# The Research on the Effect of Humidity on the Rheological Model of Swelling Rock

Authors: Ming Ji; Yi-Dong Zhang
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The research on the rheological mechanical behaviour of swelling rock in the condition of humidity field has a significant meaning for large rock engineering, such as deep coal mining, tunnelling, and slope engineering. Based on the Nishihara model, considering on the effect of humidity factor on swelling rock, the rheological elements are established and the effects of humidity on the damage of rock, the viscosity, and the elasticity are introduced. Moreover, the viscoelastic plasticity constitutive equations are established under the coupling of swelling rock stress and humidity. Afterward, considering the effects of humidity on the damage, elasticity, and viscosity, the creeping, unloading, and relaxation equations have been deduced. From this research, under the state of low stressσ3P<σS, the Nishihara model which considers the effect of humidity shows the properties of stable creeping. In addition, the unloading curves contain instantaneous elasticity, elastic after effect, and the viscosity flow induced by the humidity. However, under the state of high stress σ3P≥σS, it is an unstable creeping model. The unloading curves include instantaneous elasticity, elastic after effect, and the viscosity flow induced by the stress and the humidity. This model reflects the rheological properties of swelling rock comprehensively, and it can be used for rock rheological analysis.


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## Body

## 1. Introduction

The research on rock rheological model is an important component of the research on rock rheological mechanical theories. Moreover, it is also one of the hot and difficult topics in the field of rock mechanics. Currently, the rheological models of rock mainly have the empirical model, the component composition model, and the damage and fracture rheological model. There are plenty of studies carried out in this field in the world. For example, Xu and Xia [1] used logarithmic empirical formulas to fit creeping test results of Granite. Zhang et al. [2] performed the uniaxial compression creeping test for Gabbro rock; they mentioned the empirical formulas about the creeping law. Zhang et al. [3] used aging theory and power functions to fund nonlinear creeping equations of Mudstone. Based on traditional linear viscoelastic models, Jin and Kui-ying [4] mentioned the nonlinear viscoelastic model. Yuan-jiang [5] used the creeping body and the fractured plastic body; these two kinds of nonlinear components found a new complex rheological model to describe soft rock. Based on the creeping test results of rock, Zhi-chun et al. [6] and Xie-xing and Chen [7] pointed out the parameters of creeping modulus and the damage equations of rock creeping which can describe damage history. Yan-yi [8] studied on the process of damage rheological fractures of jointed rock; then, he mentioned the laws of delay crack initiation and expansion and instability of rock joints under constant loading; he established the damage evolution equations of jointed rock and viscoelastic plasticity constitutive equations of damage evolution coupling. Guan et al. [9] supplied a new rheological model and applied it in mountain tunnelling. Zhang and Yang [10] suggested a nonlinear unloading rheological constitutive model based on the fractional-order derivative. Zhao et al. [11] examined the creep properties of limestone under unloading triaxial creep test. Nedjar and Roy [12] suggested a creep mechanical model which could describe the whole creep process of granite with different temperatures. Brotons et al. [13] studied the mechanical characteristics of limestone in a 96 hour uniaxial compressive creep test and proposed a theological model and a creep function to study the creep behaviour of rock dependent on time. Cao et al. [14] defined a new nonlinear damage creep constitutive model of high-stress soft rock with improved Burgers model, Hooke model, and St. Venant model based on the nonlinear damage creep characteristics of rock and damage variable. Lu et al. [15] proposed a time-dependent model to simulate the creep aging damage, deformation, and fracture behaviour of brittle rock. Firme et al. [16] studied the elastoplastic creep characteristics of soft rock and provided an accurate explanation. Fahimifar et al. [17] modified a viscoplastic model to account for the creep deformation of rocks associated with the wide range of stress levels and implemented the model in numerical finite difference code (FLAC). Bazhin and Murashkin [18] studied the creep deformation and the stress relaxation characteristics of the material with loading and unloading condition under hydrostatic pressure and established a nonlinear rheological model.Based on the Nishihara model in component composition model, this paper introduced the rheological components, which consider the humidity expansion effect, to establish the swelling rock viscoelastic model that considers on the humidity, the damage, and the viscosity. The research result of this paper is a useful supplement for the theory research on rock engineering.


