A dissolution-diffusion sliding model for soft rock grains with hydro-mechanical effect

Z.Liu,C.Y.Zhou,B.T.Li,Y.Q.Lu,X.Yang

Guangdong Engineering Research Center for Major Infrastructure Safety,School of Civil Engineering,Sun Yat-sen University,Guangzhou,510275,China

1.Introduction

Deformation and failure of soft rock affected by hydromechanical(HM)effect are one of the most important issues in geotechnical engineering field.The deformation and failure of soft rock such as crack propagation,shear zone deformation,and creep are closely related to grain sliding at meso-scale.In order to understand the meso-mechanism of soft rock deformation and failure,it is necessary to understand the grain sliding of soft rock with the HM effect.Many studies have been carried out by focusing on the sliding of brittle rock grains and crystalline material grains,including elastic-,diffusion-and dislocation-coordinated grain slidings.The former two grain sliding phenomena mainly occur in low stress state and the third one mainly occurs in high stress state.This article deals with the diffusion-coordinated grain sliding in the low stress condition,in which diffusion is considered around the sliding surface.Conventionally,the grain sliding rate was the focal point in the previous studies.

Gifkins and Snowden(1966)derived the diffusion-coordinated sliding rate for the step-shaped grain sliding,but they only considered the boundary diffusion,ignoring the body diffusion.Ashby et al.(1970)and Ashby(1972)established a diffusioncoordinated grain sliding model with introduction of a sawtoothed sliding surface.In this model,the grain sliding rate with low degree of waviness was derived.Additionally,they discussed the behavior of atoms surrounding the sliding surface when the crystalline grains slid,and found that if the crystal contains significant defects,the predications of grain sliding rate and creep rate are in accordance with the classical results.Subsequently,Ashby and Verrall(1973)established adiffusion-coordinated grain deformation model in the condition of tensile stress.In this model,interaction among grains has been taken into consideration,and it was found that the grains do not exhibit apparent deformation in the direction of tensile stress.Yang and Wang(2004)assumed that the model proposed by Ashby and Verrall(1973)was not repeatable,and they proposed a nine-grain sliding model in consideration of different distances between grains.The molecular dynamics was employed for the simulation with this model,and it was observed that the grain creep rate is positively correlated with stresses.Raj and Ashby(1971)proposed a grain sliding model by taking into consideration both boundary and body diffusions,based on which the sliding and strain rates were derived.Meanwhile,a threedimensional sliding model was proposed,and the obtained results accorded with the conclusions of Gifkins and Snowden(1966).Rutter and Mainprice(1979)studied the sliding of quartzsandstone grain under the HM effect,and derived the sliding rate of quartz grains under shear stress.The theoretical results have also been veri fi ed by the stress relaxation experiments on sandstones.Morris and Jackson(2009a,b),Lee and Morris(2010)and Lee et al.(2011)modified the grain sliding model proposed by Raj and Ashby(1971),by considering the grain boundary.With this model,both elastic-and diffusion-coordinated grain slidings were discussed and the first-and second-order solutions were achieved.It was shown that the sliding rate varies with the positions of grain boundary.Jackson et al.(2014)discussed the difference and relations between the elastic-and diffusion-coordinated grain slidings,and validated the conclusions of Lee and Morris(2010)through experiments.Korla and Chokshi(2014)observed the deformation of crystalline materials as well as the sliding of internal grains under stresses using scanning electron microscope(SEM)and electron back-scattering diffraction(EBSD),and derived an continuity equation describing the grain sliding.Mancktelow and Pennacchioni(2004)found that quartz grains in aqueous conditions were different from those in anhydrous conditions,and that the quartz sliding was hindered significantly in the anhydrous conditions.

Recently,relative research has been conducted mainly in the following three respects:

(1)The impact of other substances on the grain sliding.Dobosz et al.(2012)used the finite element method(FEM)to simulate the case that the second substance was added at the boundary of grains,and they found that the grain sliding apparently enhanced the material’s hardening.In order to observe the phenomena of coupling and sliding between particles,Sch?fer and Albe(2012a)simulated the case that other substances were dispersed at the boundaries between crystal grains by the molecular dynamics theory.In addition,the uniaxial tension test was also conducted.It was shown that the solutes at the boundaries were conducive for the sliding between crystal grains.Du et al.(2011)also conducted simulations based on the molecular dynamics theory,and found that grain sliding was more significant when some other substances were present.Sch?fer and Albe(2012b)simulated the variation in solution concentration of other substances using the large-scale atomic/molecular massively parallel simulator(LAMMPS),and found that the grain sliding was thus considerably influenced.Molodov et al.(2011)studied the grain sliding within aluminium specimens and found that the increase in concentration of other substances accelerated their migration rate at the boundary of crystal grains,and thereby enhanced the sliding rate.

