*3.1. Energy Analysis*

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

According to the first law of thermodynamics, regardless of the influence of external temperature change and material exchange on the test system, the input energy *WF* is equal to the sum of the elastic strain energy *WE* and dissipation energy *WD* in the test process [14,15].

$$\mathcal{W}\_F = \mathcal{W}\_E + \mathcal{W}\_D \tag{1}$$

The input energy mainly includes the work done by the axial force and confining pressure when the concrete sample deforms, and the elastic strain energy is the energy accumulated in the interior of concrete sample when the elastic deformation occurs because (a) the elastic deformation is reversible, and (b) because the elastic strain energy is also reversible. The dissipation energy mainly includes the (a) surface energy consumed during the initiation, development and penetration of cracks, (b) plastic strain energy for irreversible plastic deformation of concrete samples, (c) heat energy generated by friction and slip between cracks, and various radiation energy [16,17].

The energy input to the concrete specimen by the external force during the test can be expressed as [18],

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

where *σ*1, *σ*<sup>3</sup> are the maximum and minimum principal stresses, respectively, *ε*<sup>1</sup> and *ε*<sup>3</sup> are axial and lateral strains, respectively, *UF* is the input energy density, *V* is the volume of the concrete sample, and *D* and *H* are the diameter and height of the concrete sample, respectively.

Similarly, the elastic strain energy and dissipation energy are obtained as follows,

$$\begin{cases} \begin{aligned} \,^\*W\_E &= \frac{\pi}{4} D^2 H U\_E = V \mathcal{U} I\_E\\ \,^\*W\_D &= \frac{\pi}{4} D^2 H U\_D = V \mathcal{U} I\_D \end{aligned} \tag{3} \end{cases} \tag{3}$$

where *UE* and *UD* are the elastic strain and dissipation energy densities, respectively. Substituting Equations (2) and (3) into Equation (1), we obtain,

$$
\mathcal{U}\_F = \mathcal{U}\_E + \mathcal{U}\_D \tag{4}
$$

According to the elastic theory (Gong et al., 2018), the elastic-strain energy density is obtained as follows,

$$\mathcal{U}\_E = \frac{1}{2} (\sigma\_1 \varepsilon\_1^\mathbf{e} + 2\sigma\_3 \varepsilon\_3^\mathbf{e}) = \frac{1}{2E} \left[ \sigma\_1^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu \sigma\_1 \sigma\_3 \right] \tag{5}$$

By substituting Equations (2) and (5) into Equation (4), the dissipation energy density is obtained as follows,

$$\mathcal{U}\_D = \int\_0^{\varepsilon\_1} \sigma\_1 \mathbf{d} \varepsilon\_1 + 2 \int\_0^{\varepsilon\_3} \sigma\_3 \mathbf{d} \varepsilon\_3 - \frac{1}{2E} \left[ \sigma\_1^2 + 2(1 - \mu)\sigma\_3^2 - 4\mu \sigma\_1 \sigma\_3 \right] \tag{6}$$

where *E* is the elastic modulus, and *μ* is the Poisson's ratio.

#### *3.2. Relationship between Energy Density and Axial Strain*

During the loading process of high-strength concrete samples, the changes of input energy, elastic-strain energy, and dissipated energy are always accompanied by the changes of initial hole compaction, elastic deformation, new crack propagation, and penetration. The energy coexisting in the specimen is not isolated from each other, but transformed with the concrete deformation process, and finally changed from the equilibrium state before the test to the new equilibrium state after the test. Based on the conventional triaxial compression test results of high-strength concrete (C60 and C70) at different confining pressures, the curves of input energy density, elastic-strain energy density and dissipation energy density with axial strain at different confining pressures are obtained according to Equations (2), (5), and (6). The results are shown in Figure 4.

