The slope of the elastic deformation phase is the modulus of elasticity. The modulus of elasticity and the trend of change of the specimens under different maintenance conditions are shown in Figure 4.When comparing the modulus of elasticity under normal maintenance, the greater the change in value, the more pronounced the softening effect of the specimen. As can be seen from Figure 5, the modulus of elasticity decreases after maintenance by the acid solution, and the longer the maintenance time, the more acidic the solution, the greater the decrease in its value.The greatest decrease in the modulus of elasticity of the specimens was approximately 34.3% after 90d of maintenance in the pH=2 solution.The longer the maintenance time, the more acidic the solution, and the more significant the softening effect of the specimen.
Figure.5 Elastic modulus under different conditions
As the load continues to increase, the specimen enters the plastic deformation stage, and the peak stress and peak strain of the specimen reflect the strength and deformation. Under acidic conditions, rock-like material minerals gradually dissolve, the longer the time, the stronger the acidity, the more dissolution precipitates minerals, microscopic manifestations of fracture development, porosity increases. After 90d of maintenance in pH=2 solution, the peak stress and strain of the specimens were 2.907MPa and 9.332×10-3 respectively, which represents a decrease in peak stress of about 26.0% and an increase in peak strain of about 1.35 times compared to that under natural maintenance.The brittle characteristics of the rock-like material decrease and the plastic characteristics increase.
Figure.6 Peak stress diagram under different conditions Figure.7 Peak strain diagram under different conditions

Damage constitutive model

The damage to the specimen after acid solution maintenance consists of two main components, the chemical damage Dc from acid corrosion of the solution and the load damageDm from the random distribution of microscopic defects during loading of the uniaxial compression test.The modulus of elasticity of the rock-like material is affected by the corrosion of the acid solution. The modulus of elasticity of the specimen under natural conditions is defined as the initial state E0 , and the modulus of elasticity after corrosion by the acid solution isEc . The deterioration of the specimen is characterised by the magnitude of the change in elastic modulus. The chemical damage Dc can be expressed as follows:
(1)
A large number of microscopic defects are randomly distributed in the rock and the defects can be considered as random damage. Based on a statistical damage mechanics approach to the study, it is assumed that the specimen micro-elements obey the Weibull distribution in uniaxial compression tests. The probability density function is given by:
(2)
where P(ε) is the rock micro-element strength distribution function; ε is the random micro-element strength distribution variable; and m and ε0 are the distribution parameters derived by solving the equation with boundary conditions.Assuming that the total number of micro-elements under load is N and the number of micro-elements with broken rings isn , the statistical damage variable Dm is expressed as:
(3)
When loaded to a certain strain ε , the number of micro-elements breaking the ring n is:
(4)
Combining equations (3) and (4) leads to:
(5)
Thus, the principal equation for damage under uniaxial compression is (6)
Rock materials have residual strength after the compressive damage process. When analysing the deformation and damage of rock materials based on the Weibull distribution function, the effect of residual strength on the strength of micro-elements after damage is not considered.Therefore, this paper considers the effect of residual strength and corrects the damage variable by defining the correction factor as δ , whose expression is:
(7)
where σr is the residual stress andσc is the peak stress.
Combined with the Lemaitre strain equivalence principle, the Weibull distribution statistical damage model for micro-fractures, taking into account residual strength corrections, has the following principal constitutive relationship:
(8)
The key to solving the above main constitutive equation is to obtain the parameters m and ε0. Deriving the strain in equation (8) and introducing the peak-point boundary condition, we have:
(9)
Combining equations (8) and (9) leads to:
(10)
(11)

Verification of principal constitutive equations

The distribution parameters m and ε0 are solved using the peak stress method and boundary conditions proposed above. The solved distribution parameters are substituted into the damage constitutive equation to obtain the theoretical curve of the constitutive equation. The uniaxial compression test curve is compared with the theoretical curve. Figure 8 shows the comparison of stress-strain curves of rock-like materials under different curing conditions.

(a) (b)

(c) (d)
Figure.8 The stress-strain curves of the first group of mixed rock-like materials under different conditions:a pH=4 kept for 60d,b pH=4 kept for 90d,c pH=2 kept for 60d,d pH=2 kept for 90d
As can be seen from Figures 8, the damage principal model established in this paper can better reflect the stress-strain relationship between the stress-elastic deformation phase, the plastic deformation phase and the post-peak phase of rock-like specimens.The test curve deviates slightly from the theoretical curve during the initial compression-density phase of the specimen because the damage variables in the theoretical curve are obtained on the basis of Lemaitre strain equivalents. The closure of randomly distributed micro-cracks in the initial compression-density phase of the specimen, a phase in which the rock-like material has an increased ability to resist deformation, can be considered as a reverse damage process, as opposed to the damage defined by the development of micro-cracks in the instantaneous equations in the uniaxial compression state, without considering the inhibiting effect of the initial micro-crack closure.

Conclusion

(1)Compared with natural maintenance, after acid solution maintenance, the initial compression-density section of the uniaxial compression test stress-strain curve becomes longer and the elastic deformation phase becomes shorter, the Young’s modulus, peak stress and residual strength of the specimen become smaller, and the peak strain and residual strain become larger, and the above characteristics become more pronounced as the maintenance time increases and the pH of the acid solution decreases.
(2)In the early stages of uniaxial compression, micro-crack closure occurs under load and there is a markedly concave section of the stress-strain curve. Relative to the natural state, the slope of the concave section decreases and the absolute value of the strain at the end of the compression-density section increases after maintenance in acid solution.
(3)After the specimens were maintained in the acid solution, the compressive density section end strain, elastic end strain, peak strain and residual strength strain all increased; the longer the specimens were maintained, the more acidic the solution, and the greater the increase.The brittle characteristics of the specimen are reduced and the plastic characteristics are increased.
(4)Based on the strain equivalence principle, the damage equation for the specimen was derived by combining a statistical damage model for the Weibull distribution of micro-cracks and considering the residual strength to correct for the damage variables.The above damage variables are also coupled by considering the effect of acid corrosion to derive a damage principal constitutive model under uniaxial compression.
(5)By introducing the residual strength correction factor and comparing the damage principal model curve and the test curve, the curve has a high degree of agreement and can effectively reflect the damage and destruction process of the specimen in the elastic phase, plastic phase and post-peak phase.

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest

Conflict of interests This study have no conflict of interest.

Authors’Contributions

Yongsheng Liu proposed the research methodology for this paper and guided the conduct of the experiment, Wang Liu was responsible for the operation of the experiment, and Cui Wang was responsible for the processing of the experimental data and the writing of the article.

Acknowledgments

We are grateful to the Jiangxi Provincial Education Department of China for funding the scientific and technological research (Grant no. GJJ2207301 and GJJ2207306 ).
.

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