(2)The coupling of grain rotating and sliding.Kim et al.(2010)proposed a rotating-sliding coupling model of regularhexagon-grain.In comparison with the sliding rates of cases with or without grain rotating,it was found that by considering the grain rotating,the diffusion-coordinated sliding rate was relatively large.Ovid’ko and Sheinerman(2013)carried out a similar study through discrete element numerical simulation,considering the coupling effect of grain rotating and sliding.According to the stress-strain curve obtained from the uniaxial compression test,the ductility of materials was obviously improved due to the coupling effect of grain rotating and sliding.Since the variations in grain rotating,sliding and deformation can be well detected by the EBSD,Bird et al.(2015)used the EBSD to observe the structures and orientations of crystalline,and put forward an improved grain sliding model to reveal the impact of grain rotating on material creep.

(3)The impact of grain sliding on crack propagation.Ovid’ko et al.(2011)proposed a new crack propagation model by taking into account the crystal grain sliding under stresses.The stress intensity factor of cracks was derived.It was indicated that the grain sliding and migration facilitate the crack propagation.Fang et al.(2014)established a twodimensional model containing initial cracks,considering the diffusion-coordinated grain sliding.It was revealed that the stress intensity factor increased with the decrease in grain size.Ovid’ko and Sheinerman(2012)established a crystalline solid model with pre-cracks,and it was found that the fracture toughness of cracks was enhanced due to the grain rotating around the cracks,which was consistent with the findings from the previous studies(Cheng et al.,2010;Liu et al.,2011).

There are distinct differences between soft and brittle rocks.The soft rock is rich in clay minerals.When it contacts with water,significant physico-chemical reactions take place,and it is thus much easier to induce disasters.With the hydro-mechano-chemical coupling effect,the sliding of soft rock grains is different from that of brittle rock grains.However,rare research has been conducted with respect to the sliding of soft rock grains under the hydro-mechano-chemical effect.

The objectives of this study are to conduct laboratory experiments on the red bed soft rock from South China,and discuss the dissolution and diffusion of soft rock grains with the HM effect.Based on the cosine sliding surface,the equations of continuity,mechanical equilibrium and chemical potential balance of the soft rock dissolution and diffusion are established.Combined with the Einstein-Stokes equation,the equation for the grain sliding rate can be established.Finally,a dissolution-diffusion sliding model of soft rock grains with the HM effect is obtained.Based on the digital image processing technology,a method for studying the relationship between the grain size of soft rock and the amplitude of sliding surface is presented.Furthermore,based on the dissolutiondiffusion sliding model of soft rock grains,the equation for creep rate of soft rock can be established.Finally,it is verified by triaxial creep tests on the soft rock with the HM effect.

Fig.1.Typical distribution and shape of red bed soft rock grains(Zhu,2009).

2.Characteristics of the dissolution and diffusion of soft rock grains with the hydro-mechanical effect

Fig.1 shows the distribution and shape of the red silty mudstone grains of a typical red bed soft rock in South China(Zhu,2009).A large number of clay minerals are filled between quartz grains.The solid lineabmarks the potential sliding surface of the clay mineral layer,and this surface matches closely with the cosine function curve indicated by the dashed lineab.The surface of clay mineral is negatively charged,thus can attract the surrounding positive charge and form a crumb structure or flocculated structure.In order to make the experiments relatively easy,the clay minerals with lumpy or flocculated structure are assumed as clay grains in this study.Compared with quartz,the strength of clay grains is much lower.Additionally,the soft rock can expand in presence of some expansible minerals such as montmorillonite and illite.Abstracted from Fig.1,Fig.2 presents the sketch of sliding of soft rock grains with the HM effect.The filling thickness of the clay minerals is assumed ast,and the amplitude of the quartz grain surface asH,thus the degree of filling by the clay minerals is defined ast/H.When relative sliding occurs between the two grains under shear force(the sliding rate˙Ukeeps constant),the sliding surface may appear in different locations such as the curveab.At the same time,the sliding surface is full of water from the solution and diffusion of soft rock minerals.