**Figure 4.** *Cont*.

**Figure 4.** Relationships between energy density and axial strain of high-strength concrete (C60 and C70) at different confining pressures. (**a**) C60, *σ*<sup>3</sup> = 0 MPa; (**b**) C70, *σ*<sup>3</sup> = 0 MPa; (**c**) C60, *σ*<sup>3</sup> = 5 MPa; (**d**) C70, *σ*<sup>3</sup> = 5MPa; (**e**) C60, *σ*<sup>3</sup> = 10 MPa; (**f**) C70, *σ*<sup>3</sup> = 10 MPa; (**g**) C60, *σ*<sup>3</sup> = 15 MPa; (**h**) C70, *σ*<sup>3</sup> = 15 MPa; (**i**) C60, *σ*<sup>3</sup> = 20 MPa; (**j**) C70, *σ*<sup>3</sup> = 20 MPa.

It can be observed from Figure 4 that the input energy density and dissipation energy density of the two strength concrete samples at different confining pressures increase as a function of the axial strain. This indicates that the external force always inputs energy to the concrete samples during the entire test process, and at the same time, the energy is gradually dissipated. At the end of the test, the increasing trend of the input energy density slows down owing to the expansion of the sample and the negative work done by confining pressure offset part of the positive work done by axial stress. The elastic-strain energy density increases first and then decreases, as a function of the axial strain, and reaches the maximum value at the peak stress. This indicates that the pre-peak stage is mainly associated with the storage process of elastic-strain energy, and the post-peak stage is mainly associated with the release process of the elastic-strain energy.

Comparing the elastic-strain energy density curves of the two types of high-strength concrete and the stress–strain curves at the same confining pressure, it is found that the change trend of elastic-strain energy density with axial strain is similar to those of concrete samples. At the beginning of the test, the micro-holes and cracks in the concrete gradually close subject to the action of load, the stiffness of the concrete increases, and the curve becomes concave. At this time, most of the work done by the external force is converted into elastic-strain energy, it is stored in the sample, and the dissipation energy is almost zero. The elastic-strain energy density curve basically coincides with the input energy density curve. When the load exceeds the elastic limit of the concrete sample, new cracks will appear in the concrete. The generation and diffusion of the new cracks need to dissipate part of the surface energy, and the crack tip produces acoustic emission energy owing to the

stress concentration effect accompanied by irrecoverable plastic strain energy and various radiation energy sources. As a result, the slope of the elastic-strain energy density curve slows down, and the dissipation energy density increases as a function of the axial strain. When the load exceeds the compressive strength of concrete, the concrete will be destroyed. At this time, the elastic-strain energy stored in pre-peak is released rapidly, and most of the input energy of external force is dissipated rapidly by the action of crack initiation, propagation and penetration, as well as the friction of fracture surface, thus resulting in an abrupt increase of dissipation energy density after the peak as a function of strain.

The results show that the storage of elastic-strain energy before the peak of concrete specimen is mainly elastic-strain energy, and constitutes the primary source of concrete failure. At the same time, with the dissipation of energy, the dissipation energy will gradually reduce the bearing capacity of concrete samples. After the peak, the elasticstrain energy is mainly released, and the released elastic-strain energy is transformed into various forms of energy dissipation, so the post-peak dissipation energy accounts for a large proportion of the input energy.

#### *3.3. Relationship between Energy Density Corresponding to Peak Stress and Confining Pressure*

Figure 5 shows the relationship between the confining pressure and input energy density *UFP* and dissipated energy density *UDP* corresponding to the peak stress of the two types of high-strength concrete samples. It can be observed from Figure 5a,b that the input energy density *UFP* and the dissipation energy density *UDP* corresponding to the peak stress of the two types of high-strength concrete samples increase as a function of the confining pressures, and the *UFP* and *UDP* of the C70 high-strength concrete samples subjected to the same confining pressure are higher than those of the C60 high-strength concrete. This is attributed to the fact that as the strength of the concrete increases, the system needs to input more energy to cause its failure. At the same time, the compressive capacity of high-strength concrete gradually decreases, and thus needs to dissipate more energy.

