A new shear rheological model for a soft interlayer with varying water content

Chong Ma,Hong-in Zhan*,Wen-min Yao,Hua-zhou Li

aSchool of Mathematics and Physics,China University of Geosciences,Wuhan 430074,China

bFaculty of Engineering,China University of Geosciences,Wuhan 430074,China

cDepartment of Geology and Geophysics,Texas A&M University,College Station,TX 77843-3115,USA

Abstract The rheological behavior of a soft interlayer is critical to understanding slope stability,which is closely related to the water content of the soft interlayer.This study used the soft interlayer of the Permian Maokou Formation in Southwest China as an example to perform ring shear creep tests with different water content amounts.The effect of water content on the creep properties of the soft interlayer was analyzed,and a new shear rheological model was established.This research produced several findings.First,the ring shear creep deformation of the soft interlayer samples varied with the water content and the maximum instantaneous shear strain increment occurred near the saturated water content.As the water content increased,the cumulative creep increment of the samples increased.Second,the water content significantly affected the long-term strength of the soft interlayer,which decreased with the increase of water content,exhibiting a negative linear correlation.Third,a constitutive equation for the new rheological model was derived,and through fitting of the ring shear creep test data,the validity and applicability of the constitutive equation were proven.This study has developed an important foundation for studying the long-term deformation characteristics of a soft interlayer with varying water content.

Keywords:Soft interlayer;Ring shear creep test;Rheological constitutive model;Water content

1.Introduction

The deformation of a soft interlayer is critical to the stability of rock mass engineering.A soft interlayer tends to become either the main pathway of or a barrier to groundwater pollutant migration,depending on its permeability tensor,which is often considerably different from those of the upper and lower strata.For instance,the permeability of the interlayer may exhibit high anisotropy,i.e.,a relatively large permeability along the bedding direction and a relatively small permeability perpendicular to the bedding.Consequently,a soft interlayer may become the main pathway for contaminant migration along the bedding direction,but it can also become an effective barrier to cross-strata contaminant migration along the direction perpendicular to the bedding.This unique feature of a soft interlayer has not been carefully investigated before.To better understand the contaminant transport through a soft interlayer,which may or may not be fully saturated,one has to understand the unsaturated permeability tensor of the layer,which is closely related to its rheological behavior with varying water content.

The rheological behavior of a rock or soil mass is a general concern in many geotechnical engineering problems(Sun,1999).Rheological tests serve as the main means of knowing the rheological and mechanical properties of a rock or soil mass(Bhat et al.,2013;Lai et al.,2014).With the rapid advancement of rheological studies in recent decades,the limitations of rheological tests on a rock or soil mass under highly simplified and sometimes unrealistically ideal conditions have become clear.

To better tackle geotechnical problems in the actual geological environment,rheological tests on a rock or soil mass under complex and realistic conditions are necessary.Fujii et al.(1999)conducted a comparative study on the circumferential creep properties of water-saturated and dry sandstones,and found that the moisture state had a strong effect on circumferential strain.Ngwenya et al.(2001)studied the rheological behavior of water-saturated sandstone under low temperature conditions and developed a modified form of the power-law constitutive equation.Zhu and Ye(2002)discussed the law of in fluence of the moisture state on the creep properties of rocks through comparative analysis of the results of rock creep tests under dry and water-saturated states and showed that the transient creep modulus and creep deformations were all affected by water content.Yang et al.(2007)performed creep tests on natural,dry,moderately moisturized,and water-saturated shales;analyzed the effects of different water content amounts on creep properties;and showed that water content had a strong in fluence on the mechanical properties of rock creep.Okubo et al.(2008)did a comparative analysis of the viscoelastic characteristics and creep models of water-saturated and airdried weathered tuffs,and proposed a new creep model capable of re flecting the changes in Young's modulus of water-saturated and air-dried rocks.Pellet et al.(2013)conducted a series of direct shear tests to examine the effect of water saturation on the mechanical properties of clayin filled discontinuities and showed that both angles of friction and cohesion decreased while the discontinuity was saturated.Liu et al.(2013)conducted creep loading and unloading tests on dry and water-saturated deep amphibolites,and found that water affected the creep properties of rocks more significantly under higher stress.Wang(2014)studied the degradation effect of the creep properties of sandstone and argillite under different water-saturation and water-loss cycles and showed that,as the number of water saturation-dehydration cycles increased,the rock's rheological properties became increasingly clear.Brantut et al.(2014)performed triaxial deformation experiments on a porous limestone saturated with water,and showed that only low levels of strain were reached during the occurrence of brittle creep failure.Ma et al.(2016)studied the triaxial rheological properties of silty mudstone and demonstrated that the rock showed different rheological properties with different water content amounts and confining pressures.In addition,Lockner(1993)and de Meer and Spiers(1995)conducted a study on the creep properties of rocks under the effect of temperature.