## 2. Free Expansion Experiment of Rock

The rock sample which was obtained from the roof of a coal mine is mudstone, which will expand but cannot disintegrate, whose dry density is 2.74 g/cm3.Shaping the rock sample into a rectangle shape with length, width, and height as 5, 5, and 10 cm. The processed sample should be heated to 105°C for 24 h in the oven to remove water completely; then, the water content of rock is 0. In the experiment, length is tested by a vernier caliper whose range is 13.5 cm and precision is 0.02 mm; height is tested by an electronic balance JM10002 whose range is 1000 g and precision is 0.01 g. Taking the dry rock sample out and putting into a desiccator for cooling, then measure and record length (x), width (y), height (z), and mass (m) of the dry sample. Lay the test area of the dry sample and measure its length, width, and height with the method getting the average value by multiple measuring.The expanding experiment was processing in the environment of 20°C. Put the sample into water and ensure it soaks completely and expands freely; then, take the sample out at an interval, dry the surface of it with absorbent paper, and measure the data: length, width, height, and mass. The standard to judge the sample whether to be water saturating is the mass of the rock increases less than 0.1 g/h one time.Record the experimental data of every time points; the result of expansion strain and the water content is shown in Table1:Table 1
Experiment results.
T (h)Expansion strain (10−2)Mass contentVolume contentεxεyεz0000000.20.07990.12010.14060.0319690.087350.40.15990.20020.22090.0426940.1163740.80.19980.24020.28120.0557190.1516651.20.23980.28020.32130.0649060.17645920.27980.32030.36150.0775940.21070140.31970.36030.40170.0850070.230556100.35970.40030.44190.0912490.247188240.39970.44040.4820.1002890.271352480.43960.48040.52220.106970.289083720.43960.48040.54230.1119950.302603960.43960.48040.54230.1119950.302603


## 3. The Relationship between Expansion Strain and Water Content

In [19–21], the concept of humidity stress field is shown, which held the point that humidity diffusion in the swelling rock has coupling relationship with water content, water absorption force, and volume deformation. Let the geotechnical expand freely, and it will bring a strain component as follows:(1)εij=αδijΔw.Among the formula,α is linear expansion coefficient,δij is a symbol of Kronecker, and Δwis a humidity changing value.Figure1 shows the relationship between expansion strain and volume water content and mass water content of lime mudstone in 3 directions at length, width, and height.Figure 1
Relationship between expansion strain and water content.From Figure1 we know that the free expansion strain has proper linear relationship with both volume water content and mass water content of the rock sample in 3 directions at length, width, and height, whose result is fit for the humidity stress field concept in [19–21]. We found the expansion coefficient of volume water content as 0.01421, 0.01603, and 0.01782 and mass water content as 0.03852, 0.04345, and 0.04831, respectively, in three directions at length, width, and height through function fitting.The anisotropy of the experimental rock sample leads to the difference of expansion coefficient in three directions at length, width, and height, but the difference is very small from the data, so the rock sample can be approximately regarded as isotropic rock mass. So, the relationship between free expansion strain and water content of lime mudstone can be shown as follows:(2)ε=Kmwm,ε=Kvwv.Among the formula,wm and wv, respectively, show the mass water content and volume water content and Km and Kv show the corresponding expansion coefficients.