**Figure 5.** Relationships between confining pressure and *UFP* and *UDP*. (**a**) Relationship between *UFP* and *σ*3; (**b**) Relationship between *UDP* and *σ*3.

Figure 6 shows the relationship between the confining pressure and elastic-strain energy density *UEP* corresponding to peak stress for two types of high-strength concrete samples. As it can be observed from the figure, *UEP* and the confining pressure are linearly related, and the correlation coefficients are all above 0.99. It can be observed from Figure 6 that as *UEP* increases, the energy release at high-confining pressures is more rapid and abrupt compared with low-confining pressures.

**Figure 6.** Relationship between confining pressure and *UEP*.

#### *3.4. Relationship between Final Energy Density and Confining Pressure*

Figure 7 shows the relationship between the confining pressure and final elastic-strain energy density *UER* of two types of high-strength concrete specimens after complete failure. It can be observed from Figure 7 that the *UER* values of the two types of high-strength concrete samples increase as a function of the confining pressure. Considering the C60 high-strength concrete sample as an example, when the confining pressure is 0, 5, 10, 15, and 20 MPa, the *UER* values are 0.0038 × <sup>10</sup><sup>6</sup> J/m3, 0.0244 × <sup>10</sup><sup>6</sup> J/m3, 0.0665 × 106 J/m3, 0.10150665 × 106 J/m3, and 0.15330665 × 106 J/m3, respectively. This shows that in the postpeak failure stage, the elastic-strain energy stored before the peak is not completely released, and a small part of the elastic strain energy is still stored in the sample. Accordingly, the larger the confining pressure is, the greater the residual elastic-strain energy is. This indicates that the confining pressure limits the release of the elastic-strain energy to a certain extent in the post-peak stage. Macroscopically, the concrete sample still has a finite bearing capacity and residual strength after failure. In addition, the larger the elastic strain energy remaining in the specimen is, the greater the residual strength of the concrete sample.

**Figure 7.** Relationship between confining pressure and final elastic-strain energy density.

#### **4. Statistical Damage Constitutive Model of High-Strength Concrete**

*4.1. Establishment of Constitutive Model*

According to Lemaitre's strain equivalent principle [19], the strain response produced by the nominal stress acting on the damaged material is equivalent to the strain response produced by the effective stress that acts on the undamaged material. Therefore, the constitutive relationship of the damaged material can be obtained by replacing the nominal stress with the effective stress.

$$
\sigma = \sigma^\*(1 - D) = E\varepsilon(1 - D) \tag{7}
$$

where *σ* and *σ*∗ are nominal and effective stresses, respectively, and *D* is the damage variable.

In the conventional triaxial compression test of concrete, *σ*1, *σ*3, and *ε*1, can be measured, and the corresponding effective stresses are *σ*∗ <sup>1</sup> and *σ*<sup>∗</sup> <sup>3</sup> . According to Equation (7) and the generalized Hooke's law [20,21], we can obtain,

$$\begin{cases} \varepsilon\_1 = \frac{1}{E} \left( \sigma\_1^\* - 2\mu \sigma\_3^\* \right) \\\\ \sigma\_1^\* = \frac{\sigma\_1}{1 - D} \\\\ \sigma\_3^\* = \frac{\sigma\_3}{1 - D} \end{cases} \tag{8}$$

The high-strength concrete sample is equivalent to the macrostructure composed of innumerable tiny microelements, wherein the microelements are small enough at the macroscale, and the microelements are large enough at the microscale. The change of mechanical properties of each microelement is equivalent to the change of the mechanical properties of the high-strength concrete sample [22]. The microelement is a linear elastic body before failure, and the stress–strain relationship satisfies Hooke's law. Because the concrete is a mixture of various mineral materials and cementitious materials, it has an obvious heterogeneity, and its mechanical properties show the characteristics of random distribution. The resulting damage is also randomly distributed in concrete materials. Therefore, the mechanical properties of concrete microelements can be described mathematically by a statistical method, based on the assumption that the ratio of dissipation energy density *UD* to dissipated energy density *UDP* at peak stress obeys the Weibull statistical law, and the probability density function can be expressed as follows,