In studies of stability problems of mine slopes and reservoir bank slopes,soft interlayers often exhibit distinctive rheological behavior due to their unique geological features,and have significant impacts on slope stability.For example,the soft interlayer of the Maokou Formation is the main sliding zone of limestone mine slopes in Sichuan Province,in China,and has become one of the major safety concerns with regard to mining activity.The water content of a soft interlayer varies over time due to changes in rainfall,reservoir water level,etc.,which will inevitably affect its creep mechanical properties.Therefore,it is particularly important to study the creep properties of the soft interlayer of the Maokou Formation with varying water content.The objective of this study was to develop a new shear rheological model and its associated constitutive equation for a soft interlayer with varying water content by performing ring shear creep tests on samples from the Maokou Formation.This study has developed an important foundation for studying the long-term variation of permeability of a soft interlayer and is signi ficant to studies of the migration and prevention of groundwater pollution.

2.Materials and methods

2.1.Ring shear creep test

The ARS fully automatic closed-loop controlled ring shear apparatus(Wille Geotechnik,Germany),as shown in Fig.1,was used in ring shear creep tests.The ring shear apparatus consists of a main unit,a main controller,and a recording and control storage unit.Driven by a highprecision motor,it can achieve linear or instantaneous changes of shear rate,shear pressure,and normal pressure.This apparatus is capable of providing a maximum axial pressure of 10 kN,a maximum shear stress of 1000 kPa,a maximum shear rate of 32 mm/min,and a maximum axial displacement of 25 mm.

Samples used in the ring shear creep test were taken from the soft interlayer of the Permian Maokou Formation of a large limestone mine slope on Mount Emei,in Sichuan Province,in China.According to the in situ moisture state of the soft interlayer and parameters such as the Atterberg limit,the samples were remolded and subjected to ring shear creep tests with five different water content amounts(15%,19%,23%,27%,and 31%)to investigate the difference in creep mechanical properties of the soft interlayer with varying water content.The choice of the five different water content amounts was based on the following characteristics of the in situ soft interlayer sample:a liquid limit of 32.64%,a plastic limit of 14.58%,and a saturated water content of 22.78%.The remolded ring shear samples are shown in Fig.2.

Fig.1.ARS ring shear apparatus.

Currently,there is a lack of reference standards for ring shear creep tests involving various water content amounts.The normal pressure during the ring shear creep test of this study was determined to be 400 kPa after taking into account various factors.According to the peak strength and residual strength of samples with different water content amounts under corresponding normal stresses, five levels of shear stress were loaded according to the residual strength,and a sixth level of stress was loaded according to 95%of the peak strength in the test.The scheme of loading different levels of stress is shown in Table 1.

In this test,the loading time for each level of shear stress was at least 10 h.When the displacement within the last 2 h at each level was less than 0.01 mm,deformation could be considered stable,and the next level of shear stress could be loaded.Each level of shear stress was loaded in sequence for 600 min,600 min,720 min,720 min,and 840 min,respectively.The failure-level(sixth-level)stress was loaded until the samples exhibited clear failure,at which point the test was completed.

2.2.Results of ring shear creep test

Fig.2.Ring shear samples.

Table 1 Level of ring shear stress load.

