Constitutive Model and Damage Evolution of Mudstone under the Action of Dry-Wet Cycles

Mudstone is a natural type of geological material, which has different ways of mechanical response between natural state and drywet cycles. According to the complex damage theory of geological materials, rock masses can be considered as a composite material consisting of the structure phase and damage phase. ,e essence of the damage of rock masses is a damage evolution process, during which the deformation energy of the structure phase converts into dissipation energy of the damage phase, and the energy dissipation from phase transformation promotes the structure phase to change into the damage phase. In this study, a customized model test container and a novel test method are applied to study the decay rate of mudstone under different temperatures and over multiple dry-wet cycles.,e decay rate and the damage variable are connected with each other and applied to the damage constitutive equation based on the energy principle to set up the damage evolution equation under the coupled action of dry-wet cycles and loads. Comparison of the proposed model with test results in a literature identifies the rationality of the established model and properly reflects the damage evolution of mudstone.


Introduction
Mudstone is a natural type of geological material.Its mechanical responses vary under the action of natural state and dry-wet cycles [1].After multiple dry-wet cycles, mudstone is easy to weather and disintegrate.For example, the water level variation of reservoir bank for a long time and the underground water variation caused by rainfall and pumping may lead to slope failure or ground subsidence.erefore, researches on the physical and mechanical characteristics of mudstone under the action of dry-wet cycles increase gradually.Cantón et al. [2] found that the number of dry-wet cycles had the strongest influence on weathering.Guo et al. [3] studied the effect of alternation of heat and water on the slaking phenomenon of redbeds and concluded that the disintegration of mudstone was the result of the comprehensive action of wetting, heating, and loading.Qi et al. [4] concluded that the main mechanisms of slaking of red strata mudstone were water absorption, dehydration shrinkage during drying, and mineral dissolution.Even so, currently there are few researches about the damage of mudstone under the action of dry-wet cycles and the damage constitutive model under the coupled action of dry-wet cycles and loads.
Damage evolution of rock is an important issue in the study of rock mass damage mechanics [5][6][7][8][9].e rock material is anisotropic, in which there are microcracks and macroscopic defects such as cracks, bubbles, cavities, and joints.Crack propagation makes the rock material presents complex stress-strain relations under stress, and the continuum damage mechanics, which focuses on studying the continuous deterioration of these defects, is the basic theory to solve this problem [10].In the continuum damage mechanics, rock masses are regarded as a complex [11][12][13][14][15], which is a composite material consisting of the structure phase and damage phase.With effective bearing area of the structure phase as the basis of the definition of the damage variable, the decrease of the effective bearing area is the damage evolution process of materials; namely, the damage variable can be defined as a ratio of the area of the damage phase to the original area of a cross section area.Zhou and Liu [16] obtained the reasonable synthetic mode of stress and strain for a complex theory of geomaterial damage based on the basic concept of continuum mechanics, provided the simpli ed formulas of the synthetic mode of stress and strain for isotropic damage, and put forward a constitutive model for isotropic damage of geomaterial.Zhou et al. [17] connected the damage variable with cross section areas of the structure phase and damage phase, de ned the external force damage power as damage dissipation energy, and introduced it into the soil energy balance equation.Finally, a damage evolution equation is derived based on the energy consumption under the conditions of structural damage.
e damage of rock masses is the process of energy dissipation [18][19][20], and rock damage after dry-wet cycles also can be regarded as the process of energy dissipation.However, at present, there are few researches about the damage evolution and damage constitutive equation of rock masses under the action of dry-wet cycles.Zhu and Zhou [21] analyzed the forming mechanism of the dissipative structure during the softening process of saturated soft rocks based on the dissipative structure theory.Chen et al. [22] set up damage evolution equation of granite based on the principle of energy dissipation by the uniaxial compression test under multiple dry-wet cycles.According to the irreversible thermodynamics theory, the work done by the external force partly changes into phase transformation damage dissipation energy due to the transformation from the structure phase to the damage phase.
ere is also a certain energy relationship during the process of dry-wet cycles, and a deeper understanding of the damage characteristics of rock masses under the action dry-wet cycles can be realized by analyzing the energy change law of rock damage.erefore, a customized model test container and a novel test method are applied to conduct the dry-wet cycle test on mudstone in Chongqing, and the change law of the decay rate under di erent temperatures and over multiple dry-wet cycles is obtained.According to the experimental results and damage evolution equation based on the energy principle, the damage evolution equation under the coupled action of dry-wet cycles and loads is set up.By comparing the test results from a literature with the proposed model, it shows that the model can properly re ect the damage evolution of mudstone.Finally, the e ect of the number of dry-wet cycles on damage is analyzed.