Fig.3 gives an enlarged view of ZoneOindicated in Fig.2,illustrating the dissolution and diffusion processes.The clay minerals can be dissolved by chemical reactions on the grain surface and then be transported by diffusion of solutes from higher to lower concentration area.A variety of solutes(ionic forms)can be generated by dissolving the soft rock minerals.In order to simplify the processes,the solute unit is assumed as the relevant mineral crystal cell(e.g.kaolinite mineral crystal cell as Al4［Si4O10］（OH）8to be diffused).

Fig.4 shows a cosine shape sliding surface extracted from Fig.1.The periodicity and amplitude of the cosine surface are defined asλ andh,respectively.The upper grains slide from left to right,which means that the sliding surface slides from the solid lineabto the dashed lineAB.The lower grains slide from right to left,suffering a shear forceτfrom the left.As the sliding surface is smooth,the normal stressσnperpendicular to the sliding surface should be added to balance the shear stressτ.According to the principle of force interaction,the normal stress on the segmentcbof the sliding surface can be divided into the compressive stress on the segmentcdand the tensile stress on the segmentdb.The solutes diffuse from the compressive stress zone(slash area)to the tensile stress zone(dotted area),which indicates the movement of the sliding surface.In the following sections,the characteristics of the dissolution and diffusion of soft rock grains with the HM effect are introduced,including the mineral dissolution and the relationship among hydration film,disjoining pressure,and chemical potential.

Fig.2.Sketch of sliding of soft rock grains with the hydro-mechanical effect.

Fig.3.Enlarged view of mineral dissolution-diffusion at Zone O in Fig.2.

Fig.4.Solute diffusion-migration and position change of sliding surface(revised after Raj and Ashby,1971).

2.1.Characteristics of soft rock mineral dissolution

According to composition test(Zhu,2009),the main minerals of a typical silty mudstone in South China are kaolinite,illite,quartz,and a certain percentage of montmorillonite.The free expansion rate of this silty mudstone in the vertical direction of bedding is low,between 0.197%and 0.562%(Zhou et al.,2003,2005a).The clay mineral can be dissolved and hydrogen ion(H+)can be separated out,which can result in low pH solution(as low as 4)(Zhou et al.,2004).At the same time,some soluble cationsMN+such as Na+and K+can be released into the solution to form low concentration saline solution.Studies(Dove,1994;Luo et al.,2001)show that acidic solution can increase the solubility of clay minerals,but will have limited effect on the dissolution of quartz.The saline water,however,could increase the dissolution rate and solubilityof quartz by the “salt effect”(Xie and Walther,1993;Davis et al.,2011).The mineral solubilitySis also affected by the temperatureT,pressureP,and stress σ,which can be expressed asS(P,T,σ,pH,MN+),and shorten asS′.

2.2.Relationship between thickness of hydration film and disjoining pressure of the dissolution and diffusion of soft rock grains

Water molecules can be strongly attached onto the surface of hydrophilic minerals,such as montmorillonite,illite and kaolinite,to form an orderly aligned boundary layer known as the hydration film in the soft rock(Peng et al.,2012).Also in the soft rock,the deprotonated Si-O-can attract hydrogen ion to form hydroxyl on the surface of quartz mineral.The hydrogen bonds composed by hydroxyl and water molecules can arrange water molecules into the hydration film(Dove,1994).The sliding surface in Fig.2 is actually the hydration film.It can transfer the grain normal and shear stresses and is the carrier for the dissolution and diffusion of soft rock mineral.The dissolution of minerals is controlled by surface chemical reaction and diffusion.The dissolution in the film is“diffusion-controlled”,namely it reaches the chemical reaction equilibrium.The concentrationC′at a certain point on the sliding surface is equal to the solubility of the mineralS′at the same point with corresponding normal stress,namelyC′=S′.

The hydration film is squeezed when two mineral particles approach each other to create resistance or disjoining pressureΠ(Derjaguin,1955).With the consideration of the pore water pressure,the disjoining pressureΠis similar to the effective stress(Heidug,1995):

whereσcis the stress between grains,andPWis the pore water pressure.The thickness of the hydration filmδrelates closely to the disjoining pressureΠbetween the mineral grains.Experimental and fitting curves of the disjoining pressure versus the thickness of the hydration film on the mica and quartz grains are presented in Fig.5.In a certain range,with the increase ofΠ,δwill arrive at a lower valueδ0(Fig.5),andΠ is inversely correlated withδ:

whereaandbare the fitting parameters.Mica,illite and kaolinite are layered silicate minerals,thus the relationship betweenδandΠ of the clay mineral grains of soft rock also fits Eq.(2).

