crystals

Article Energy Evolution Analysis and Evaluation of High-Strength Concrete Considering the Whole Failure Process

Ruihe Zhou 1,* , Hua Cheng 1,2, Mingjing Li 1, Liangliang Zhang 1 and Rongbao Hong 1

1 School of Civil Engineering and Architecture, Anhui University of Science and Technology, Huainan 232001, China; [email protected] (H.C.); [email protected] (M.L.); [email protected] (L.Z.); [email protected] (R.H.) 2 School of Resources and Environmental Engineering, Anhui University, Hefei 230022, China * Correspondence: [email protected]; Tel.: +86-187-5602-3671

 Received: 23 October 2020; Accepted: 27 November 2020; Published: 30 November 2020 

Abstract: In this work, we aimed to solve the problems that exist in the brittleness evaluation method of high-strength concrete through a triaxial test of C60 and C70 high-strength concrete. Then, the relationship between the energy evolution of its elastic energy, dissipative energy, pre-peak total energy and additional energy and its axial strain, confining pressure, and concrete strength grade was analyzed. Taking the accumulation rate of pre-peak elastic strain energy and the dissipation rate of dissipative energy, and the release rate of post-peak elastic energy, as the evaluation indicators to characterize the brittleness of high-strength concrete. A brittleness evaluation method that reflects the whole failure process of high-strength concrete is proposed and verified by experiments. The results show that with the increase of the confining pressure, the proportion of elastic energy in the whole process of high-strength concrete failure gradually decreases. The storage rate of pre-peak elastic energy and the release rate of post-peak elastic energy are gradually reducing, the brittleness index gradually decreases, and the confining pressure inhibits the brittleness of high-strength concrete. Under the same confining pressure, the brittleness index of C70 concrete is greater than that of C60 concrete, which indicates that, with the increase of the strength grade, the brittleness level of concrete gradually increases and the decreases. These findings have a certain theoretical significance for the scientific design of high-strength concrete structures and the improvement of their safety in the future.

Keywords: high-strength concrete; energy evolution; elastic strain energy; brittleness evaluation index

1. Introduction In recent years, high-strength concrete has been widely used in civil engineering, transportation, water conservancy, and municipal engineering. Concrete is a quasi-brittle material, and its failure mode is not only affected by the strength level of concrete, but is also closely related to its state [1–5]. Studies have shown that, compared with ordinary concrete, high-strength concrete presents obvious brittle characteristics [6–8]. To date, domestic and foreign scholars have carried out a great deal of research on the characterization methods of concrete brittleness. Jenq and Shah [9] proposed a critical material constant Q that was positively related to the brittleness index based on the and elastic modulus, showing that the larger the value of Q, the greater the brittleness of the material. Tasdemir et al. [10] used the energy method to evaluate the brittleness of concrete and proposed the ratio of the recoverable elastic energy to unrecoverable plastic deformation energy as

Crystals 2020, 10, 1099; doi:10.3390/cryst10121099 www.mdpi.com/journal/crystals Crystals 2020, 10, 1099 2 of 16

the brittleness evaluation index B1 of concrete. Zhang et al. [11] proposed to use the ratio of the pre-peak recoverable elastic energy to the total energy as the brittleness evaluation index B2 of concrete. Yan et al. [12] used mechanical parameters and area methods to evaluate the brittleness of concrete and proposed that the ratio of the fracture energy to nominal stress σN should be taken as the brittleness index B3; furthermore, they established the brittleness evaluation index B4 based on the roughness of the fracture surface [13]. Their results showed that these two brittleness indicators were linearly correlated, and it was considered that the greater the roughness of the fracture surface, the greater the unrecoverable deformation energy consumed by fracture failure and the lower the brittleness of concrete. Yao et al. [14] used the strength method to evaluate the brittleness of concrete and proposed that the tension–compression ratio of concrete specimens should be taken as their brittleness index B5; furthermore, they considered that the smaller the tension–compression ratio, the greater the brittleness of concrete. Guo and Huang et al. [15–17] considered the size effect of strength and established a brittleness index η that was proportional to the size of the plastic zone; furthermore, they considered that the smaller the value of η, the greater the brittleness of the material. Han et al. [18] proposed the evaluation of the brittleness of concrete by a morphology method based on the analysis of the insufficient brittleness of the concrete cross-sectional area ratio and established a new brittleness evaluation method of concrete based on the area fraction of section stones. In summary, most of the existing concrete brittleness evaluation indicators are suitable for low-strength concrete, and there are few studies on the brittleness evaluation of high-strength concrete. The physical meaning of some concrete brittleness evaluation indicators is not clear, and the evaluation indicators cannot be obtained through conventional mechanical tests of concrete; thus, their practical application and promotion have certain limitations. Some concrete brittleness evaluation indicators only consider the pre-peak or post-peak stage and cannot fully reflect the brittleness characteristics of the entire failure process of concrete. Based on the experimental results of C60 and C70 high-strength concrete under different confining pressures, the evolution of elastic strain energy, dissipative energy, and input energy with axial strain during the deformation process is analyzed in this paper. Combined with these three kinds of energy, a brittleness evaluation index which can comprehensively reflect the whole deformation and failure process of high-strength concrete is established. Furthermore, a brittleness evaluation method for high-strength concrete is proposed. This has a certain theoretical significance for the scientific design of high-strength concrete structures and the improvement of their safety in the future.

2. Triaxial Compression Test of C60 and C70 High-Strength Concrete

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/m3. 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 Table1).