## 4. The Constitutive Model

The fundamental components of the conventional rheological model only considered the effect of the load; however, the effect of humidity factor also has obvious impact on the mechanical properties of swelling rock. Considering the effect of humidity factor and combining with the humidity stress field theory, the following hypotheses have been mentioned (Table2).Table 2
Rheological elements in the humidity field.
Fundamental componentsSymbolsExplanationHooker modelσ Is the stress; E and a are the elasticity coefficient and the humidity expansion coefficient of Hooker model, respectively; Δw is the humidity difference of before and after two states; moreover, assume that the changes of the humidity have no effect on the Hooker law of the elastic modelPlastic modelWhenσ<σs, there is no strain changes; when σ>σs, the plastic model yield; σs is the yield limit of the material; when the stress meets the yield limit, the yield continues, no-slip in friction surface and have no relationship with the time; in the coupling of humidity and stress, the yield stress of the material is σs(w) and the yield surface is f (σs, w), both of them have the relationships with the humidity; moreover, assume that the yield stress is a given constantNewton modelThe stress and strain obey to sticky laws in the Newton model; there is a proportional relationship between the stress and the strain rate; the formulation isσ=ηε˙, η is the sticky coefficient; under the coupling of the humidity and the stress, the sticky coefficient is expressed by η(w) and it has the relationship with the humidity; moreover, assume that the sticky coefficient is a given constantThe Nishihara model is series connected by the Hooker model, the Kelvin model, and the ideal viscoplastic model. In this paper, based on the Nishihara model, the humidity effect is introduced; then, as shown in Figure2, the Nishihara model, which considers the effect of the humidity, has been found.Figure 2
The Nishihara model under the humidity effect.Under the coupling of the humidity-stress, the stress-strain relationships in this model can be shown as follows:(3)σ=σ1=σ2=σ3,ε=ε1+ε2+ε3.In these formulas,σ1=E1ε1−E1α1Δw,(4)σ2=σ2E+σ2η,σ2E=E2ε2E−E2α2Δw,σ2η=η2wε˙2η,σ3=σ3P+σ3η,σ3P=0,σ3P<σs,σs,σ3P≥σs,σ3η=η3wε˙3η,ε2=ε2E=ε2η,ε3P=0,σ3P<σs,ε3,σ3P≥σs,ε3=ε3P=ε3η.In these formulas,σ is the total stress of this model; ε is the total strain of this model; Δwis the amount of the humidity changes between two states; σ2 and ε2 are the total stress and the total strain in the Kelvin model, respectively; σ3 and ε3 are the total stress and the total strain in the ideal viscoplastic model, respectively; σ1, E1, ε1, and α1 are the stress, the elastic module, the strain, and the wet expansion coefficient in the Hooker model respectively; σ2E, E2, ε2E, and α2 are the stress, the elastic module, the strain, and the wet expansion coefficient in the Kelvin model, respectively; σ2η, ε2η, and η2(w) are the stress of the viscous model, the strain, and the wet viscosity coefficient of the Kelvin model, respectively; σ3η, ε3η, and η3(w) are the stress of the viscous model, the strain, and the wet viscosity coefficient of the ideal viscoplastic model, respectively; σ3P, ε3P, and σ3S are the stress of elastic components, the strain, and the yield limit of the ideal viscoplastic model, respectively.Under the state of low stressσ3P<σS, the ideal viscoplastic model has no effect and the model is the generalized Kelvin model, as shown in Figure 3.Figure 3
The generalized Kelvin model with the humidity effect.In Figure2, σ1=σ2=σ, ε1+ε2=ε, and ε˙1+ε˙2=ε˙.For the Hooker model,σ=E1ε˙1−E1α1Δw˙ and σ˙=E1ε˙1−E1α1Δw˙.For the Kelvin model,σ=E2ε2−E2α2Δw+η2wε˙2.Thus,σ=E2ε−ε1−E2α2Δw+η2wε˙−ε˙1.Then, the constitutive equation, which considers the effect of the humidity on the elasticity, can be deduced:(5)η2wE1σ˙+1+E2E1σ+η2wα1Δw˙+E2α1+α2Δw=η2wε˙+Eε2.Under high-stress stateσ3P≥σS, as shown in Figure 1, this model has three components. Moreover, the constitutive equation can be deduced with the similar process:(6)σ¨+E1η2w+E1η3w+E2η2wσ˙+E1E2η2wη3wσ−σs+E1α1Δw¨+E1E2α1+α2η2wΔw˙=E1ε¨+E1E2η2wε˙.The humidity and the humidity content have dramatic effects on the creeping properties of the rock. Generally, the rock strength will decrease with the increase of the humidity; however, the rate of rock creeping will increase. According to [22], the viscosity coefficient will decrease with the increase of the humidity; moreover, it is almost a linear change. If ξ2 and ξ3 are presumed as the corresponding viscosity attenuation coefficient, η2′w=η2w−ξ2Δw and η3′w=η3w−ξ3Δw. Then, the constitutive relationship is(7)η2w−ξ2ΔwE1σ˙+1+E2E1σ+η2w−ξ2Δwα1Δw˙+E2α1+α2Δw=η2w−ξ2Δwε˙+E2ε,σ3P<σs,σ¨+E1η2w−ξ2Δw+E1η3w−ξ3Δw+E2η2w−ξ2Δwσ˙+E1E2η2w−ξ2Δwη3w−ξ3Δwσ−σs+E1α1Δw¨+E1E2α1+α2η2w−ξ2ΔwΔw˙=E1ε¨+E1E2η2w−ξ2Δwε˙,σ3P≥σs.Large amounts of experiments have identified that the properties of swelling rock will be reduced after it is contacted with water; moreover, the elastic modulus will also be decreased with the increase of the humidity [23]. Thus, introducing two damage factors (D1 and D2) for the Hooker element, then E1′=E11−D1 and E2′=E21−D2, and the constitutive equations are(8)η2w−ξ2ΔwE11−D1σ˙+1+E21−D2E11−D1σ+η2w−ξ2Δwα1Δw˙+E21−D2α1+α2Δw=η2w−ξ2Δwε˙+E21−D2ε,σ3P<σs,σ¨+E11−D1η2w−ξ2Δw+E11−D1η3w−ξ3Δw+E21−D2η2w−ξ2Δwσ˙+E11−D1E21−D2η2w−ξ2Δwη3w−ξ3Δwσ−σs+E11−D1α1Δw¨+E11−D1E21−D2α1+α2η2w−ξ2ΔwΔw˙=E11−D1ε¨+E11−D1E21−D2η2w−ξ2Δwε˙,σ3P≥σs.