Fig.5.Relationship between the disjoining pressureΠand the thickness of hydration filmδof mineral grains(Pashley,1982;Gee et al.,1990).

2.3.Chemical potential of the dissolution and diffusion of soft rock grains

The chemical potential gradient is the essential driving force of diffusion.Although the definition of chemical potential for solids is still controversial,according to the definition of the Gibbs free energy and the Helmholtz free energy(Zhu,2009),the chemical potential of one-component material can be expressed as where μ is the chemical potential,f′is the partial molar Helmholtz free energy,pis the pressure,andvis the partial molar volume.The definition of chemical potential is used for liquid and gas,and the pressures are scalars.With the external force,the internal stress state of solid presents as tensor σijand its strain tensor as εij.The chemical potential of solid is difficult to be defined.Kamb(1961)argued that only when the solid phase of the tensor stress is locally balanced with the same component liquid phase,then it can be applied for the definition of chemical potential of the substance on the solid surface.

Under low stresses at room temperature,local equilibrium is reached between the surface of one-component grains(such as grains of kaolinite,illite and quartz)and the hydration film.Thus,the chemical potential of the mineral on the surface of soft rock grains,μs,is similar to that of the same component solute in the hydration film,μl.An improved expression of the chemical potential is proposed by Liu and Peng(2000):

wheref0,0.5∑（σijεijv0），EPlasticityandvsare the molar Helmholtz free energy without normal stress,the elastic strain energy,the plastic strain energy with normal stress,and the molar volume of the mineral on the surface of one-component soft rock grains with normal stress,respectively.Under low stresses at room temperature,there is no plastic deformation of the soft rock grains and the elastic strain energy is much smaller thanσnvs.The partial molar volumevsis stable,thus it can be expressed as the molar volumev0.Therefore,Eq.(4)can be simplified as

3.Dissolution-diffusion sliding model for soft rock grains with the hydro-mechanical effect

Laboratory data showed that the sliding surface of typical red bed soft rock grains can be described bythe cosine shape.Equations of continuity,mechanical equilibrium and chemical potential balance of the dissolution and diffusion of soft rock were established with the assumption of cosine shape surface.

3.1.Shape and half-wavelength waviness of the sliding surface of soft rock grains

According to the shape and distribution of the typical red bed soft rock grains,the shape of the sliding surface of soft rock grains can be expressed as a cosine function(Fig.4):

The half-wavelength waviness of the sliding surface can be defined as 2h/λ,and for quartz grains,it is equal to 2H/λ.

There are few studies on the half-wavelength waviness of the sliding surface,2h/λ,and the position of damage.Chen et al.(2005)conducted several sets of direct shear tests on the muddy interlayer and found that the rupture surface was located at the interface between the muddy interlayer and the rock surface,or in the muddy interlayer,indicating that the sliding surface of soft rock grains may occur at the interface between quartz and clay mineral layers or in the clay mineral layer.The half-wavelength waviness of the sliding surface,2h/λ,relates closely with the filling degreet/H.Goodman et al.(1968)added crushed mica fillings with different thicknesses between a man-made saw-toothed structural surface to conduct shear tests,and they found that the larger the filling degree,the lower the friction coefficient of structural plane.Thus,for the soft rock grains,the friction coefficient of sliding surfacefcan be written as a function of the filling degree of the clay mineral within the quartz grains:

Eq.(7)merely shows that the friction coefficient of sliding surfacefis related to the filling degreet/H.It cannot determine the range off.Here the friction coefficient of soft rockf1is assumed asf,namelyf=f1=tanθ,where θis the internal friction angle of soft rock.But with the same half-wavelength waviness,the friction coefficient of the cosine sliding surface is different from that of the saw-toothed sliding surface.