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

Strength Cement Admixture Cementitious Sand Gravel Water Sand Dosage of Grade (kg/m3) (kg/m3) Materials (kg/m3) (kg/m3) (kg/m3) (kg/m3) Rate Admixture C60 415 135 550 620 1105 175 0.36 32.5% C70 425 145 570 615 1095 170 0.36 32.5% Crystals 2020, 10, 1099 3 of 16

Crystals 2020, 10, x FOR PEER REVIEW 3 of 19 The conventional triaxial loading equipment for high-strength concrete specimens was the The conventional triaxial loading equipment for high-strength concrete specimens was the ZTCR-2000ZTCR-2000 low-temperature low-temperature rock rock triaxial triaxial system system (Figure(Figure 1).1). During During the the test, test, the concrete the concrete specimens specimens were firstwere preloaded first preloaded to 0.5 MPa,to 0.5 andMPa, then and confiningthen confining pressure pressure was was applied applied to theto the predetermined predetermined value at the loadingvalue speed at the ofloading 50 N/ sspeed according of 50 N/s to the according “load” to control the “load” mode. control The testmode. confining The test pressureconfining was set at 0, 5, 10,pressure 15, and was 20 set MPa. at 0, 5, After 10, 15, the and confining 20 MPa. Af pressureter the confining reached pressure the predetermined reached the predetermined value, the pressure was stabilizedvalue, the for pressure 30 s, and was finally, stabilized the for axial 30 s, pressure and finally, was the applied axial pressure at the was loading applied speed at the of loading 0.05 mm /min speed of 0.05 mm/min according to the “displacement” control mode until the concrete specimens according to the “displacement” control mode until the concrete specimens was destroyed. was destroyed.

FigureFigure 1. 1.Rock Rock mechanicsmechanics test test system. system.

2.2. Analysis2.2. Analysis of Test of Results Test Results from high to low. This shows that there are obvious differences in the brittleness characteristics Figure2 shows the stress–strain curves of C60 and C70 high-strength concrete specimens under of concrete under different confining pressures. different confining pressures. It can be seen from Figure2 that, when the confining pressure is 0 MPa, there is no obvious 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 theCrystals increase 2020, 10 of, xconfining FOR PEER REVIEW pressure, the brittleness of high-strength concrete develops from4 of high 19 to low. This shows that there are obvious differences in the brittleness characteristics of concrete under different confining pressures.

140 140

120 120 20MPa 100 20MPa ） 100 ） 15MPa MPa 15MPa MPa 80 80 （ （ 10MPa 10MPa 60 60

40 40 5MPa 5MPa Deviatoric Stress Deviatoric Stress Deviatoric 20 0MPa 20 0MPa 0 0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -3 -2 -1 0 1 2 3 （ ） Radial Strain % Axial Strain（%） Radial Strain（%） Axial Strain（%）

(a) (b)

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

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.

() () −3 −3 μ σ 1p MPa σ MPa ε ()10 ε ()10 E ()GPa () 1r 1p 1r σ3 MPa C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 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 confining pressure can limit the brittleness of concrete.

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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 Table2.

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

3 3 σ1p (MPa) σ1r (MPa) ε1p (10− ) ε1r (10− ) E (GPa) µ σ3 (MPa) C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 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

Figure3 shows the variation of the concrete peak strain with confining pressure. It can be seen from Figure3 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 theCrystals brittleness 2020, 10, x of FOR concrete. PEER REVIEW 5 of 19

1.2

0.8 ） %

（ C60 C70

0.4 Peak Strain

0.0 0 5 10 15 20 （ ） Confining Pressure MPa Figure 3. Variation of peak strain with confiningconfining pressure.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 Assuming that there is no heat exchange between the concrete system and the external environment, Assuming that there is no heat exchange between the concrete system and the external according to the first law of thermodynamics, the energy WF input to the system by the external environment, according to the first law of thermodynamics, the energy WF input to the system by force during the concrete deformation and failure process is equal to the elastic strain energy WE the external force during the concrete deformation and failure process is equal to the elastic strain plus the energy WD dissipated by the concrete specimen during the test [20–22]. The energy input by theenergy external WE forceplus tothe the energy system W mainlyD dissipated includes by thethe workconcrete done specimen by the axial during force the when test the[20–22]. concrete The specimenenergy input undergoes by the external axial deformation force to the and system the work mainly done includes by the the confining work done pressure by the when axial radial force deformationwhen the concrete occurs. specimen The dissipative undergoes energy axial is mainly deformation used for theand development the work done of internal by the damageconfining or plasticpressure deformation when radial in the deformation concrete, including occurs. theThe surface dissipative energy energy consumed is duringmainly theused generation, for the 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 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): =+ WWWFED (1) The work done by the axial force and confining pressure during the test is [24]:

π εε13 WDH=+=2 ( σεd2 σε d) VU (2) F1133F4 00 σσ εε where 13， are the axial pressure and confining pressure, respectively; 13， are the axial and radial strain, respectively; DH， are the diameter and height of the concrete specimen,

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

 π WDHUVU= 2 =  EEE4  (3) π WDHUVU= 2 =  DDD4

Crystals 2020, 10, 1099 5 of 16 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): WF = WE + WD (1) The work done by the axial force and confining pressure during the test is [24]:

Z ε1 Z ε3 ! π 2 WF = D H σ1dε1+2 σ3dε3 = VUF (2) 4 0 0 where σ1, σ3 are the axial pressure and confining pressure, respectively; ε1, ε3 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 UF is the input energy density. In the same way, the elastic strain energy and dissipative energy are as follows:

 π 2  WE = D HUE = VUE  4 (3)  π 2  WD = 4 D HUD = VUD where UE is the elastic strain energy density and UD is the dissipative energy density. According to the elastic theory [24], the elastic strain energy density is:

1  U = σ εe + σ εe + σ εe (4) E 2 1 1 2 2 3 3 The three-dimensional constitutive relationship of concrete is:

e 1 + µ µ εij = σij σkkδij (5) Eij − Eij

e ( = ) = where εij i, j 1, 2, 3 is the elastic strain in the direction of the main stress, σij(i, j 1, 2, 3) is the main stress, σkk = σ1 + σ2 + σ3; δij is the Kronecker tensor and Eij(i, j = 1, 2, 3) is the unloading modulus of —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 WE:

1   W = σ2 + σ2 + σ2 2µσ σ 2µσ σ 2µσ σ V (6) E 2E 1 2 3 − 1 2 − 1 3 − 2 3

In the conventional triaxial compression test, where σ2 = σ3, Equation (6) can be simplified to:

1 h i W = σ2 + 2(1 µ)σ2 4µσ σ V (7) E 2E 1 − 3 − 1 3 Substituting Equation (7) into Equation (1) and combining this with Equation (2) to obtain the dissipative energy results in the following: (Z Z ) ε1 ε3 1 h i 2 ( ) 2 WD = σ1dε1+2 σ3dε3 σ1 + 2 1 µ σ3 4µσ1σ3 V (8) 0 0 − 2E − −

The energy conversion diagram of the concrete failure process is shown in Figure4. When the load reaches the yield stress σ1c, the elastic strain energy accumulated inside the concrete is WE(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 Crystals 2020, 10, 1099 6 of 16

the pre-peak stage is WE(B). One part of the work WF(pre) done by the external testing machine is converted into dissipative energy WD, 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:

WF(pre) = WE(B) + WD (9)

Z ε1p Z ε3p ! Z ε1p ! WF(pre) = σ1dε1+2 σ3dε3 V = σ1dε1+2σ3ε3p V (10) 0 0 0 1 h i W = σ2 + 2(1 µ)σ2 4µσ σ V (11) E(B) 2E 1p − 3 − 1p 3 Z ! ε1p 1 h i 2 ( ) 2 WD = σ1dε1+2σ3ε3p V – σ1p + 2 1 µ σ3 4µσ1pσ3 V (12) Crystals 2020, 10, x FOR PEER REVIEW0 2E − − 7 of 19 where WF(pre) is the area S1 + S2 enclosed by the stress–strain curve and the axial strain axis from the unloading curve and the axial strain axis, and is the area S between the loading curve and initial loading point to the peak strength, WE(B) WisD the area S2 enclosed1 by the peak point unloading thecurve unloading and the axial curve. strain axis, and WD is the area S1 between the loading curve and the unloading curve.

Figure 4. SchematicSchematic diagram diagram of of energy energy conver conversionsion during concrete failure.

When the the concrete concrete stress stress reaches reaches the the peak peak strength strength,, the specimen the specimen entersenters the failure the failurestate, and state, its and its internal storage of elastic strain energy WE(B) is not sufficient to support its complete failure: internal storage of elastic strain energy WE(B) is not sufficient to support its complete failure: therefore, external work is required to provide additional energy WF(post). When the concrete stress therefore,reaches the external residual work strength, is requ theired internal to provide storage additional of elastic energy strain W energyF(post) . WhenWE(B )theand concrete the external stress additional energy WF(post) are converted into the energy Wf required for the failure of concrete, and at reaches the residual strength, the internal storage of elastic strain energy WE(B) and the external this time, a part of the elastic strain energy Wr will remain inside the concrete specimen. The energy additional energy W are converted into the energy required for the failure of concrete, relationship in the post-peakF(post) stage is as follows: Wf and at this time, a part of the elastic strain energy will remain inside the concrete specimen. The Wr W = W ( ) + W ( ) Wr (13) energy relationship in the post-peak stagef is Faspost follows:E B − Z Z  Z   ε1r WW=+−ε3r  W Wε1r   W =  + frFpost()V =EB() + V (13) F(post)  σ1dε1 2 σ3dε3  σ1dε1 2σ3 ε3r ε3p  (14) ε1p ε3p ε1p − εε1r 3r ε 1r W ()=+σεd2h σε dV =+−σε d2i σε() ε V (14) Fpost εε111 2 332  ε 11 33r3p 1pWr = σ 3p+ 2(1 µ)σ 1p 4µσ σ V (15) 2E 1r − 3 − 1r 3 Z ε 1   1r W =214 σμσμσσ22+−() 1 h − V   i (15) W =  σ dε +r1r31r32σ εε V + σ2 σ2 4µ σ σ σ V (16) f  1 1 23E 3r 3p  E 1p 1r 1p 1r 3 ε1p − 2 − − − ε =+−+−−−1r ()1 22 () Wf σε 1d2 1 σε 3 3r ε 3pV σσ 1p 1r 4 μσσσ 1p 1r 3 V (16) ε1p 2E 

energy with the axial strain of high-strength concrete under different confining pressures are obtained, as shown in Figures 5 and 6.

Crystals 2020, 10, x FOR PEER REVIEW 7 of 17

εε1r 3r ε 1r W =+σεd2 σε dV =+−σε d2 σε() ε V Fpost()11 33  11 33r3p (14) εε1p 3p ε 1p 1 W =214σμσμσσ22+−() − V (15) r1r31r32E 

ε =+−+−−−1r ()1 22 () Wf σε 1d2 1 σε 3 3r ε 3pV σσ 1p 1r 4 μσσσ 1p 1r 3 V (16) ε1p 2E  Crystals 2020, 10, 1099 7 of 16

where WFpost() is the area S3 enclosed by the stress–strain curve between the peak strength and the

residual strength and the axial strain axis, W is the area S4 enclosed by the unloading curve of where WF(post) is the area S3 enclosed by the stress–strainr curve between the peak strength and the the residual strength point and the axial strain axis, and is the area shown by the orange area in residual strength and the axial strain axis, Wr is the areaWSf 4 enclosed by the unloading curve of the residualthe figure. strength point and the axial strain axis, and Wf is the area shown by the orange area in the figure. 3.2. Relationship between Energy Evolution and Axial Strain 3.2. Relationship between Energy Evolution and Axial Strain By substituting the triaxial compression test results into Equations (2), (7), and (8), the Byevolution substituting curves theof the triaxial input compressionenergy, elastic test ener resultsgy and intodissipative Equations energy (2), wi (7),th andthe axial (8), thestrain evolution of curveshigh-strength of the input concrete energy, under elastic different energy confining and dissipative pressures energy are obtained, with the as axial shown strain in Figures of high-strength 5 and concrete6. under different confining pressures are obtained, as shown in Figures5 and6.