### 4.1. Creeping Equations

Assuming that the humidity is constant after the absorbent of the rock; then,Δw˙=0.Under the state of low stressσ3P<σS, it has a constant value, σ˙=0. Then, the model is the generalized Kelvin model. For the Hooker model, when σ=σ0, then ε1t=σ0/E1+α1Δw. For the Kelvin model, σ0=E2ε2t−E2α2Δw+η2wdε2t/dt; then, ε2t=σ0+E2α2Δw/E2+Ae−E2/η2wt; in this equation, A is the integration constant. When t = 0, ε2=0; then, A=−σ0+E2α2Δw/E2. The creeping equation is(9)εt=ε1t+ε2t=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2wt.Whent tends to infinity, εt=σ0/E1+α1Δw+σ0/E2+α2Δw is the horizontal asymptote of the creeping equation and this model is under stable creeping.Under the state of high stressσ3P≥σS, the rock has plastic deformation. For the Hooker model, ε1t=σ0/E1+α1Δw. For the Kelvin model, ε2t=σ0+E2α2Δw/E21−e−E2/η2wt. For the ideal viscoplastic model, ε3t=σ0−σs/η3wt. The creeping equation is(10)εt=ε1t+ε2t+ε3t=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2wt+σ0−σsη3wt.This model will become larger as the time goes; the strain will be reduced gradually; this model will be under unstable creeping and the creeping will be instability finally.The creeping equation considers the effects of humidity on elasticity and viscosity as follows:(11)εt=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2w−ξ2Δwt,σ3P<σs,σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2w−ξ2Δwt+σ0−σsη3wt,σ3P≥σs.The creeping equation considers the effects of humidity on elasticity, damage, and viscosity as follows:(12)εt=σ0E11−D1+α1Δw+σ0+E21−D2α2ΔwE21−D21−e−E21−D2/η2w−ξ2Δwt,σ3P<σs,σ0E11−D1+α1Δw+σ0+E21−D2α2ΔwE21−D21−e−E21−D2/η2w−ξ2Δwt+σ0−σsη3w−ξ3Δwt,σ3P≥σs.