For the saw-toothed sliding surface(Fig.6),the friction coefficient is defined as

wherefsawis the friction coefficient of the saw-toothed sliding surface that can be expressed as the half-wavelength waviness 2h/λ;and θ1is the angle between the saw-toothed sliding surface andy-axis,which equals the internal friction angle of soft rock.The friction coefficient of the cosine sliding surface(Fig.6)can be defined as

wherefcosineis the friction coefficient of the cosine sliding surface,θyis the angle between the tangent at a point on the cosine sliding surface andy-axis,andxyis the derivative ofxwith respect toy.The right side of Eq.(9)is the elliptic integral which cannot be expressed by an elementary function,as it needs to be solved by numerical integration.The obtained results are presented in Table 1.When the internal friction angle of soft rock θ＜ 25°(the inner friction angle of the saturated silty sandy mudstone is generally between 13.3°and 24.3°(Zhou et al.,2005b)),we have Eq.(13)indicates the connection between the half-wavelength waviness of the sliding surface of soft rock grains and the internal friction angle of soft rock.Whenhtakes the value ofHin Eq.(13),the friction coefficient of the cosine sliding surface is equal to that of sliding surface between clay and quartz grains.

3.2.Continuity of soft rock grain sliding with the hydro-mechanical effect

Fig.7 shows an enlarged view of the element E in Fig.4.At room temperature,the boundary diffusion is much more intense than the bulk diffusion.Thus,this study focuses only on the boundary diffusion.

As shown in Fig.7,the boundary diffusion flux isJB,which represents the mass that passes vertically through unit area within unit time.The width of the element E(ABCD)isΔsand its angle withy-axis isθy.The mineral density is assumed asρ.The element E moves right integrally with the sliding surface at a rate˙U,namely during thetimeΔt,the element E movesfora distance of˙UΔt.There are two coordinate systems in Fig.7,the first one is the whole coordinate systemx-yand the second one is the partial coordinate systemn-t.During the timeΔt,the sliding surface(boundary)moves to the right and the variation in the mineral mass of the element E is defined as

Solute diffuses andmigrates in the hydration film,and the solute diffusion fluxes of the sectionsABandCDof the hydration film areJB(t)andJB(t+Δs),respectively.The continuity of steady sliding requires that the diffusion mass through the sliding surface isΔmE.During the timeΔt,the mass variation within the element E is

Combined with the chemical potential,Fick’s diffusion equation can be expressed as the Onsager diffusion equation(Liu and Peng,2000):

Fig.6.Sketch of saw-toothed sliding surface(left)and cosine sliding surface(right).

Table 1Shear strengths of saw-toothed and cosine sliding surfaces.

where μ (x,y)is the chemical potential field(J/mol); ?μ(x,y)/?tis the chemical potential gradient ont-axis(J/(mol m));C′0is the mineral saturated concentration(kg/m3)without any stressσ(solubility),at certain temperatureT,pressurePW,pH value,and ionic compositionMN±(in Eq.(16),the original expression isDC′/（RT）（?μ/?t），due to the limited variation ofC′at low stress and normal temperature conditions,it can be replaced byC′0);Dis the diffusion coefficient(m2/s);Risthe gasconstant,R=8.314 J/(K mol);andL=DC′/（RT）is the Onsager diffusion coefficient.

According to the definition of directional derivative,μx=?μ（x，y）/?x，μxx= ?2μ（x，y）/?x2，μy= ?μ（x，y）/?y，μyy= ?2μ（x，y）/?y2，thus Eq.(15)can be expressed in thex-ycoordinate system as

Fig.7.Detailed view of diffusion on the sliding surface element E(revised after Raj and Ashby,1971).

where μxx｜Γand μyy｜Γare the second-order partial derivatives ofμ(x,y)toxandy,respectively,on the sliding surface Γ.Combining Eqs.(14),(17)and(18),the following equation can be obtained:

3.3.Mechanical equilibrium equation of grain sliding of soft rock with the hydro-mechanical effect

The normal stress of grains on the sliding surface,σn,also has the sine function form within the periodic wavelength(Raj and Ashby,1971):

Considering the balance of force on the parallel sliding direction,the shear stress between particles with periodic variation τλis expressed as

Combining Eqs.(10),(20)and(21),the following equation is obtained:

whereαsis the influence coefficient of half-wavelength waviness on the normal stress,which can be calculated according to Eq.(22)using numerical integration.The results are listed in Table 2.

