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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