40 15 200 Input Energy 24 Input Energy Dissipative Energy Dissipative Energy Elastic Energy ） ） Elastic Energy J J 32 12

（ （ 150 18 ） ） J 24 9 J （ （ gy 100 12 gy

16 6 Dissipative Energy , Elastic Ener Elastic Elastic Ener 50 6 8 3 Input Energy,Dissipative Energy Energy,Dissipative Input Input Energy Input

0 0 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Axial Strain（%） Axial Strain（%） (a) (b)

400 30 Input Energy Input Energy 500 Elastic Energy 36 Dissipative Energy Dissipative Energy Elastic Energy

） 30 ） J 320 24

J 400 （ （

） 24 J ）

240 18 J

（ 300 （ 18

160 12 200 12 Elastic Energy

Elastic Energy 80 6 100 6 Input Energy,Dissipative Energy Energy,Dissipative Input Input Energy,Dissipative Energy Energy,Dissipative Input

0 0 0 0 0.0 0.4 0.8 1.2 1.6 2.0 0.00.40.81.21.62.02.4 Crystals 2020, 10, x FOR PEER REVIEW 8 of 17 Axial Strain（%） Axial Strain (%) (c) (d)

750 Input Energy 50 Dissipative Energy Elastic Energy ）

J 600 40 （ ） 450 30 J （

300 20 Elastic Energy

150 10 Input Energy,Dissipative Energy

0 0 0.0 0.5 1.0 1.5 2.0 2.5 Axial Strain (%) (e)

FigureFigure 5. Energy 5. Energy evolution evolution curves curves of C60 of high-strength C60 high-stren concretegth concrete under under different different confining confining pressures. (a) 0 MPa;pressures. (b) 5 (a MPa;) 0 MPa; (c) ( 10b) MPa;5 MPa; ( d(c)) 1510 MPa;MPa; ( (de)) 15 20 MPa; MPa. (e) 20 MPa.

54 Input Energy 24 250 30 Dissipative Energy Input Energy Elastic Energy Dissipative Energy 45 20 25 200 Elastic Energy

36 16 20 150

27 12 15

100 18 8 10 Elastic Energy (J) Elastic EnergyElastic (J)

50 9 4 5 Input Energy,Dissipative Energy (J) Input Energy,Dissipative Energy (J) Energy Energy,Dissipative Input

0 0 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.3 0.6 0.9 1.2 1.5 1.8 Axial Strain (%) Axial Strain (%) (a) (b)

Input Energy 350 Input Energy Elastic Energy 40 700 42 Dissipative Energy Dissipative Energy Elastic Energy 300 35 32 560

250 28 24 420 200 21 150

16 Dissipative Energy (J) 280 , 14 Elastic Energy (J) 100 Elastic Energy (J)

8 140 7 50 Input Energy,Dissipative Energy (J) Energy Energy,Dissipative Input Input Energy Input

0 0 0 0 0.0 0.3 0.6 0.9 1.2 1.5 1.8 0.0 0.5 1.0 1.5 2.0 2.5 Axial Strain (%) Axial Strain (%) (c) (d)

Crystals 2020, 10, x FOR PEER REVIEW 8 of 17

750 Input Energy 50 Dissipative Energy Elastic Energy ）

J 600 40 （ ） 450 30 J （

300 20 Elastic Energy

150 10 Input Energy,Dissipative Energy

0 0 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 Crystalspressures.2020, 10, 1099 (a) 0 MPa; (b) 5 MPa; (c) 10 MPa; (d) 15 MPa; (e) 20 MPa. 8 of 16

54 Input Energy 24 250 30 Dissipative Energy Input Energy Elastic Energy Dissipative Energy 45 20 25 200 Elastic Energy

36 16 20 150

27 12 15

100 18 8 10 Elastic Energy (J) Elastic EnergyElastic (J)

50 9 4 5 Input Energy,Dissipative Energy (J) Input Energy,Dissipative Energy (J) Energy Energy,Dissipative Input

0 0 0 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.3 0.6 0.9 1.2 1.5 1.8 Axial Strain (%) Axial Strain (%) (a) (b)

Input Energy 350 Input Energy Elastic Energy 40 700 42 Dissipative Energy Dissipative Energy Elastic Energy 300 35 32 560

250 28 24 420 200 21 150

16 Dissipative Energy (J) 280 , 14 Elastic Energy (J) 100 Elastic Energy (J)

8 140 7 50 Input Energy,Dissipative Energy (J) Energy Energy,Dissipative Input Input Energy Input

0 0 0 0 Crystals0.0 2020, 10 0.3, x FOR 0.6 PEER 0.9REVIEW 1.2 1.5 1.8 0.0 0.5 1.0 1.5 2.0 2.59 of 17 Axial Strain (%) Axial Strain (%) (c) (d)

900 Input Energy 50 Dissipative Energy Elastic Energy 750 40

600 30

450

20 300 Elastic EnergyElastic (J)

10 150 Input Energy,Dissipative Energy (J) Energy Input Energy,Dissipative

0 0 0.00.51.01.52.02.53.0 Axial Strain (%) (e)

Figure 6.6.Energy Energy evolution evolution curves curves of C70 of high-strengthC70 high-stren concretegth concrete under diunderfferent confiningdifferent pressures.confining (pressures.a) 0 MPa; (ba) 0 5 MPa; ( bc)) 105 MPa; MPa; ( (cd) )10 15 MPa; MPa; (d (e) )15 20 MPa; MPa. (e) 20 MPa.

