*5.5. Verification of Damage Constitutive Model*

In order to verify the applicability and rationality of the damage constitutive model that considers the rock residual strength proposed in this paper, the conventional triaxial compression tests of siltstone under different confining pressure conditions (5 MPa, 10 MPa, 15 MPa and 20 MPa) are carried out. The mechanical parameters of the rock samples are as follows: average uniaxial compressive strength *σ<sup>c</sup>* = 49.4 MPa, cohesion *c* = 12.94 MPa, internal friction angle *ϕ* = 22◦ and Poisson's ratio *µ* = 0.25. The parameters of the improved three-shear energy criterion are obtained by linear regression, *<sup>a</sup>* <sup>=</sup> 1.42 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , *b* = −0.035. The calculated parameters of the constitutive model are shown in Table 3. The parameters of the damage statistical model are substituted into the formula, and the theoretical curve is created, which is compared with the test curve of the conventional triaxial compression under four confining pressures, as showed in Figure 5.

**Table 3.** Parameter values of constitutive model under different confining pressures.


Figure 5 presents the comparison results of the theoretical curves and the test curves under different confining pressures. It can be seen from Figure 5 that the damage constitutive equation of siltstone proposed in this paper can better reflect the actual mechanical behavior of rock, and its initial deformation modulus and peak strength are approximately the same as the test results. The consistency between the constitutive model curve and the test curve is high, which overcomes the defect in some constitutive models that cannot describe the residual strength in the post peak stage, improves the accuracy of the model, and illustrates that the constitutive model based on the energy principle is more reasonable than the traditional constitutive model.

In order to further verify the rationality of the damage constitutive model of siltstone that has been established in this paper, the deviation between the conventional triaxial stress–strain curve and the model curve of siltstone under four different confining pressures is analyzed, and the calculation formula is shown in Formula (48).

$$\begin{cases} \quad \eta = \sqrt{\frac{\sum\_{i=1}^{\eta} \left(\sigma\_{s} - \sigma\_{l}\right)^{2}}{n-1}}\\ \qquad f = \frac{\eta}{\sigma\_{0}} \end{cases} \tag{48}$$

3 /MPa

> where, *η* is the standard deviation, *f* is the relative standard deviation, *σ<sup>s</sup>* , *σ<sup>l</sup>* are the test value and theoretical value respectively, *σ*<sup>0</sup> is the mean value of the test value and *n* is the number of samples. models that cannot describe the residual strength in the post peak stage, improves the accuracy of the model, and illustrates that the constitutive model based on the energy principle is more reasonable than the traditional constitutive model.

> Figure 5 presents the comparison results of the theoretical curves and the test curves under different confining pressures. It can be seen from Figure 5 that the damage constitutive equation of siltstone proposed in this paper can better reflect the actual mechanical behavior of rock, and its initial deformation modulus and peak strength are approximately the same as the test results. The consistency between the constitutive model curve and the test curve is high, which overcomes the defect in some constitutive

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**Table 3.** Parameter values of constitutive model under different confining pressures.

 3.28 × 10−5 −7.10 × 10−5 1.54 × 10−4 −1.94 × 10−8 11,250 −2.28 2.89 × 10−5 −6.25 × 10−5 1.35 × 10−4 −1.71 × 10−8 20,533 −3.65 2.75 × 10−5 −5.95 × 10−5 1.29 × 10−4 −1.63 × 10−8 10,435 −2.19 2.68 × 10−5 −5.81 × 10−5 1.26 × 10−4 −1.59 × 10−8 7784 −1.76

**A B C H**

*b* 0.035

improved three-shear energy criterion are obtained by linear regression,

triaxial compression under four confining pressures, as showed in Figure 5.

. The calculated parameters of the constitutive model are shown in Table 3. The

parameters of the damage statistical model are substituted into the formula, and the theoretical curve is created, which is compared with the test curve of the conventional

4

*a* 1.42 10 ,

*C*<sup>0</sup>

**Figure 5.** Comparison between model curve and test curve: (**a**) 5 MPa; (**b**) 10 MPa; (**c**) 15 MPa; (**d**) 20 MPa. **Figure 5.** Comparison between model curve and test curve: (**a**) 5 MPa; (**b**) 10 MPa; (**c**) 15 MPa; (**d**) 20 MPa.