Thus,the average compressive stress can be expressed as

According to the discussion in Section 2.2,the thickness of the hydration film δis related to the disjoining pressureΠ.Different normal stresses at different points result in differentδvalues,for which further studies are needed.The thickness of the hydration film between grains with the compressive stress is thinner than that of the hydration film between grains with the tensile stress.The hydration film with the thinner thinness is essential for the solute diffusion rate on the sliding surface.Therefore,it is of practical significance to replaceδin Eq.(19)with the average thickness of the hydration filmδΠunder the compressive stress.If σcin Eq.(1)is replaced by the average compressive stress σc，the average disjoining pressure Πcan be expressed as

3.4.Chemical potential equilibrium equation of grain sliding of soft rock with the hydro-mechanical effect

During steady-state sliding,the solute is neither created nor absorbed within the two grains,which means that the solute just migrates to other locations of the border.Therefore,the divergence of the diffusion flux on the sliding surface▽·J→(whereJ→is the diffusion flux)is zero,whereby the chemical potential fieldμ(x,y)satisfies the Laplace equation:

In order to solveμ(x,y),two boundary conditions are required according to Eq.(5):

Table 2Influence coefficients of different half-wavelength wavinesses on the normal stress and rate.

whereμ(∞,y)is the chemical potential without normal stress at the location ofx→∞;and μ（x，y）｜Γis the chemical potential of the mineral on the surface of soft rock grain or the chemical potential of the same component solute in hydration film.The question in Eq.(26)with the boundary conditions(Eqs.(27)and(28))is known as the de Lickley question of the Laplace equation.If only the shape of boundary is certain and the function is smooth and continuous,it has the unique solution.Eqs.(6),(27)and(28)ensure the boundary shape and the smooth and continuous function,thus Eq.(26)has a unique solution.

3.5.Dissolution-diffusion sliding model for soft rock grains with the hydro-mechanical effect

Whenh? λ，we have θy→0,then Eq.(25)can be simplified as

Meanwhile,the boundary condition(i.e.Eq.(28))can be changed as

According to variables separation,Eq.(26)has a solution with the boundary conditions(Eqs.(27)and(30)):

Assumingx=0,and substituting Eqs.(8)and(9)into Eq.(29),the sliding rate of grains˙Uh?λcan be obtained whenh?λ:

Substituting Eq.(31)into Eq.(25),the sliding rate of the grains with different half-wavelength wavinesses can be obtained:

The sliding rate is different for each point,thus˙Uis replaced by the average sliding rate˙U:

whereαhis the influence coefficient of half-wavelength waviness on the sliding rate.The influences of the half-wavelength waviness on the sliding rate and the normal stress can be unified withα,namely α = αsαh,as shown in Table 2.When θis less than 25°,the influence of half-wavelength waviness on the sliding rateU˙ is low.UsingU˙ instead ofU˙,we have

wherekis the Boltzmann constant,k=1.38×10-23J/K;ηis the liquid viscosity(Pa s);andr′is the radius of the solute unit(m).In addition,the molar volume is v0=N0［4/（3πr3）］，whereN0is the Avogadro’s number,N0=6.02 × 1023;andris the equivalent radius of the mineral crystal cell(m),r=N0k.The solute unit diffuses in the form of the relevant mineral crystal cell,thusr′=r，and then Eq.(36)can be rewritten as

Eq.(38)is the dissolution-diffusion sliding model for soft rock grains with the HM effect,and it is also the formula for the calculation of grain sliding rate.Some parameters can vary more significantly,such as the amplitude of the sliding surface,the shear stress and the average thickness of the hydration film,which indicates that these three parameters affect the sliding rate of soft rock grains more significantly.

Eq.(38)indicates the sliding rate between two groups of soft rock grains with the same component.If the sliding surface occurs between different clay grains,such as kaolinite and illite grains,the average value ofrcan be applied due to the limited difference.If the sliding surface occurs between clay grains and quartz grains,due to the smaller equivalent radius and sliding rate,the relative sliding rate between the two grains is controlled by the sliding rate of the quartz sliding rate.