It cancan bebe seen seen from from Figures Figures5 and 5 and6 that 6 underthat under di fferent different confining confining pressures, pressures, both the both input the energy input andenergy the and dissipative the dissipative energy increaseenergy increase with the with increase the increase of the axial of the strain, axial andstrain, the and elastic the strainelastic energy strain firstenergy increases first increases and then and decreases. then decreases. In the initial In the stage initial of stage loading, of loading, the initial the pores initial inside pores the inside concrete the graduallyconcrete gradually close under close the under action the of action external of external load, and load, most and of most the work of the done work by done external by external load is convertedload is converted into elastic into energy, elastic whichenergy, is which stored is inside stored the inside specimen. the specimen. The elastic The strain elastic energy strain gradually energy increasesgradually with increases the axial with strain, the theaxial dissipative strain, the energy dissip isative very energy low at thisis very stage low and at the this evolution stage and curves the ofevolution input energy, curves elasticof input energy energy, and elastic dissipative energy energyand dissipative all show energy an upward all show concave an upward trend. concave As the trend. As the 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 WFpost() 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 WFpost() 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.

Crystals 2020, 10, 1099 9 of 16 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 Table3. It can be seen from Table3 that when the confining pressure is 0 MPa, the additional energy WF(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 WF(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.

WE(B) (J) WF(pre) (J) WD (J) WF(post) (J) Wf (J) Wr (J) σ3 (MPa) C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 0 14.52 22.94 21.38 28.80 6.86 5.86 14.65 21.83 28.42 44.14 0.75 0.64 5 21.58 29.05 54.81 52.22 33.23 23.17 129.47 155.86 146.53 179.13 4.51 5.78 10 27.22 36.21 97.30 118.63 70.08 82.42 205.88 227.01 219.23 247.66 13.86 15.56 15 33.43 40.73 142.69 214.17 109.26 173.44 320.11 382.98 333.00 398.29 20.54 25.41 20 42.08 48.57 251.92 255.41 209.84 206.83 400.11 593.85 411.51 611.59 30.68 30.84

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 Figure7. It can be seen from Figure7 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, 1099 10 of 16 Crystals 2020, 10, x FOR PEER REVIEW 11 of 19

450 700 W WF(pre) F(pre) 400 W WE(B) 600 E(B) W 350 WF(post) F(post) ） J W Wf 500 f 300 （ Wr Wr W 250 WD 400 D

200 300 150 200 Characteristic Energy (J)

100 Energy Characteristic

50 100

0 0 0 5 10 15 20 0 5 10 15 20 Confining Pressure (MPa) Confining Pressure (MPa) (a) (b)

Figure 7. The characteristic energy of high-strength concrete under different confining pressures. (a) Figure 7. The characteristicC60; (b) C70. energy of high-strength concrete under different confining pressures. (a) C60; (b) C70. 3.4. Relationship between Energy Evolution and Concrete Strength Grade 3.4. Relationship betweenCompared Energy with Figures Evolution 5 and 6, and it can Concrete be seen that Strengthunder the same Grade confining pressure, the slope of the elastic strain energy evolution curve of C70 concrete is greater than that of C60 concrete, Compared withindicating Figures that the5 energyand6 storage, it can rate be of C70 seen concrete that in underthe pre-peak the stage same and confiningthe energy release pressure, the slope rate in the post-peak stage are both higher than that of C60 concrete. Figure 8 shows the of the elastic straincharacteristic energy energy evolution evolution trend curve of high-strengt of C70h concrete concrete with different is greater strength thangrades. It that can of C60 concrete, indicating that thebe seen energy from Figure storage 8 that, ratewith the of increase C70 concreteof concrete strength in the grade, pre-peak the elastic strain stage energy, and the the energy release pre-peak total energy and the fracture energy all increase, while the residual elastic strain energy rate in the post-peakshows stage little difference. are both When higher the confining than thatpressure of is C60 10 MPa, concrete. the elastic Figure strain energy8 shows WEB() the characteristic energy evolutionstored trend before of the high-strength peak of C60 and C70 concrete concrete is 27.22 with J and di 36.21fferent J, respectively, strength and the grades. proportion It can be seen from of energy in the total energy before the peak is 28% and 31%, respectively. This shows that with the Figure8 that, withincrease the of increase the strength of grade, concrete the storage strength capacity of grade,the pre-peak the elastic elastic energy strain of the concrete energy, the pre-peak total energy and thespecimen fracture has increased. energy After all reaching increase, the peak while strength, the the residual elastic elastic strain energy strain of the energy shows little C60 and C70 concrete specimens is 13.86 J and 15.56 J, respectively, and the elastic strain energy difference. Whenrelease the confiningrate is 49% and pressure57%, respectively, is 10 indicating MPa, that the theelastic elastic strain strain energyenergy accumulatedW inE C70(B) stored before the peak of C60 andconcrete C70 concrete is released more is 27.22completely. J and The releasable 36.21 elastic J, respectively, energy of C60 and and C70 concrete the proportion is 13.36 J of energy in and 20.65 J, respectively, which accounts for 6% and 8%, respectively, of the fracture energy. This the total energy beforeshows that the thepeak C70 concrete is 28% specimen and 31%,has a greater respectively. ability to maintain This self-fracturing shows that and withits the increase of the strength grade,brittleness the storage is higher. Since capacity the energy of evolution the pre-peak characteristics elastic under the energy five confining of thepressures concrete are specimen has the same, this section only takes a confining pressure of 10 MPa as an example, and other confining increased. Afterpressure reaching levels thewill not peak be repeated. 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 confiningCrystals pressure 2020, 10, x FOR of PEER 10 REVIEW MPa as an example, and other confining12 of 19 pressure levels will not be repeated.