Test Process.
e experimental materials selected are typical mudstone from the mudstone layers of the Suining Group (its lithostratigraphic unit is marked by J3s), which is located in the Jinfeng district of Chongqing, southwestern China (29 °30′46.05″N,106 °18′55.03″E).A total of 9 samples from the same type of mudstone were subjected to nine treatments.To ensure homogeneity, the samples for each treatment were selected from the same mudstone block.And they were cut into similar-sized cubes (180 mm × 180 mm × 180 mm) with an electric saw and drilled through to make to a 15 mm diameter borehole with an electric drill, which was the channel of inlet water and drainage from the centre of the top surface to the bottom surface [23].
e customized testing device, as shown in Figure 1, includes a container, a thermostat-controlled heating panel, and a water supply and drainage unit.e sample is placed in the container, and lateral restraints for the sample are provided by tightening the nuts of the activity steel plate.e heating panel connects in series with a temperature controller, a thermocouple, and a relay, which can control the temperature and maintain a constant temperature.
e dry-wet cycle test is conducted by using the heating plate to heat the sample and by using the top-down drilling hole to maintain water supply and drainage.e test process is as follows: heating for 24 h at 60 °C → water exposure for 8 h → heating for 24 h at 60 °C → water exposure for 8 h.e period of each cycle is 32 h and so forth.
After dry-wet cycles, the decay rate of rock was calculated as follows [24]: 2 Advances in Civil Engineering where DR � decay rate (%), W ini � initial weight of a sample (grams), and W rem � weight of the largest remaining fragment of a sample (grams).

Test Results.
e effect of constant temperature on the decay rate of rock is shown in Table 1.When the number of dry-wet cycles is equal, the higher temperature the rocks experience, the higher the decay rate.
In Table 1, ( e effect of the number of dry-wet cycles on the decay rate is shown in Table 2.When the heating temperature is equal, the decay rate increases with the increase of the number of dry-wet cycles.If the number of dry-wet cycles continuously increases, the decay rate will increase to over 90% until the sample is completely broken.
e relationship between the decay rate and the number of dry-wet cycles in Table 2 is fitted by a function as follows: (3)

Framework of Damage Evolution Equation.
e whole evolution process during dry-wet cycles has inseparable connection with its surroundings, including energy, materials, and information exchange.During dry-wet cycles, the complex mechanism of the action between water and rock is stored in the whole system in the form of energy, which embodies the communication between rock and its surroundings [21].e research on the law of energy transformation in the process of rock damage is helpful for getting a deeper understanding of the damage of rock masses under the action of dry-wet cycles.e area of the representative volume element of a rock unit is A, and the density is ρ. e rock unit is composed of the structure phase element and damage phase element.Under a certain stress state, the area of the structure phase element is A n and that of the damage phase element is A d .During the process of dry-wet cycles, the damage of the structure phase leads to the release of elastic energy.When the structure phase element changes into the damage phase element, the strain increment produced by the structure phase is dε n and stress is σ n .
ere is a certain relationship between the decay rate (DR) defined in this paper and the damage variable (D).Taking the unit thickness of the rock element, according to the classical continuum damage mechanics, the damage variable is defined as erefore, (3) can be expressed as (5) Equation ( 5) is the effect of the number of dry-wet cycles on the decay rate at a certain temperature.Introducing the temperature influence coefficient α(T) into ( 5) can lead to the result of e damage increment based on the energy principle is as follows [17]: e combined action between dry-wet cycles and loads accelerates the total damage and shows obvious nonlinear characteristics.Dry-wet cycles can lead to local damage inside rock masses, but the slippage and dislocation between rock grains under the action of loads can limit the action of pore water in the rock masses to some extent, showing that the total damage will weaken under the coupling effect of dry-wet cycles and loads.erefore, total damage increment under the coupling effect of dry-wet cycles and loads can be expressed as [25] where dDdD N is the coupling phase.Substituting ( 6) into (8) leads to the result of at is, at is, Introducing ( 2), ( 5), and ( 7) into (11) leads to Advances in Civil Engineering 64.8e 0.22(T−111.98) 1 + e 0.22(T−111.98) 21 + e −0.3(n−13.7)dT + 27e −0.3(n−13.7) 1 + 4.23e 0.22(T−111.98) 1 + e −0.3(n−13.7) 2 1 + e 0.22(T−111.98) Equation (12) shows that the total damage under the coupling e ect of dry-wet cycles and loads changes along with the temperature, the number of dry-wet cycles, and the strain, re ecting the mutually coupled and e ected characteristics of the temperature, the number of dry-wet cycles, and the strain to the damage expansion of materials and properly revealing the damage mechanics and damage extension law during dry-wet cycles.