4.Steady-state creep rate of soft rock and its relation with the diameter of grains and amplitude of sliding surface

4.1.Steady-state creep rate of soft rock based on the dissolutiondiffusion sliding model of grains

In consideration of the correlation between the soft rock steadystate creep incrementΔeand the soft rock grain sliding,the diameter of the soft rock grain is assumed asd,and˙UΔt′/dis the ratio of the sliding distance of soft rock grain during the time Δt′to the grain diameter.The relationship between˙UΔt′/dandΔecan be expressed as(Rutter and Mainprice,1978):

where˙eis the soft rock steady-state creep rate with the HM effect,andγis the derivative of the fissure coefficient.Generally,γis equal to 1 when the soft rock sample is intact,otherwise it is greater than 1(Ma et al.,2010).Substituting Eq.(38)into Eq.(39)can obtain:

Eq.(40)indicates thatdh2is a key factor to control the strain rate.In order to determine the strain rate,the diameter of grain and the amplitude of sliding surface should be determined first.However,their values are uncertain due to various sizes of soft rock grains which also control the amplitude of sliding surface.As shown in Fig.8,at the initial scale,if the diameters of the grains are 10 mm and 10 μm,the surface amplitudes are 0.1 mm and 0.1 μm,respectively.If the grain with the diameter of 10 mm is magnified 1000 times,the amplitude can reach 0.1μm.Here the amplitude at a smaller scale(e.g.h′=0.1 μm)is called the second-level amplitude(Fig.8).

Generally,the larger the grain size,the greater the corresponding amplitude.The relationship between these two parameters is

Eq.(42)indicates that if the variation ofβis not significant,then the diameter of soft rock grains is an essential factor for the strain rate.Therefore,it is necessary to do some research onβ.

4.2.Relationship between the soft rock grain size and the amplitude of sliding surface based on the digital image processing

A method for studyingβis discussed here using the digital image processing technology.In orderto determine the influence of different scales onβ,SEM is utilized to observe nine different zones of the surface of the silty mudstone rock with 1000,3000 and 5000 magnifications,respectively(if the magnification is too small,the structure of the grain cannot be observed clearly;if it is too large,the number of grains in the view is too small,without statistical significance).Afterwards,the SEM images require further image processing in order to obtain more information.Here the digital image processes,including contrast enhancement processing,binarization processing,expansion and corrosion processing,and watershed processing,were conducted by MATLAB,and the binarization figure(Fig.9)was generated finally.The irregular black lines are the contact zone(sliding surface)between the white grains(Fig.9).The smallest unit of the half-wavelength waviness is the unitof the pixel sizew,thus the amplitudeh=w.Assuming that the diameter of thei-th grain isdi,theβvalue of this grain isβi=di/h=di/w=di.Consequently,the average βof the observed zone is

Fig.8.Relationship between grain size and amplitude of the sliding surface at different scales.

Fig.9.Sketch of grains contact.

whereNis the number of grains,anddis the average diameter of grains.Several groups of thin slices of silty mudstone were scanned by the electron microscope,and the results obtained from the digital image processing and Eq.(43)are listed in Table 3.It is indicated that the βvalue of the silty mudstone varies between 26 and 32.71,and the average diameter varies between 0.857μm and 3.406μm.

Replacingβanddin Eq.(42)with βandd，respectively,we have

5.Verification by experiments

In order to verify the dissolution-diffusion sliding model of soft rock grains with the HM effect,the triaxial creep experimental data of soft rock with the HM coupling effect were adopted to verify the reliability of Eq.(44)for the steady-state creep rate of soft rock,and also to verify the relationship between the inversion average disjoining pressure and the average thickness of the hydration film according to Eq.(2).Moreover,it can verify the reliability of the established model.

5.1.Triaxial creep experimental data of soft rock with the hydromechanical coupling effect

In order to determine the creep characteristics of soft rock,Qiu(2012)conducted a soft rock creep experiment with the HM coupling effect.The rock sample was placed in a water- filled pressure chamber directly.Here water can be applied as the confining pressure,which simulates the natural states of soft rock and water.The results of the experiment are presented in Table 4.

5.2.Verification of Eq.(44)for soft rock steady-state creep rate

The values of the parameters in Eq.(44)have a certain range.If the results obtained from Eq.(44)are in the same magnitude as the soft rock creep rate shown in Table 4,it is considered to be rational.The shear stress is replaced with the deviatoric stress,i.e.τ=σ1-σ3,and the average disjoining pressure can be calculated according to Eqs.(23)and(24):

Table 3Values of βandd obtained from digital image processing and Eq.(43).

Table 4Results of triaxial creep tests with the hydro-mechanical coupling effect(Qiu,2012).

Table 5Results of Π，δande˙ of soft rock grains obtained from the established model.

Fig.10.Experimental and fi tting relations between ΠandδΠ.