Fig.3 shows the whole-process curves of ring shear creep under the normal pressure of 400 kPa and different amounts of water content.As can be seen in Fig.3,the curves with different amounts of water content all undergo the four stages of instantaneous elastic shear deformation,decelerated creep,stable creep,and accelerated creep.Elastic shear deformation means the instantaneous strain as the stress is applied.This is evident in Fig.3,in which the curve suddenly increases within a very short period of time.Then,after a short decelerated creep stage,the curves demonstrate a horizontal line or a line with a certain constant slope.During the last level of the shear stress,the strain rate suddenly increases within a very short period of time as well,which is the accelerated creep.The accelerated creep means that the rock or the soil is damaged.During the test,the soft interlayer samples exhibited distinctive creep properties and the change trend of strain with time was clear.All creep curves show some manner of nonlinearity and ductile failure characteristics.The soft interlayer samples with different water content amounts are significantly different from one another in terms of creep properties.That is to say,a higher water content leads to more distinctive creep properties under the same normal pressure,which means that the change trend of strain with time is more pronounced.For remolded samples,the nonlinearity of creep is mostly caused by grain sliding or dislocation of the soft interlayer,and the geometric characteristics of the internal damage also change.Different water content amounts would result in changes in pore structure and pore pressure,leading to further deformation nonlinearity.

3.Analysis of results of ring shear creep test

3.1.Effect of water content on ring shear strain

Based on Fig.3,some calculations were made to obtain accurate data of some specific strains,which are shown in Table 2 and can be used for accurate and detailed analysis.Table 2 lists the ring shear strains after loading different levels of shear stress without accelerated failure with different amounts of water content under the normal pressure of 400 kPa.

Fig.3.Soft interlayer strain versus time with different water content amounts under normal pressure of 400 kPa.

Table 2 Strain behavior of samples with different water content amounts under each level of ring shear stress.

As can be seen in Table 2,the instantaneous ring shear strain increases as the level of stress applied increases,and the increment of instantaneous strain varies with the water content.The instantaneous ring shear strains of samples with five different water content amounts under the first level of stress were compared with those under the fifth level of stress;the results show that the instantaneous ring shear strains of samples with the water content amounts of 15%,19%,23%,27%,and 31%under the fifth level of stress increaseby85.97%,89.77%,110.83%,102.98%,and 95.41%,respectively,compared with those under the first level of stress.Through comparative analysis,it can be concluded that when the water content is lower than the saturated water content of the in situ sample(about 23%),the instantaneous shear strain increment increases with the increase of the stress level as the water content increases;the instantaneous shear strain increment decreases with the increase of water content when the water content is higher than 23%,and the maximum increment occurs when the water content is near 23%.

Each sample exhibits the minimal shear creep deformation after application of the first level of shear stress.At the following levels of stress,the stage shear modulus increases with the stress level,suggesting that a higher ring shear stress will lead to more distinctive shear creep properties.To investigate the effect of water content on stage creep deformation,the stage deformations of samples with different amounts of water content under the fifth level of stress were compared with those under the second level of stress;the results show that the stage creep deformations of samples with the water content amounts of 15%,19%,23%,27%,and 31%under the fifth level of stress increase by 96.36%,409.62%,224.61%,37.89%,and 580.17%,respectively,compared with those under the second level of stress.The strain increment is not significantly correlated with the water content,but the largest creep deformation is found in the sample with the highest water content(31%).For cumulative creep deformation,the cumulative creep deformations of samples with the water content amounts of 15%,19%,23%,27%,and 31%under the fifth level of stress increase by 113.52%,162.15%,125.82%,151.31%,and 213.49%,respectively,compared with those under the first level of stress.It can be concluded that,as the water content increases,the cumulative creep increment of samples also increases.In addition,based on the ratio of deformation to transient elastic strain when the creep curve reaches a steady state,it can be seen that,as the ring shear stress level increases,the ratio of deformation to transient elastic strain during steady-state creep gradually increases.This suggests that,as the stress level increases,the proportion of creep deformation in total deformation increases as well.The analysis above shows that creep properties of the soft interlayer samples of the Maokou Formation have become more distinctive with the increase of water content and the ring shear stress level.

3.2.Effect of water content on long-term strength of soft interlayer

The isochronous stress-strain curve method and the steadystate creep rate method were employed to determine the longterm strength of the soft interlayer with varying water content.The results are listed in Table 3.