$$\begin{cases} \chi = \frac{\underline{U}\_D}{\underline{U}\_{Dp}}\\\ P[\chi] = \frac{k}{m} \left(\frac{\chi}{m}\right)^{k-1} \exp\left[-\left(\frac{\chi}{m}\right)^k\right] \end{cases} \tag{9}$$

where *P*[*χ*] is the function of probability density, *m* is the scale parameter, and *k* is the shape parameter of the distribution.

The dissipation energy increases as a function of the axial strain. When the dissipation energy density reaches a certain level, the number of damaged microelements in concrete samples can be expressed as,

$$N\_D = N \int P[\chi] \mathbf{d}(\chi) = N \int \frac{k}{m} (\frac{\chi}{m})^{k-1} \exp\left[-\left(\frac{\chi}{m}\right)^k\right] \mathbf{d}(\chi) = N \left\{1 - \exp\left[-\left(\frac{\chi}{m}\right)^k\right]\right\} \tag{10}$$

where *ND* is the number of damaged microelements, and *N* is the number of total microelements.

The damage variable *D* is the ratio of damaged microelements to the total number of microelements [23], and is expressed as follows,

$$D = \frac{N\_D}{N} \tag{11}$$

By substituting Equation (11) into Equation (10), the statistical damage variables describing the damage characteristics of concrete are obtained as follows,

$$D = 1 - \exp\left[-\left(\frac{\chi}{m}\right)^k\right] \tag{12}$$

The statistical damage constitutive relation of high-strength concrete based on Weibull distribution can be obtained by substituting Equation (12) into Equation (8),

$$
\sigma\_1 - 2\mu \sigma\_3 = E \varepsilon\_1 \exp\left[-\left(\frac{\mathcal{X}}{m}\right)^k\right] \tag{13}
$$

#### *4.2. Verification of Constitutive Model*

The key to establish the statistical damage constitutive relation of high-strength concrete is to determine the Weibull distribution parameters *m* and *k*. Applying the logarithms twice on Equation (13), the following equation is obtained:

$$\ln\left[-\ln\left(\frac{\sigma\_1 - 2\mu\sigma\_3}{E\varepsilon\_1}\right)\right] = k\ln(\chi) - k\ln m \tag{14}$$

This is a linear equation with a slope coefficient equal to *k* and an intercept equal to −*k* ln *m*. Therefore, parameters *m* and *k* can be easily determined based on a linear regression analysis on a set of triaxial test data of concrete samples, as shown in Table 4. The statistical damage constitutive relation curves of high-strength concrete are obtained by substituting *m* and *k* into Equation (13). The results are shown in Figure 8.

**Table 4.** Weibull distribution parameters *m* and *k* at different confining pressures.


**Figure 8.** Comparison of constitutive model and test results. (**a**) C60 and (**b**) C70.

According to the comparison results of the statistical damage constitutive model curves and test curves of high-strength concrete (C60 and C70) at different confining pressures in Figure 8, it can be observed that the theoretical curves of the statistical damage constitutive model of high-strength concrete are in good agreement with the experimental curves in both the pre-peak elastic strain energy storage stage and the post-peak elastic strain energy release stage, and the correlation coefficient is above 0.96. Conversely, this constitutive model overcomes the defect of low correlation between concrete constitutive model and test results in the post-peak stage, and improves the accuracy of the model. Conversely, it is verified that the statistical damage constitutive model established from the energy theory and statistical damage theory is suitable to describe the constitutive behavior of high-strength concrete. Although the theoretical curve of the constitutive model established in this paper is in good agreement with the experimental curve, because the model parameters are obtained by fitting the experimental curve, whether the model can predict the total stress–strain curve of high-strength concrete needs further study.