The half-wavelength waviness of grains on the sliding surface,2h/λ,can be figured out by the internal friction angle of the saturated silty mudstone in the range of 13.3°-24.4°.Here it is assumed as 20°,namelyf=tan20°=0.36,and the corresponding αs=1.114,andα=0.62.The calculated average disjoining pressure is presented in Table 5.

According to Eq.(39),γ=1.According to the crystal cell of kaolinite,illite and quartz,r=(2.4-4)×10-10m.At the normal temperature and pressure,the saturated concentration of the neutral pH solutionC′0=10-5g/L=10-5kg/m3(Luo et al.,2001).According toTable 3,the range of the average diameter of soft rock grains is 0.857-3.406μm.The diameter of clay mineral is generally sm aller than 2μm,namelyd=2×10-6m,andβ=28.There is limited variation in the viscosity of water,η=1×10-3Pa s(Renard et al.,1997).The mineral densityρ=2.5×103kg/m3.According to Fig.5 and Eq.(45),whenΠ >10 MPa,namely(σ1-σ3)>4.5 MPa,δ≈1 nm.Substituting all the above parameters into Eq.(44),the results of the steady creep rate can be obtained,as presented in Table 5,whicharein the same magnitude(10-7s-1)as those shown in Table 4.It proves the rationality of the dissolution-diffusion sliding model of soft rock grains with the HM effect.

5.3.Veri fi cation of the relationship between the thickness of hydration film and the disjoining pressure of soft rock grains

Section 5.2 indicates that the thickness of the hydration film calculated by the following equation is also in the magnitude of nanometer:

The correlation coefficientR2of Eq.(47)is equal to 0.950459.Eqs.(47)and(2)have the consistent pattern,which proves the rationality of the dissolution-diffusion sliding model of soft rock grains with the HM effect from another perspective.

6.Conclusions

In order to reveal the deformation and failure of red bed soft rock in South China,the meso-mechanism of soft rock grain sliding with the HM effect was investigated.In this article,we discussed the characteristics of dissolution and diffusion of soft rock grains with the HM effect.By combining the continuity equation,mechanical equilibrium equation and chemical potential equilibrium equation of the dissolution-diffusion of soft rock grain sliding,a dissolution-diffusion sliding model for soft rock grains has been established in consideration of the HM effect.Based on this model,the equation characterizing the soft rock steady creep rate was derived and validated by experiments.The conclusions can be drawn as follows:

(1)The characteristics of the dissolution and diffusion of soft rock grains with the HM effect,including the dissolution of soft rock mineral,the relationship between the thickness of hydration film and the disjoining pressure,and the chemical potential of dissolution and diffusion,are the foundation of the dissolution-diffusion sliding model for soft rock grains with the HM effect.

(2)According to the shape and distribution of the red bed soft rock grains,the cosine sliding surface was applied for this study.Combined with the equations for continuity,mechanical equilibrium and chemical potential balance of the dissolution and diffusion of soft rock,the dissolutiondiffusion sliding model for soft rock grains with the HM effect was established.The calculation equation for the sliding rate was proposed,which indicated that the amplitude of the sliding surfaceh,the shear stressτand the average thickness of the hydration filmδΠaffected the sliding rate of soft rock grains with the HM effect significantly.

(3)Based on the dissolution-diffusion sliding model for soft rock grains with the HM effect,the digital image processing was applied in order to discuss the relationship between the diameter of soft rock grain and the amplitude of sliding surface.The equation for soft rock steady-state creep rate was derived,which indicated that the average diameter of the soft rock grain affected the soft rock steady-state creep rate significantly.

(4)The triaxial creep experimental data of soft rock with the HM

coupling effect were used to verify the reliability of the equation for the soft rock steady-state creep rate and the relationship between the inversion average disjoining pressure and the average thickness of the hydration film according to the theoretical formula.It proved the rationality of the dissolution-diffusion sliding model for soft rock grains with the HM effect.

Conflicts of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The research is supported by the National Key Research and Development Project(Grant No.SQ2017YFSF060085),the National Natural Science Foundation of China(NSFC)(Grant Nos.41472257,41530638,and 41372302),the Special Fund Key Project of Applied Science and Technology Research and Development in Guangdong(Grant No.2016B010124007),and the Special Support Program for High Level Talents in Guangdong(Grant No.2015TQ01Z344).

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