As can be seen in Table 3,the long-term sample strength gradually decreases as the water content increases;all values of the relative error of the long-term strength|τ1- τ2|/τ2are less than 5.45%.In all cases except for the sample with the water content of 19%,the long-term strength determined with the isochronous stress-strain curve method is higher than that determined by the steady-state creep rate method,but the difference is not significant,suggesting that the determined long-term strength is probably reasonable.

To further investigate the correlation between the long-term strength and the water content,the averages of two long-term strength values of samples with different water content amounts were taken into consideration,as shown in Fig.4.

According to Fig.4,there is a negative linear correlation between the long-term strength and the water content between 15% and 31%,with a linearregressiveequation of y=229.153-4.837x,where x and y represent the water content and the long-term strength,respectively,and with a coefficient of determination(R2)of 0.98633.This relationship thus provides a basis for studying the long-term stability of slopes with a soft interlayer with varying water content.

Table 3 Statistical values of long-term strength of samples with different water contents.

Fig.4.Curve of long-term strength versus water content.

4.Ring shear creep constitutive equation

The study of rheological constitutive models has been a key part of rock and soil mass rheological studies.There are mainly three typesofrheologicalconstitutive models:empirical models,element combination models,and models based on fracture mechanics,damage mechanics,and endochronic theory considering creep mechanisms.The greatest advantage of element combination models is that they are based on physical elements with sound physical meanings.Afterthenonlinearimprovementofthoseparameters,element combination models can fit the experimental data better and can be very helpful for the later development of numerical simulations and further applications.The empirical models based on some specific formulas like power function are only applicable for the study materials.As for models based on fracture mechanics,damage mechanics,and endochronic theory,they are more proper for a rock mass than for a soft interlayer or soil.Considering the research objective and potential for further study of the constitutive model's application,this study focused on a new method of establishing element combination models based on fractional derivation.In an element combination model,the mechanical properties of rock and soil materials are made equivalent to different combinations of the mechanical properties of elastic,viscous,and plastic materials(Xia et al.,2008),while rheological deformation is a combination of elastic,viscous,and plastic deformations described by the Hook solid,Newtonian fluid,and plastic body theories,respectively.This model is widely used in rock and soil mass rheological studies because of its clear physical meaning.

4.1.Identification of rheological models

There are mainly three methods of identifying the rheological constitutive model:the direct screening method,posterior exclusion method,and comprehensive analysis method(Huang,2010).The comprehensive analysis method is used for model identification against the results of ring shear creep tests in a water-saturated state.As can be seen in Fig.3,at the moment when each level of shear stress is applied,all soft interlayer samples have significant instantaneous shear strains,suggesting that the model should contain elastic elements.The strain increases over time.At low shear stress levels,the strain rate decreases and gradually approaches zero,and the strain converges to a constant value,presenting viscosity and viscoelasticity variations.Therefore,the model should have viscosity and viscoelasticity.At high shear stress levels,the strain rate begins to converge to a nonzero constant value.As the strain increases continuously,the viscoplastic deformation characteristic is presented,suggesting that the rheological model should also contain viscoplastic elements.Based on this analysis,it is possible to conclude that the element model of the shear creep process curve should contain elastic,viscoelastic,and viscoplastic elements.

4.2.Establishment of modified three-element model

Among classical creep constitutive models,the Hoek-Kelvin(H-K)shear model is a typical constitutive model that can describe the elastic and viscoelastic shear rheological behavior.Its elements are combined in a form such as that shown in Fig.5.

The constitutive equation for the H-K shear model is written as

where τ is the shear stress;ε is the total shear strain;G1and G2are the shear moduli of the first and second elastic bodies,respectively;η is the viscosity coefficient;and t is time.

The Abel dashpot based on fractional calculus proposed by Zhou et al.(2011)has been widely used in the study of rheological constitutive models,since it can better re flect the process of nonlinear gradual change than other methods.For the viscous element in a three-element model,the Abel dashpot was used in this study to replace the viscous body,so as to improve the three-element model,as shown in Fig.6.

The modified three-element model consists of an elastic body and an improved viscoelastic body connected in a series.According to the series and parallel connections among the elements,the following equation can be obtained:

Fig.5.H-K model.