50 280 C60 C60 C70 240 C70 40

200

30 160

120 20

80 Elastic Strain Energy (J) 10 (J) Energy Total Pre-peak 40

0 0 0 5 10 15 20 0 5 10 15 20 Confining Pressure (MPa) Confining Pressure (MPa) (a) (b)

630 32 C60 C60 540 C70 28 C70

24 450

20 360 16 270 12

Fracture Energy (J) 180 8

90 Residual Elastic Strain Energy (J) Strain Elastic Residual 4

0 0 0 5 10 15 20 0 5 10 15 20 Confining Pressure (MPa) Confining Pressure (MPa) (c) (d)

Figure 8. The characteristic energy of high-strength concrete with different strength grades. (a) Figure 8. The characteristicElastic energystrain energy; of (b) high-strength pre-peak total energy; concrete(c) fracture energy; with (d) differentresidual elastic strengthstrain energy. grades. (a) Elastic strain

energy; (b) pre-peak4. Evaluation total energy; Method of ( cConcrete) fracture Brittleness energy; (d) residual elastic strain energy.

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

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4. Evaluation Method of Concrete Brittleness

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:

2 + 2(1 ) 2 4 W WE(B) σ1p µ σ3 µσ1pσ3 B = 1 D = = − − (17) pre R ε1p  − WE(B) + WD WE(B) + WD + 2E 0 σ1dε1 2σ3ε3p where WE(B) is the elastic strain energy at the peak point and WD is the dissipative energy before the peak. Bpre 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, Bpre = 1, whereas when the concrete is in the fully plastic state, Bpre = 0. Therefore, the larger the value of Bpre, 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

W ( ) W Wr W Wr B = 1 F post = E(B)− = E(B)− post W +W Wr W +W Wr W − E(B) F(post)− E(B) F(post)− f σ2 σ2 4µ(σ σ )σ (18) 1p− 1r− 1p− 1r 3 = " #   R ε1r 2 2 2E σ1dε1+2σ3(ε3r ε3p) + σ σ 4µ(σ1p σ1r)σ3 ε1p − 1p− 1r− − where WF(post) is the additional energy provided by the external testing machine, Wr is the residual elastic strain energy, and Wf is the fracture energy. Bpost 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, Bpost = 0; when concrete is in an ideal brittle state, Bpost = 1. Therefore, the larger the value of Bpost, the more obvious the brittleness of concrete. Crystals 2020, 10, 1099 12 of 16

4.2. Brittleness Evaluation Method

Since both Bpre and Bpost 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 Bpre and Bpost is used to characterize the brittleness index of the whole stress–strain process as follows: B = BpreBpost (19) The value range of the brittleness index B is (0, 1). For the ideal brittle state of concrete, B = Bpre = Bpost = 1; for the ideal plastic state of concrete, B = Bpre = Bpost = 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 Table4. 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. Figure9 shows the variation of brittleness index with confining pressure; the brittleness indexes Bpre, Bpost 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.

σ 3 2 C60 : B = 0.008 + 0.32 e− 2.36 ,R = 0.999 (20) · σ3 C70 : B = 0.008 + 0.39 e− 2.81 (21) ·

Table 4. Calculation results of different brittleness evaluation indexes.

Bpre Bpost BB1 B2 B3 σ3/MPa C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 C60 C70 0 0.679 0.796 0.484 0.505 0.329 0.402 2.116 3.911 0.403 0.453 0.442 0.518 5 0.394 0.556 0.116 0.130 0.046 0.072 0.649 1.254 0.117 0.140 1.777 1.795 10 0.280 0.305 0.061 0.083 0.017 0.025 0.388 0.439 0.090 0.105 2.238 2.154 15 0.234 0.190 0.039 0.038 0.009 0.007 0.306 0.235 0.072 0.068 2.941 3.115 20 0.167 0.190 0.028 0.029 0.005 0.006 0.201 0.235 0.065 0.057 3.187 4.305 Notes: the standard deviation of brittleness evaluation indexes for C60 and C70 is 0.0029 and 0.0044, respectively. Crystals 2020, 10, 1099 13 of 16

Comparing the brittleness evaluation indexes calculated by different evaluation methods in Figure9, we can see that the brittleness index B in this paper has a high consistency with the trend of B1 and B2, and the values of B, B1, and B2 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 B B Crystalswith 2020 the, 10 confining, x FOR PEER pressure. REVIEW However, 1 and 2 only consider the impact of recoverable elastic energy15 of 19 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 increases with the increase of confining pressure, indicating that B is positively correlated with reflect the brittle behavior of the whole process of concrete failure, which3 indicates that these two the confiningbrittleness pressure, characterization and the methods value rangeB1 and ofB2 haveB3 is certain larger, limitations. which is not The conducive brittleness levelto measuring reflected the brittlenessby B3 is level contrary of the to material, reality; the as value the ofnumeratorB3 gradually (fracture increases energy) with the and increase denominator of confining (nominal pressure, stress) indicating that B is positively correlated with the confining pressure, and the value range of B is in the expression do3 not belong to the same order of magnitude, so the physical meaning of3 B is larger, which is not conducive to measuring the brittleness level of the material, as the numerator 3 not sufficiently(fracture energy) clear. and The denominator brittleness (nominal evaluation stress) index in the expressionB proposed do not in belong this paper to the sameis established order basedof on magnitude, the whole so theprocess physical of meaningconcrete of energyB3 is not evol suffiutionciently with clear. a Theclear brittleness physical evaluation meaning: index the Bvalue rangeproposed is (0, 1), in and this the paper value is established of B is basedcontinuous on the and whole monotonous, process of concrete so it has energy good evolution adaptability. with In summary,a clear physicalthe brittleness meaning: level the repr valueesented range is by (0, the 1), and brittleness the value index of B is continuousB is more and consistent monotonous, with the physicalso it hastest goodresults, adaptability. and it has In a summary, better evaluation the brittleness effect level on representedthe brittleness by the of brittlenesshigh-strength index concreteB is materialsmore consistentunder differen with thet confining physical testpressures. results, and it has a better evaluation effect on the brittleness of high-strength concrete materials under different confining pressures.

