. Relationship between Energy Evolution and Confining Pressure*

Based on the triaxial compression test data of high-strength concrete under different confining pressures, the characteristic energy of high-strength concrete is calculated by substituting Equations (9)–(16), and the results are shown in Table 3. It can be seen from Table 3 that when the confining pressure is 0 MPa, the additional energy *W*F(post) provided from the outside for the post-peak fracture of C60 and C70 concrete specimens is 14.65 J and 21.83 J, respectively, accounting for 52% and 49% of the fracture energy, respectively, which indicates that the elastic energy stored before the peak is the source power of the specimen failure. When the confining pressure is 20 MPa, the additional energy *W*F(post) required for the post-peak fracture of C60 and C70 concrete specimens is 400.11 J and 593.85 J, respectively, accounting for 97% of the fracture energy, indicating that the energy required for the failure of C60 and C70 concrete specimens at this time is mainly provided by external work, that the self-sustaining fracture ability of the specimens is poor and that the brittleness level is low.


**Table 3.** Calculation results of the characteristic energy of C60 and C70 high-strength concrete.

The evolution laws of elastic energy, dissipative energy, pre-peak total energy, additional energy, fracture energy, and residual elastic strain energy with confining pressure are shown in Figure 7. It can be seen from Figure 7 that, with the increase of confining pressure, the six kinds of energy increase at different rates, and the storage rate of elastic energy gradually increases, which indicates that the confining pressure has a more obvious limiting effect on crack propagation, thus keeping the concrete from reaching the failure state and finally reducing the brittleness of concrete. When the confining pressure is 0 MPa, the elastic strain energy stored inside the specimen is released suddenly at the peak value, which can make the specimen fully fracture, and the additional energy consumed is very small; at this time, the self-sustaining fracture ability of concrete is stronger and the brittleness is higher. When the confining pressure is 20 MPa, the additional energy consumed by the specimen failure is much greater than the releasable elastic energy stored before the peak, and the further failure after the peak of the specimen will require more energy than the elastic deformation energy accumulated inside; this means that the concrete specimen will not only have no sudden release of energy, but will also require more external work to further destroy it, and the specimen will show obvious ductility characteristics.

*Crystals* **2020**, *10*, x FOR PEER REVIEW 11 of 19

**Figure 7.** The characteristic energy of high-strength concrete under different confining pressures. (**a**) C60; (**b**) C70. *3.4. Relationship between Energy Evolution and Concrete Strength Grade*  **Figure 7.** The characteristic energy of high-strength concrete under different confining pressures. (**a**) C60; (**b**) C70.

of the elastic strain energy evolution curve of C70 concrete is greater than that of C60 concrete,

### Compared with Figures 5 and 6, it can be seen that under the same confining pressure, the slope *3.4. Relationship between Energy Evolution and Concrete Strength Grade*

indicating that the energy storage rate of C70 concrete in the pre-peak stage and the energy release rate in the post-peak stage are both higher than that of C60 concrete. Figure 8 shows the characteristic energy evolution trend of high-strength concrete with different strength grades. It can be seen from Figure 8 that, with the increase of concrete strength grade, the elastic strain energy, the pre-peak total energy and the fracture energy all increase, while the residual elastic strain energy shows little difference. When the confining pressure is 10 MPa, the elastic strain energy *W*E B( ) stored before the peak of C60 and C70 concrete is 27.22 J and 36.21 J, respectively, and the proportion of energy in the total energy before the peak is 28% and 31%, respectively. This shows that with the increase of the strength grade, the storage capacity of the pre-peak elastic energy of the concrete specimen has increased. After reaching the peak strength, the residual elastic strain energy of the C60 and C70 concrete specimens is 13.86 J and 15.56 J, respectively, and the elastic strain energy release rate is 49% and 57%, respectively, indicating that the elastic strain energy accumulated in C70 concrete is released more completely. The releasable elastic energy of C60 and C70 concrete is 13.36 J and 20.65 J, respectively, which accounts for 6% and 8%, respectively, of the fracture energy. This shows that the C70 concrete specimen has a greater ability to maintain self-fracturing and its brittleness is higher. Since the energy evolution characteristics under the five confining pressures are the same, this section only takes a confining pressure of 10 MPa as an example, and other confining pressure levels will not be repeated. Compared with Figures 5 and 6, it can be seen that under the same confining pressure, the slope of the elastic strain energy evolution curve of C70 concrete is greater than that of C60 concrete, indicating that the energy storage rate of C70 concrete in the pre-peak stage and the energy release rate in the post-peak stage are both higher than that of C60 concrete. Figure 8 shows the characteristic energy evolution trend of high-strength concrete with different strength grades. It can be seen from Figure 8 that, with the increase of concrete strength grade, the elastic strain energy, the pre-peak total energy and the fracture energy all increase, while the residual elastic strain energy shows little difference. When the confining pressure is 10 MPa, the elastic strain energy *W*E(B) stored before the peak of C60 and C70 concrete is 27.22 J and 36.21 J, respectively, and the proportion of energy in the total energy before the peak is 28% and 31%, respectively. This shows that with the increase of the strength grade, the storage capacity of the pre-peak elastic energy of the concrete specimen has increased. After reaching the peak strength, the residual elastic strain energy of the C60 and C70 concrete specimens is 13.86 J and 15.56 J, respectively, and the elastic strain energy release rate is 49% and 57%, respectively, indicating that the elastic strain energy accumulated in C70 concrete is released more completely. The releasable elastic energy of C60 and C70 concrete is 13.36 J and 20.65 J, respectively, which accounts for 6% and 8%, respectively, of the fracture energy. This shows that the C70 concrete specimen has a greater ability to maintain self-fracturing and its brittleness is higher. Since the energy evolution characteristics under the five confining pressures are the same, this section only takes a confining pressure of 10 MPa as an example, and other confining pressure levels will not be repeated. *Crystals* **2020**, *10*, x FOR PEER REVIEW 12 of 19

**Figure 8.** The characteristic energy of high-strength concrete with different strength grades. (**a**) Elastic strain energy; (**b**) pre-peak total energy; (**c**) fracture energy; (**d**) residual elastic strain energy. **4. Evaluation Method of Concrete Brittleness Figure 8.** The characteristic energy of high-strength concrete with different strength grades. (**a**) Elastic strain energy; (**b**) pre-peak total energy; (**c**) fracture energy; (**d**) residual elastic strain energy.

*4.1. Brittleness Evaluation Index* 

parameters.

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

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

## **4. Evaluation Method of Concrete Brittleness**