### 4.2. Unloading Equations

Under the state of low stressσ3P<σs,, unloading at the time of t1, the stress is zero. For the Hooker model, ε1t=α1Δw.For the Kelvin model,dε2t/dt+E2ε2t/η2w=E2α2Δw/η2w; then, ε2t=α2Δw+Be−E2/η2wt. In this equation, B is integration constant. A the time of t1, ε2=ε21=σ0+E2α2Δw/E21−e−E2/η2wt1; then, B=ε21−α2Δwe−E2/η2wt1. The unloading equation is(13)εt=ε1t+ε2t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t.Whent tends to infinite, εt=α1+α2Δw. Thus, because of the effect of humidity, the deformation of the model is not recovered to zero:(14)εt=ε1t+ε2t+ε3t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t+σ0−σsη3wt1.Under the state of high stressσ3P≥σs, for the Hooker model, ε1t=α1Δw. For the Kelvin model, ε2t=α2Δw+ε21−α2Δwe−E2/η2wt1−t. For the ideal viscoplastic model, the model remains at the time of t1 after unloading, the strain is ε3=σ0−σs/η3wt1, and the deformation is the eternal plastic deformation. The unloading equation is(15)εt=ε1t+ε2t+ε3t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t+σ0−σsη3wt1.Whent tends to infinite, εt=α1+α2Δw+σ0−σs/η3wt1.Considering about the effects of the humidity on the elasticity and the viscosity, the unloading equation is as follows:(16)εt=α1+α2Δw+Be−E2/η2w−ξ2Δwt,σ3P<σs,α1+α2Δw+Be−E2/η2w−ξ2Δwt+σ0−σsη3w−ξ3Δwt1,σ3P≥σs.In this equation,B=ε21−α2Δwe−E2/η2w−ξ2Δwt1.Considering the effects of the humidity on elasticity, damage, and viscosity, the unloading equation is as follows:(17)εt=α1+α2Δw+Be−E21−D2/η2w−ξ2Δwt,σ3P<σs,α1+α2Δw+Be−E21−D2/η2w−ξ2Δwtσ0−σsη3w−ξ3Δwt1,σ3P≥σs.In this equation,B=ε21−α2Δwe−E21−D2/η2w−ξ2Δwt1.


### 4.3. Relaxation Equations

Under the state of low stressσ3P<σs, and ε=ε0. That is, ε˙=0. Then, σ˙+E1+E2/η2wσ=E1/η2wE2ε0−E2α1+α2Δw, introducing t=0 and σ=σ0 into it; then, σ0=E1ε0:(18)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2wt+E1E2E1+E2ε0−α1+α2Δw.Under the state of high stressσ3P≥σs, for the ideal viscoplastic model, the relaxation deformation is ε = ε0 (ε˙=0). That is, when t = 0, σ = σS and σ′=σs; then, the relaxation equation of the Nishihara model as follows:(19)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2wt+E1E2E1+E2ε0−α1+α2Δw+σs.The below is the relaxation equation which considers the effects of humidity on elasticity and viscosity:(20)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2w−ξ2Δwt+E1E2E1+E2ε0−α1+α2Δw,σ3P<σs,−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2w−ξ2Δwt+E1E2E1+E2ε0−α1+α2Δw+σs,σ3P≥σs.The below is the relaxation equation which considers the effects of humidity on elasticity, damage, and viscosity.(21)σ=−E11−D1E11−D1+E21−D2E11−D1ε0+E21−D2α1+α2Δwe−E11−D1+E21−D2/η2w−ξ2Δwt+E11−D1E1−D22E11−D1+E21−D2ε0−α1+α2Δw,σ3P<σs,−E11−D1E11−D1+E21−D2E11−D1ε0+E21−D2α1+α2Δwe−E11−D1+E21−D2/η2w−ξ2Δwt+E11−D1E1−D22E11−D1+E21−D2ε0−α1+α2Δw+σs,σ3P≥σs.Figures4 and 5 show the creeping curves and the unloading curves of the model under two states of the stress, respectively. The creeping curve 1 and the unloading curve 1 indicate that the effect of the humidity on the elasticity is only considered on. The creeping curve 2 and the unloading curve 2 express that the effects of the humidity on the elasticity and the viscosity are considered about. The creeping curve 3 and the unloading curve 3 show that the effects of humidity on elasticity, damage, and viscosity are considered about.Figure 4
Model curves of creeping and unloading whileσ3P<σS.Figure 5
Model curves of creeping and unloading whileσ3P≥σS.According to Figures4 and 5, under three situations, when σ3P<σS, the model has a relatively fast deformation at the beginning; afterward, and it tends to stable gradually. Based on the unloading curve, the model has the instantaneous elastic deformation, elastic after effect, and the viscous flow which are caused by the humidity. At the situation of σ3P≥σS, it transfers to the unstable creeping. According to the unloading curve, it has the instantaneous elastic deformation, elastic after effect, and viscosity flow induced by both of humidity and stress. This model indicated two creeping states of many rocks under the effect of the humidity.From these two curves, the effects of humidity on damage, elasticity, and viscosity are introduced into this model. It can indicate the rheological properties of rock comprehensively. With the increase of humidity, viscosity, and strength of the rock will be decreased; however, the damage and the plastic will be increased. Thus, it will cause larger creeping deformation. The model in this paper can describe the rheological properties of rock which is sensitive to the change of the humidity.