Fig.6.Modified H-K model.

where ε1and ε2are,respectively,the strains of the two parts in a series connection;β is the parameter of the Abel dashpot;τpis the shear stress of the elastic body in the second part of the series connection;and τAis the shear stress of the soft-matter element.The meaning of β in the Abel dashpot remains unclear,due to the primary use of fractional calculus in the field of rheological mechanics.This is an issue that requires further experimental and theoretical investigation to determine the physical meaning of β.

Solving the stress-strain relationship in Eq.(2)provides the creep constitutive equation for the modified three-element model,which is written as

where Γ（z）is the gamma function,and Ea，b（z）is the Mittag-Lef fler function,defined,respectively,as

where a and b are the parameters of the Mittag-Lef fler function.When β=1,Eq.(3)can be transformed to Eq.(1),suggesting that the previous three-element model is a special case of the modified three-element model,and the latter can better describe the rheological viscoelastic properties.

4.3.Establishment of nonlinear rheological model

Rheological deformation is a complex process of change in which elasticity,viscoelasticity,plasticity,and viscoplasticity coexist.To comprehensively describe the process of nonlinear gradual change in rheological behavior and the characteristics of accelerated creep,the modified threeelement model outlined in the previous section was used to re flect the characteristics of elasticity and viscoelasticity variations.To do so,a power-function viscoplastic body was connected in the series to re flect the characteristics of viscoplasticity variation at the accelerated creep stage,and thus to establish a new shear creep constitutive model,as shown in Fig.7.

The creep constitutive equation for the viscoplastic body in the third part of series connection under a constant shear stress is where τsis the shear yield stress of rock and soil materials;η2is the viscoelasticity coefficient;and n is a parameter to be determined.ε1,ε2,and ε3in Fig.7 are,respectively,the shear strain of each part in the series connection in the modified three-element model,and the total shear strain ε can be expressed as

Fig.7.Nonlinear visco-elasto-plastic rheological model.

Fig.8.Rheological curve and fitting results.

According to Eqs.(3),(6)and(7)as well as the series and parallel connections among the elements,a constitutive equation for the new visco-elasto-plastic rheological constitutive model can be written as

The new constitutive model was used to fit the results of the ring shear creep test on a sample with a water content of 27%.Based on the Boltzmann superposition principle,loading curves of different stress levels a water content of 27%are listed according to Fig.3.In the Origin software,the new rheological constitutive model was used to fit the test data,the fitting results are shown in Fig.8,and the model parameters obtained from such fitting are listed in Table 4.Meanwhile,to justify the accuracy and the reasonableness of the new model,the sevencomponent model established by Xu et al.(2005)was chosen to fit the testdata,as shown in Fig.8.The model fits well,as can be seen from thevalues of the coefficients of determination(R2)in Table 4,which are close to 1.0.

As can be seen in Fig.8,the new constitutive model can simulate the ring shear creep process curve as well as the seven-component model;the goodness offit of both models(Table 4)shows that a higher stress level leads to a better degree offitness.For the accelerated creep stage,the newly established model reaches a coefficient of determination of 0.987,proving the validity and applicability of the new nonlinear creep constitutive model.

This study also has a few limitations.For instance,it cannot answer the question of whether the water content affects the model parameters or not,which will be answered in future investigation.

5.Conclusions

(1)Water content has a significant impact on ring shear creep deformation.When the water content is lower than the saturated water content,the instantaneous shear strainincrement increases with the stress level,as the water content increases.When the water content is higher than the saturated water content,the instantaneous shear strain increment decreases with the increase of water content,and the maximum increment occurs near the saturated water content.As the water content increases,the cumulative creep increment of the samples also increases.

Table 4 Rheological constitutive model parameters.

(2)The long-term strength is negatively correlated with the water content.A higher water content will lead to lower longterm strength of the soft interlayer.

(3)The classical three-element model was improved based on the soft-matter element model,and a new nonlinear rheological constitutive model and equation have been established through a series connection with a power-function viscoplastic body.Excellent agreement of the proposed new constitutive equationwiththecreepcurveofasamplewithawatercontentof 27%proves the validity and applicability of the new equation.