Constitutive Relation of the Structure Phase.
e structure phase is characterized by the elastic state and can be described with the linear elastic model, which is the general Hooke's law.ere are principal stresses σ 1 , σ 2 , and σ 3 in the principal stress space, which satisfy the relationship of σ 1 ≥ σ 2 ≥ σ 3 , and the stress increment can be expressed as where α 1 K + 4G/3 and α 2 K − 2G/3, in which K is the bulk modulus and G is the shear modulus.

Constitutive Relation of the Damage Phase.
e damage phase possesses the elastic-plastic energy dissipation mechanism, and the elastic-plastic model can be used to describe its stressstrain relationship.In most cases, the relationship between the residual strength and con ning pressure is linear, which can be described by the Mohr-Coulomb condition.Applying the di erence between the total strain increment and plastic strain increment to represent the elastic strain increment can obtain where dλ is a nonnegative proportion coe cient and N φ (1 + sin φ)/(1 − sin φ).

Numerical Implementation of the Coupled Model
By using the damage constitutive calculation program developed by FLAC 3D , simulations of triaxial compression tests are performed, and the numerical results have been compared with the experimental results in a published literature [26].e test process in the published article is introduced as follows.
Mudstone with relatively good integrity and uniformity is chosen as the research object.
e average density of mudstone is 2.14 g/cm 3 , the natural compressive strength with a small variation range is from 2.948 MPa to 3.108 MPa, and the moisture content is 5.1%∼7.68%.Mudstone is processed into cylinders with 75 mm diameter and 150 mm height.e test scheme is as follows: (1) Based on the buried depth and in situ stresses of mudstone, con ning pressures of the triaxial compression test are 0 MPa, 1 MPa, 1.5 MPa, 2.0 MPa, and 2.5 MPa, respectively.(2) Keep con ning pressures constant throughout the test and apply displacement control with a rate of 0.5 mm/min to control the test until the sample is broken.(3) Record the whole stress-strain curves under di erent con ning pressures in real time.
e complete stress-strain curve is shown in Figure 2.

Parameter Determination of the Structure Phase.
When ignoring the bearing capacity of damage parts such as microdefects, the intact rock shows an elastic damage energy dissipation mechanism and re ects its elastic state before damage.In the initial state, the material is all composed of the structure phase; in this case, the elastic modulus and Poisson's ratio of the structure phase can be considered approximately as the elastic parameters of the intact rock in the initial state.

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According to the published article [26], the relationship between the elastic modulus and con ning pressures is shown in Table 3.Based on (15), the bulk modulus and shear modulus under di erent con ning pressures can be calculated, as shown in Table 3: In the process of axial compression, the energy absorbed by the rock material is the work done by the tester to the sample.When the sample deforms under the e ect of conning pressures, the work is also done by con ning pressures to the sample.In the conventional triaxial compression experiment, the work is done by the sample in the form of circumferential expansion to the hydraulic oil in the triaxial pressure cylinder, so the actual energy absorbed by rock materials is less than the work done by the tester to the sample in the process of axial compression.
erefore, the actual absorbed energy in the test is where the unit of K is MJ/m 3 , which is equivalent to the unit of stress MPa, and ε 1 and ε 3 are the axial strain and circumferential strain, respectively.e circumferential strain is negative, and according to Poisson's ratio e ect, it obtains According to Figure 2, the relationship between the dissipation energy and the con ning pressure can be calculated based on (16), as shown in Table 4.