0.75 3.2 0.8 4.5 Bpre

Bpost B 0.60 Bpre B 2.4 0.6 2 Bpost 2 2 B1 B B B 3.0 B 、 0.45 、 3

B 3 B B 2 3 B B 、 、 、 B 1.6 、 0.4 1 1 1 post post B B B B B3 、 、 0.30 pre pre 1.5 B B 0.8 0.2 0.15

0.00 0.0 0.0 0.0 0 5 10 15 20 0 5 10 15 20 Confining Pressure (MPa) Confining Pressure (MPa) (a) (b)

FigureFigure 9. Relationship 9. Relationship between between brittlenessbrittleness evaluation evaluation index in anddex confining and confining pressure ofpressure high-strength of high-strengthconcrete. concrete.(a) C60; (a) ( bC60;) C70. (b) C70. 4.3.2. Analysis of Influencing Factors 4.3.2. Analysis of Influencing Factors The relationship between the characteristic strength of concrete under triaxial compression and the brittlenessThe relationship evaluation between index B theis showncharacteristic in Figure streng 10. Theth figure of concrete shows thatunder the triaxial brittleness compression evaluation and the brittlenessindex B is negatively evaluation correlated index withB is the shown peak strength in Figure and 10. residual The strength,figure shows showing that an exponentialthe brittleness evaluationfunction index relationship. B is negatively With the increasecorrelated of thewith peak the strength peak strength and residual and residual strength, strength, the brittleness showing an exponentialevaluation indexfunctionB gradually relationship. decreased, With and the theincrease brittleness of the level peak of concretestrength gradually and residual decreased, strength, the brittlenessindicating that evaluation the confining index pressure B gradually can inhibit decreased, the initiation and and the propagation brittleness of micro-crackslevel of concrete in graduallyconcrete decreased, and increase indicating the threshold that stress the forconfining micro-crack pressure initiation can and inhibit propagation the ininitiation concrete, and propagationthereby improving of micro-cracks the load-bearing in concrete capacity and ofincrea concretese the and threshold reducing itsstress brittleness for micro-crack level. initiation and propagation in concrete, thereby improving the load-bearing capacity of concrete and reducing its brittleness level.

Crystals 2020, 10, x FOR PEER REVIEW 16 of 19 Crystals 2020, 10, 1099 14 of 16

0.36 0.42

σ 0.36 0.30 x 1p σ − 1p y =+0.007 551.30 ⋅ e 8.64 σ 1r x σ 2 − R0.999= 0.30 y =+0.007 12289.15 ⋅ e 8.23 1r 0.24 R2 = 0.999

− x 0.24 y =+⋅0.009 1.17 e 11.31 0.18 − x 17.69 B 2 = B y =+⋅0.007 0.89 e R0.999 0.18 R0.9992 = 0.12 0.12

0.06 0.06

0.00 0.00 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 160 Strength Parameter (MPa) Strength Parameter (MPa) (a) (b)

FigureFigure 10. Relationship10. Relationship between between brittleness brittleness evaluation evaluation index index and and strength strength parameters.parameters. ( (aa)) C60; C60; ( (bb) ) C70. C70. The relationship between the brittleness index B and the elastic energy stored before the peak, the total energyThe relationship before the between peak, the the brittleness additional index energy B and and the the elastic fracture energy energy stored is before shown the in peak, Figure 11. It canthe be total seen energy from before Figure the peak,11 that the theadditional brittleness energy evaluation and the fractureB has energy an exponential is shown in Figure function 11. and 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, 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 the additional energy, and the fracture energy. In the pre-peak stage, with the increase of confining pressure,pressure, the the load-bearing load-bearing capacity capacity ofof thethe concreteconcrete increases. increases. At Atthis this time, time, the thepropagation propagation and and penetration of the micro-cracks inside the specimen needs to dissipate more energy, so WD gradually penetration of the micro-cracks inside the specimen needs to dissipate more energy, so WD increases with the increase of confining pressure. The ratio of WD to WF(pre) Wincreases gradually, gradually increases with the increase of confining pressure. The ratio of WD to F(pre) increases while that of WE(B) to WF(pre) decreases gradually and the storage capacity of elastic energy decreases gradually, while that of WE(B) to WF(pre) decreases gradually and the storage capacity of elastic gradually, resulting in the gradual weakening of the brittleness level of concrete. In the post-peak energy decreases gradually, resulting in the gradual weakening of the brittleness level of concrete. In W W W stage, as the confining pressure increases, r and F(post) graduallyW increases and the ratio of F(post) the post-peak stage, as the confining pressure increases, Wr and F(post) gradually increases and to Wf gradually increases. This is because the confining pressure increases the threshold stress for the ratio of WF(post) to W gradually increases. This is because the confining pressure increases the crackCrystals initiation 2020, 10, x and FOR improvesPEER REVIEWf the residual strength of the concrete specimen at the post-peak17 of stage; 19 threshold stress for crack initiation and improves the residual strength of the concrete specimen at furthermore, the self-sustaining fracture ability of the specimen weakens at the post-peak stage, the post-peak stage; furthermore, the self-sustaining fracture ability of the specimen weakens at the thus weakening its brittleness. post-peak stage, thus weakening its brittleness.