Standard deviation and relative standard deviation can measure the deviation between the experimental and theoretical model results [24]. To verify the accuracy of the statistical damage constitutive model of high-strength concrete, the standard deviation and relative standard deviation of triaxial stress–strain test curve and the theoretical curve of two types of high-strength concrete under five confining pressures are calculated based on Equation (15). The results show that the relative standard deviations between the statistical damage constitutive model results and the test results of C60 high strength concrete are 7.56%, 3.54%, 2.45%, 2.40%, and 2.26%, respectively, at the confining pressures of 0, 5, 10, 15, and 20 MPa, and the average relative standard deviation is only 3.64%. Additionally, the relative standard deviation between the statistical damage constitutive model results and the test results of C70 high-strength concrete are 6.49%, 2.27%, 3.84%, 4.30%, and 3.03%, respectively, and the average relative standard deviation is only 3.99%. The error analysis further shows that the statistical damage constitutive model of high-strength concrete is reasonable and feasible.

$$\begin{cases} \quad \eta = \sqrt{\frac{\sum\_{i=1}^{n} \left(\sigma\_{s} - \sigma\_{l}\right)^{2}}{n}}\\ \quad f = \frac{\eta}{\sigma\_{c}} \end{cases} \tag{15}$$

where *η* is the standard deviation, *f* is the relative standard deviation, *σs*, *σ<sup>l</sup>* are the test values and theoretical values, respectively, *σ<sup>c</sup>* is the compressive strength, and *n* is the data volume.

#### *4.3. Damage Analysis of High-Strength Concrete*

The Weibull distribution parameters *m* and *k* in Table 3 are substituted into Equation (12) to obtain the relationship between the damage variable and axial strain, as shown in Figure 9.

It can be observed from Figure 9 that the damage evolution curves of C60 and C70 high-strength concrete at different confining pressures are similar to the "S" curves. At the initial stage of loading, the damage weakening is not obvious. Owing to the continuous closure of micro-pores and micro-cracks in concrete samples subjected to pressure, the concrete gradually transforms from discontinuous medium to a quasi-continuous medium. With the increase of axial strain, the concrete microdefects are further compacted and closed, and the concrete enters the elastic deformation stage. Because the stress level at this stage is not enough to cause the crack to begin to expand, the microdefects will not decrease after their closure, that is, the damage cannot occur in the real linear elastic stage. However, in the elastic deformation stage of concrete, the damage variable is still increasing slowly. This indicates that the concrete at this time not only includes the elastic deformation, but also the mutual sliding of closed cracks that shows the existence of nonlinearity at low-stress levels. When the stress of concrete exceeds a certain level or its deformation reaches a certain value, new micro-cracks begin to sprout and expand slowly between the relatively

weak particle boundaries, the concrete yields and produces plastic deformation, and the damage of concrete begins to evolve and expands steadily. With the increase of the stress level, the micro-cracks in the concrete are concentrated, expanded and penetrated locally, thus forming macro-cracks, and the concrete damage develops rapidly. The main fracture surface is formed by the ladder connection of the macro-cracks that leads to the sudden release of stress and the rapid decrease of concrete strength, thus resulting in damage. However, owing to the incomplete release of elastic strain energy, the damage variable is slightly less than one. With the increase of confining pressure, the rate of change of damage with strain decreases. This indicates that the increase of confining pressure can effectively inhibit the release of elastic-strain energy and the development of damage and will improve the stress state of concrete.

**Figure 9.** Relationship between damage variable *D* and axial strain. (**a**) C60 and (**b**) C70.

#### **5. Conclusions**


improved the accuracy of the model. It was also verified that the statistical damage constitutive model established from the energy theory and statistical damage theory was suitable in describing the constitutive behavior of high-strength concrete.

**Author Contributions:** Conceptualization, L.Z. and J.L.; methodology, L.Z.; validation, X.W. and L.G.; writing—original draft preparation, L.Z.; writing—review and editing, H.C.; funding acquisition, H.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (51874005).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used to support the findings of this study are available from the corresponding author upon request.

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