Parameter Determination of the Damage Phase.
After the complete damage of the structure phase, the rock material turns into a deformation stage of residual strength, and the damage phase bears all the bearing capacity.
e damage phase of mudstone shows an elastic-plastic energy dissipation mechanism, and the stress-stain relationship of the damage phase can be described by the elastic-plastic model.e Mohr-Coulomb criterion can be applied to describe the linear relationship between residual strength and con ning pressure.
e damage phase re ects the stress-strain relationship of the residual strength, so the cohesion and internal friction angle of the damage phase can be obtained by using the regression analysis of experimental data of the residual strength.e elastic modulus and Poisson's ratio of the damage phase can be obtained approximately by using the parameters of class VI surrounding rock.
e residual strength under di erent con ning pressures based on the experimental results is shown in Figure 3.Because there is no stress decrease to determine the peak strength in Figure 2, the approximate constant stress with the increase of strain is de ned as the maximum stress σ 1 .e test results show that the residual strength is linearly related to the conning pressure.According to the Mohr-Coulomb criterion, the maximum principal stress of materials can be expressed as σ 1 σ 3 N + σ c , in which N and σ c are According to the test results in Figure 3, the tting equation is σ 1 4.5493σ 3 + 2.4794.Based on (18), the strength parameter of the damage phase is C 0.581 MPa and φ 39.76 °.Assume that the elastic parameters of the damage

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phase are approximately equal to those of class VI surrounding rock; namely, the bulk modulus is K n � 0.17 GPa and the shear modulus is G n � 0.04 GPa.
All the parameters of the model are listed in Table 5.

Numerical Simulations for Triaxial Tests.
e model dimension is in agreement with that of the published article [26], in which the sample is a cylinder with 75 mm diameter and 150 mm height.e model is divided into 1024 units, as shown in Figure 4. Vertical constraints are imposed at the bottom of the model, and a velocity of 0.5 mm/min is imposed on the top.
By introducing the parameters in Table 5 to the damage constitutive calculation program developed by the FLAC 3D , simulations of triaxial compression tests are performed, and the numerical results are compared with experimental results in the published article [26].e confining pressures σ 3 are 0 MPa, 1 MPa, 1.5 MPa, 2 MPa, and 2.5 MPa, respectively.Because the dry-wet cycle test is not involved in the article [26], the temperature T � 25 °C and the number of dry-wet cycles n � 0 are chosen as the parameters in the numerical simulation.e test results and numerical results under different confining pressures are shown in Figure 5.In the whole simulation process, there is no decrease of stress, which is consistent with the test results.With the increase of confining pressure, the load the damage phase bears increases gradually under the action of confining pressures; thus, there is no obvious decrease of the ultimate bearing capacity after the structure phase element breaks and transforms into the damage phase element, which shows a state of strain hardening, and the residual strength is the peak strength.erefore, the damage model proposed in this paper can better simulate the characteristics of mudstone materials.

Effect of the Number of Dry-Wet Cycles.
In the dry-wet cycle test, with the increase of the number of dry-wet cycles, the decay rate of mudstone increases, but the elastic modulus, cohesion, and internal friction angle decrease.As shown in Tables 6  and 7, the elastic modulus, cohesion, and internal friction angle under different dry-wet cycles at 105 °C are obtained based on [27,28] and parameters in this paper.e stress-strain curves and the damage variables with different numbers of dry-wet cycles are shown in Figures 6  and 7.With the increase of the number of dry-wet cycles, the peak deviatoric stress decreases, and the damage develops quickly and early.With the increase of confining pressures, the variation of the peak deviatoric stress and the damage development increases.

Conclusion
According to the experimental results of dry-wet cycles and the damage evolution equation under the conditions of structural damage energy dissipation, the damage evolution equation under the coupled action of dry-wet cycles and loads is set up. e numerical results are compared with the test results in a published article, which proves the accuracy and validity of the model.
e main conclusions can be drawn as follows: (1) According to the complex damage theory of geotechnical materials, rock masses can be considered as a composite material constituting the structure phase and damage phase.Damage occurs after the elastic deformation energy stored in the internal structure element increases and becomes higher than the critical value.With the continual expansion of the damage phase, the strain energy density dissipated by the material unit increases until the element is completely broken.(2) According to the effects of temperatures and the number of dry-wet cycles on decay rate, combined with the damage evolution equation based on the energy principle, the damage evolution equation under the coupled action of dry-wet cycles and loads

Table 1 :
Effect of constant temperature on rock decay rates (%).

Table 2 :
Effect of the number of dry-wet cycles on rock decay rates (%).

Table 4 :
Dissipation energy under di erent con ning pressures.

Table 3 :
Bulk modulus and shear modulus under di erent con ning pressures.

Table 5 :
Material properties of the model.

Table 6 :
Strength parameters under di erent dry-wet cycles at 105 °C.

Table 7 :
Elastic parameters under di erent dry-wet cycles at 105 °C.