## 4.1. Creeping Equations

Assuming that the humidity is constant after the absorbent of the rock; then,Δw˙=0.Under the state of low stressσ3P<σS, it has a constant value, σ˙=0. Then, the model is the generalized Kelvin model. For the Hooker model, when σ=σ0, then ε1t=σ0/E1+α1Δw. For the Kelvin model, σ0=E2ε2t−E2α2Δw+η2wdε2t/dt; then, ε2t=σ0+E2α2Δw/E2+Ae−E2/η2wt; in this equation, A is the integration constant. When t = 0, ε2=0; then, A=−σ0+E2α2Δw/E2. The creeping equation is(9)εt=ε1t+ε2t=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2wt.Whent tends to infinity, εt=σ0/E1+α1Δw+σ0/E2+α2Δw is the horizontal asymptote of the creeping equation and this model is under stable creeping.Under the state of high stressσ3P≥σS, the rock has plastic deformation. For the Hooker model, ε1t=σ0/E1+α1Δw. For the Kelvin model, ε2t=σ0+E2α2Δw/E21−e−E2/η2wt. For the ideal viscoplastic model, ε3t=σ0−σs/η3wt. The creeping equation is(10)εt=ε1t+ε2t+ε3t=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2wt+σ0−σsη3wt.This model will become larger as the time goes; the strain will be reduced gradually; this model will be under unstable creeping and the creeping will be instability finally.The creeping equation considers the effects of humidity on elasticity and viscosity as follows:(11)εt=σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2w−ξ2Δwt,σ3P<σs,σ0E1+α1Δw+σ0+E2α2ΔwE21−e−E2/η2w−ξ2Δwt+σ0−σsη3wt,σ3P≥σs.The creeping equation considers the effects of humidity on elasticity, damage, and viscosity as follows:(12)εt=σ0E11−D1+α1Δw+σ0+E21−D2α2ΔwE21−D21−e−E21−D2/η2w−ξ2Δwt,σ3P<σs,σ0E11−D1+α1Δw+σ0+E21−D2α2ΔwE21−D21−e−E21−D2/η2w−ξ2Δwt+σ0−σsη3w−ξ3Δwt,σ3P≥σs.


## 4.2. Unloading Equations

Under the state of low stressσ3P<σs,, unloading at the time of t1, the stress is zero. For the Hooker model, ε1t=α1Δw.For the Kelvin model,dε2t/dt+E2ε2t/η2w=E2α2Δw/η2w; then, ε2t=α2Δw+Be−E2/η2wt. In this equation, B is integration constant. A the time of t1, ε2=ε21=σ0+E2α2Δw/E21−e−E2/η2wt1; then, B=ε21−α2Δwe−E2/η2wt1. The unloading equation is(13)εt=ε1t+ε2t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t.Whent tends to infinite, εt=α1+α2Δw. Thus, because of the effect of humidity, the deformation of the model is not recovered to zero:(14)εt=ε1t+ε2t+ε3t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t+σ0−σsη3wt1.Under the state of high stressσ3P≥σs, for the Hooker model, ε1t=α1Δw. For the Kelvin model, ε2t=α2Δw+ε21−α2Δwe−E2/η2wt1−t. For the ideal viscoplastic model, the model remains at the time of t1 after unloading, the strain is ε3=σ0−σs/η3wt1, and the deformation is the eternal plastic deformation. The unloading equation is(15)εt=ε1t+ε2t+ε3t=α1+α2Δw+ε21−α2Δwe−E2/η2wt1−t+σ0−σsη3wt1.Whent tends to infinite, εt=α1+α2Δw+σ0−σs/η3wt1.Considering about the effects of the humidity on the elasticity and the viscosity, the unloading equation is as follows:(16)εt=α1+α2Δw+Be−E2/η2w−ξ2Δwt,σ3P<σs,α1+α2Δw+Be−E2/η2w−ξ2Δwt+σ0−σsη3w−ξ3Δwt1,σ3P≥σs.In this equation,B=ε21−α2Δwe−E2/η2w−ξ2Δwt1.Considering the effects of the humidity on elasticity, damage, and viscosity, the unloading equation is as follows:(17)εt=α1+α2Δw+Be−E21−D2/η2w−ξ2Δwt,σ3P<σs,α1+α2Δw+Be−E21−D2/η2w−ξ2Δwtσ0−σsη3w−ξ3Δwt1,σ3P≥σs.In this equation,B=ε21−α2Δwe−E21−D2/η2w−ξ2Δwt1.