0.36 0.42 W W Fpre () Fpre () W 0.30 W 0.35 EB() EB() x − W F() post =+⋅3.37 W Fpost() x y 0.007 23.78 e − W =+ ⋅3.41 f 2 W y 0.009 328.47 e 0.24 R0.999= f 0.28 R0.9982 = − x 15.61 y =+⋅0.009 1.26 e − x y =+⋅0.013 3.90 e 12.51 0.18 R2 = 0.999 0.21 B B R0.9962 =

− x − x y =+⋅0.004 0.53 e 74.62 0.12 y =+⋅0.006 0.42 e 54.98 0.14 2 = 2 = R0.999 R 0.999 − x y =+⋅0.006 0.53 e 56.61 − x 0.06 0.07 y =+⋅0.004 0.72 e 75.04 2 R= 0.999 R0.9992 =

0.00 0.00 0 60 120 180 240 300 360 420 0 80 160 240 320 400 480 560 640 Energy Parameter (J) Energy Parameter (J) (a) (b)

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

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high-strength concrete specimen is damaged, the ratio of the additional energy WF(post) provided by the outside world to the fracture energy is proportional to the confining pressure. 2. Based on the energy evolution law in the full stress–strain process of high-strength concrete, the pre-peak and post-peak brittleness indexes Bpre and Bpost are defined; then, according to the positive correlation between Bpre, Bpost and the brittleness of the concrete, a method to characterize the brittleness index B of the whole stress–strain process by the product of Bpre, Bpost is proposed. 3. The brittleness evaluation index B is negatively correlated with the peak strength and residual strength, showing an exponential function relationship, and it has an exponential function and negative correlation with the elastic energy stored before the peak, the total energy before the peak, additional energy, and fracture energy. The comparative analysis of different brittleness evaluation methods shows that the brittleness evaluation index method proposed in this paper presents a continuous and monotonous brittleness index and can better characterize the brittleness level of high-strength concrete under different stress conditions.

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.

References

1. Morley, A. ; Longmans Green and Company: London, UK, 1944; pp. 71–72. 2. Hetenyi, M. Handbook of Experimental Stress Analysis; John Wiley: New York, NY, USA, 1966; pp. 23–25. 3. Ramsay, J.G. Folding and Fracturing of Rocks; McGraw-Hill: London, UK, 1967; pp. 44–47. 4. Lawn, B.R.; Marshall, D.B. , toughness, and brittleness: An indentation analysis. J. Am. Ceram. Soc. 1979, 62, 347–350. [CrossRef] 5. Quinn, J.B.; Quinn, G.D. Indentation brittleness of : A fresh approach. J. Mater. Sci. 1997, 32, 4331–4346. [CrossRef] 6. Li, X.; Xue, W.; Fu, C. Mechanical properties of high-performance -fibre- and its application in underground mine engineering. Materials 2019, 12, 2470. [CrossRef][PubMed] 7. Chen, Z.; Tang, J.; Zhou, X.; Zhou, J.; Chen, J. Interfacial bond behavior of high strength concrete filled steel tube after exposure to elevated temperatures and cooled by fire hydrant. Materials 2019, 13, 150. [CrossRef] [PubMed] 8. Eid, R.; Muravin, B.; Kovler, K. Acoustic emission monitoring of high-strength concrete columns subjected to compressive axial loading. Materials 2020, 13, 3114. [CrossRef][PubMed] 9. Jenq, Y.; Shah, S.P. Two parameter fracture model for concrete. J. Eng. Mech. 1985, 111, 1227–1241. [CrossRef] 10. Tasdemir, C.; Tasdemir, M.A. Effects of silica fume and aggregate size on the brittleness of concrete. Cem. Concr. Res. 1996, 26, 63–68. [CrossRef] 11. Zhang, B.; Bicanic, N.; Pearce, C.J. Relationship between brittleness and moisture loss of concrete exposed to high temperatures. Cem. Concr. Res. 2002, 32, 363–371. [CrossRef] 12. Yan, A.; Wu, K.; Zhang, D.; Yao, W. The relationship between the brittleness and fracture surface characteristics of high-strength concrete. J. Tongji Univ. 2002, 30, 66–70. (In Chinese) 13. Lange, D.A.; Hamlin, M.; Surendra, P.S. Relationship between fracture surface roughness and fracture behavior of cement paste and mortar. J. Am. Ceram. Soc. 1993, 76, 589–597. [CrossRef] 14. Yao, M.; Zhan, B.; Huang, X. Experimental study on the effect of curing on the brittleness of high performance concrete. J. Hefei Univ. Technol. 2004, 11, 1438–1442. (In Chinese) 15. Guo, X. Study on the Brittleness Evaluation Method and Toughening Measures of High-Strength Concrete. Ph.D. Thesis, Wuhan University, Wuhan, China, 2005. (In Chinese). 16. Huang, Y.; Qian, J. Brittleness and strength size effect of high-strength and ultra-high-strength concrete. Ind. Constr. 2005, 35, 15–17. (In Chinese) Crystals 2020, 10, 1099 16 of 16

17. Huang, Y.; Qian, J.; Zhou, X. Study on the brittleness index of quasi-brittle materials based on the strength size effect. Eng. Mech. 2006, 23, 38–42. (In Chinese) 18. Han, J.; Yan, P. A new method of concrete brittleness evaluation-fractured stone area fraction method. Concrete 2011, 2, 1–3. (In Chinese) 19. Ministry of Housing and Urban-Rural Development of China. Specification for Design of Concrete Mixture Ratio; JGJ55-2011; China Construction Industry Press: Beijing, China, 2011. (In Chinese) 20. Xie, H.P.; Ju, Y.; Li, L.Y.; Peng, R.D. Energy mechanism of deformation and failure of rock masses. Chin. J. Rock Mech. Eng. 2008, 27, 1729–1740. (In Chinese) 21. Zhang, L.; Gao, S.; Wang, Z.; Cong, Y. Analysis of marble failure energy evolution under loading and unloading conditions. Chin. J. Rock Mech. Eng. 2013, 32, 1572–1578. (In Chinese) 22. Tian, Y.; Yu, R. Energy analysis of limestone during triaxial compression under different confining pressures. Rock Soil Mech. 2014, 35, 118–122. (In Chinese) 23. Xie, H.P.; Li, L.Y.; Ju, Y.; Peng, R.D.; Yang, Y.M. Energy analysis for damage and catastrophic failure of rocks. Sci. China Technol. Sci. 2011, 54, 199–209. [CrossRef] 24. Wu, J. Elasticity; Higher Education Press: Beijing, China, 2011. (In Chinese)

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