The calculation results show that when the confining pressure is at 5, 10, 15 and 20 MPa, the relative standard deviations between the constitutive model results and the test results are 3.75%, 4.77%, 4.05% and 4.83% respectively, and the average relative standard deviation is 4.35%, indicating that the difference between the constitutive model results and the test results is minute, which further proves the rationality of the constitutive model that has been established in this paper.

The change curve of the damage variable with axial strain is obtained by substituting the model parameters into Equation (39), as shown in Figure 6. It can be seen from Figure 6 that the damage evolution curve is approximately an "S" curve. There is no obvious damage accumulation in the initial stage of the curve, then there is a rapid rise stage, indicating that alongside the gradual axial loading, the internal microcracks rub and squeeze each other, and the damage accumulates and converges continuously. Finally, the curve gradually flattens, indicating that the internal structure of the rock is completely destroyed. Along with the increase of the confining pressure, the development trend of cumulative damage slows, because the confining pressure inhibits the development of damage and improves the stress state of rock.

**Figure 6.** Variation curve of damage variable with axial strain. **Figure 6.** Variation curve of damage variable with axial strain.

### **6. Conclusions 6. Conclusions**

where,

is the number of samples.

According to the stress–strain curves of siltstone under different confining pressures, the energy evolution characteristics of siltstone samples under conventional triaxial loading are analyzed, the damage constitutive model of siltstone is established, and the rationality of the model established in this paper is verified by using the conventional triaxial test data of siltstone under different confining pressures. The following According to the stress–strain curves of siltstone under different confining pressures, the energy evolution characteristics of siltstone samples under conventional triaxial loading are analyzed, the damage constitutive model of siltstone is established, and the rationality of the model established in this paper is verified by using the conventional triaxial test data of siltstone under different confining pressures. The following conclusions are drawn:

In order to further verify the rationality of the damage constitutive model of siltstone that has been established in this paper, the deviation between the conventional triaxial stress–strain curve and the model curve of siltstone under four different confining

*s l*

 

1

*n*

1

*i*

*n*

0

0

The calculation results show that when the confining pressure is at 5, 10, 15 and 20 MPa, the relative standard deviations between the constitutive model results and the test results are 3.75%, 4.77%, 4.05% and 4.83% respectively, and the average relative standard deviation is 4.35%, indicating that the difference between the constitutive model results and the test results is minute, which further proves the rationality of the constitutive

The change curve of the damage variable with axial strain is obtained by substituting the model parameters into Equation (39), as shown in Figure 6. It can be seen from Figure 6 that the damage evolution curve is approximately an "S" curve. There is no obvious damage accumulation in the initial stage of the curve, then there is a rapid rise stage, indicating that alongside the gradual axial loading, the internal microcracks rub and squeeze each other, and the damage accumulates and converges continuously. Finally, the curve gradually flattens, indicating that the internal structure of the rock is completely destroyed. Along with the increase of the confining pressure, the development trend of cumulative damage slows, because the confining pressure inhibits the development of

2

is the relative standard deviation,

(48)

are the

, *s l*

is the mean value of the test value and *n*

pressures is analyzed, and the calculation formula is shown in Formula (48).

*f*

*f*

 

is the standard deviation,

test value and theoretical value respectively,

model that has been established in this paper.

damage and improves the stress state of rock.

conclusions are drawn: Under different confining pressures, the input energy and dissipative energy of siltstone samples increase with the increase of axial strain, and the elastic strain energy increases at first and then decreases. When the specimen reaches the peak strength, the elastic strain energy gradually decreases and the dissipative energy gradually increases until the specimen is damaged and reaches the maximum and minimum values respectively.

Considering the internal friction characteristics and hydrostatic pressure effect of rock materials, based on the test results of siltstone samples, the three-shear energy yield criterion is improved, the functional relationship between the sum of shear strain energy and hydrostatic pressure is established, and the improved three-shear energy yield criterion is obtained.

Based on the continuous damage theory, the damage evolution equation of rock is derived using the minimum energy consumption principle and the improved three-shear energy yield criterion, and the damage constitutive model of siltstone under complex stress state is established. The model overcomes the defect of some existing damage constitutive models that cannot simulate the residual strength. By comparing the model curve with the test curve, it has been found that the margin of error is small, and the relative standard deviation is 4.35%, which verifies the rationality of the model established in this paper.

**Author Contributions:** Conceptualization, R.Z.; methodology, R.Z.; validation, R.Z. and L.G.; data curation, R.H.; writing—original draft preparation, R.Z.; writing—review and editing, R.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

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

### **References**