## 4.3. Relaxation Equations

Under the state of low stressσ3P<σs, and ε=ε0. That is, ε˙=0. Then, σ˙+E1+E2/η2wσ=E1/η2wE2ε0−E2α1+α2Δw, introducing t=0 and σ=σ0 into it; then, σ0=E1ε0:(18)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2wt+E1E2E1+E2ε0−α1+α2Δw.Under the state of high stressσ3P≥σs, for the ideal viscoplastic model, the relaxation deformation is ε = ε0 (ε˙=0). That is, when t = 0, σ = σS and σ′=σs; then, the relaxation equation of the Nishihara model as follows:(19)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2wt+E1E2E1+E2ε0−α1+α2Δw+σs.The below is the relaxation equation which considers the effects of humidity on elasticity and viscosity:(20)σ=−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2w−ξ2Δwt+E1E2E1+E2ε0−α1+α2Δw,σ3P<σs,−E1E1+E2E1ε0+E2α1+α2Δwe−E1+E2/η2w−ξ2Δwt+E1E2E1+E2ε0−α1+α2Δw+σs,σ3P≥σs.The below is the relaxation equation which considers the effects of humidity on elasticity, damage, and viscosity.(21)σ=−E11−D1E11−D1+E21−D2E11−D1ε0+E21−D2α1+α2Δwe−E11−D1+E21−D2/η2w−ξ2Δwt+E11−D1E1−D22E11−D1+E21−D2ε0−α1+α2Δw,σ3P<σs,−E11−D1E11−D1+E21−D2E11−D1ε0+E21−D2α1+α2Δwe−E11−D1+E21−D2/η2w−ξ2Δwt+E11−D1E1−D22E11−D1+E21−D2ε0−α1+α2Δw+σs,σ3P≥σs.Figures4 and 5 show the creeping curves and the unloading curves of the model under two states of the stress, respectively. The creeping curve 1 and the unloading curve 1 indicate that the effect of the humidity on the elasticity is only considered on. The creeping curve 2 and the unloading curve 2 express that the effects of the humidity on the elasticity and the viscosity are considered about. The creeping curve 3 and the unloading curve 3 show that the effects of humidity on elasticity, damage, and viscosity are considered about.Figure 4
Model curves of creeping and unloading whileσ3P<σS.Figure 5
Model curves of creeping and unloading whileσ3P≥σS.According to Figures4 and 5, under three situations, when σ3P<σS, the model has a relatively fast deformation at the beginning; afterward, and it tends to stable gradually. Based on the unloading curve, the model has the instantaneous elastic deformation, elastic after effect, and the viscous flow which are caused by the humidity. At the situation of σ3P≥σS, it transfers to the unstable creeping. According to the unloading curve, it has the instantaneous elastic deformation, elastic after effect, and viscosity flow induced by both of humidity and stress. This model indicated two creeping states of many rocks under the effect of the humidity.From these two curves, the effects of humidity on damage, elasticity, and viscosity are introduced into this model. It can indicate the rheological properties of rock comprehensively. With the increase of humidity, viscosity, and strength of the rock will be decreased; however, the damage and the plastic will be increased. Thus, it will cause larger creeping deformation. The model in this paper can describe the rheological properties of rock which is sensitive to the change of the humidity.


## 5. Conclusions

(1)
The expansion strain is directly proportional to the mass or volume moisture content(2)
A new Nishihara model was established by introducing the effect of humidity to improve components(3)
The constitutive equations, creeping equations, and unloading equations considering the change of damage and viscosity coefficient by the effect of humidity are deduced(4)
Under the state of low stress, the model shows stable creeping; however, when it is under high stress, the model expresses unstable creeping(5)
The elastic deformation, elastic after effect, and viscous flow are occurred in the unloading curveProblems: the model in the paper only qualitatively shows the rheological properties of the swelling rock under the effects of humidity and humidity-relative parameters theoretically. More relative experiments and field test verification should be followed up.



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*Source: 8811071-2020-08-14.xml*