Improved Nonlinear Nishihara Shear Creep Model with ...downloads.hindawi.com/journals/ace/2020/7302141.pdf · 0 1 2 3 4 5 6 7 8 Shear stress (MPa) 0 5 10152025 Shear displacement

Research ArticleImproved Nonlinear Nishihara Shear Creep Model with VariableParameters for Rock-Like Materials

Hang Lin 1 Xing Zhang 1 Yixian Wang 2 Rui Yong 3 Xiang Fan4 Shigui Du3

and Yanlin Zhao 5

1School of Resources and Safety Engineering Central South University Changsha Hunan 410083 China2School of Civil Engineering Hefei University of Technology Hefei 230009 China3Ocean College Zhejiang University Zhoushan Zhejiang 316000 China4School of Highway Changrsquoan University Xirsquoan Shannxi 710064 China5School of Energy and Safety Engineering Hunan University of Science and Technology Xiangtan Hunan 411201 China

Correspondence should be addressed to Yixian Wang wangyixian2012hfuteducn

Received 19 September 2019 Revised 13 January 2020 Accepted 3 February 2020 Published 27 February 2020

Academic Editor Flavio Stochino

Copyright copy 2020 Hang Lin et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Creep property is an important mechanical property of rocks Given the complexity of rock masses mechanical parameterschange with time in the creep process In this work a nonlinear function for describing the time-dependent change of parameterswas introduced and an improved variable-parameter nonlinear Nishihara shear creep model of rocks was established By creatingrock-like materials the mechanical properties of rocks under the shear creep test condition were studied and the deformationcharacteristics and long-term shear strength of rocks during creep were analyzed e material parameters of the model wereidentified using the creep test results Comparison of the modelrsquos calculated values and experimental data indicated that the modelcan describe the creep characteristics of rocks well thus proving the correctness and rationality of the improved model Duringshear creep the mechanical properties of rocks have an aging effect and show hardening characteristics under low shear stressFurthermore according to the fact that Gk of the nonlinear model can characterize the creep deformation resistance a method todetermine the long-term shear strength is proposed

1 Introduction

Creep property is an important mechanical property ofrocks and the main factor affecting long-term stability ofgeotechnical engineering [1ndash7] Obtaining the stress anddeformation characteristics of rock masses from the per-spective and method of their rheology is consistent withreality [8ndash10] us studying the creep characteristics ofrocks is crucial e creep behavior of rocks can be dividedinto three stages namely attenuated stable and acceleratedcreep In the initial creep stage the strain increases withtime In the steady-state creep stage the strain increasesuniformly with time and the duration of this stage is rel-atively long In the accelerated creep stage the strain rateincreases with time and the rock is destroyed after reachingthe critical fracture point Numerous studies have shown

that rocks exhibit typical nonlinear characteristics duringcreep in which the accelerated creep stage is the mostobvious [4 11 12] To reveal the mechanical properties ofrock rheological behavior scholars have conducted targetedin-depth research and obtained abundant research results[13ndash15] By summarizing existing rheological models Xiaet al [16] proposed a unified rheological mechanical modelRelevant studies have shown that model parameters varyduring rock creep and are related to stress time temper-ature and other factors [17 18] Referring to damage me-chanics theory [19 20] Jia et al [21] proposed a nonlinearelastic-viscoplastic damage model based on the modifiedMohrndashCoulomb criterion Moghadamaab [22] used -con-stitutive model to describe expansion and short- and long-term damages in the creep process of rock salt and estab-lished a numerical model of the rock salt excavation

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 7302141 15 pageshttpsdoiorg10115520207302141

underground chamber as it changes with time Yang and Li[11] replaced the viscous element of the Nishihara modelwith a new element by using fractional calculus andestablished a nonlinear Nishihara model that describes rockrheology Shao et al [23] established a full stress-strain creepmodel from the perspective of microstructure evolution byconsidering various factors ese models effectively de-scribe the nonlinear characteristics of model parametersduring rock creep but focus on the creep mechanicalproperties of rocks under uniaxial and triaxial compressionResearch on rock characteristics under direct shear creepconditions is relatively scarce e primary failure mode ofrocks under natural conditions is shear failure [24ndash30]erefore the shear creep failure behavior of rocks should bestudied e classical Nishihara model can describe themechanical properties of rocks in decay and stable creepstages [11 31 32] but its parameters are constant and cannotreflect the nonlinear characteristics of creep To reveal themechanism of rock mechanical behavior under shear creepconditions and establish a corresponding shear creep modelthis study investigated the shear rheological mechanicalcharacteristics of rocks with plastic-elastic deformation byperforming a shear test on a similar sandstone material Atime-varying factor was introduced to the Nishihara modelto describe the functional relationship between the viscositycoefficient and time On the basis of this factor an improvednonlinear Nishihara shear creep model that can reveal theunsteady and nonlinear characteristics of parameters wasestablished e modelrsquos rationality was verified through acomparison with experimental values

2 Specimen Preparation and Test Methods

Natural rock masses contain many micro- and macrocrackswhich lead to different physical and mechanical charac-teristics [33ndash41] Miller divided the stress-strain curves ofrocks with different physical and mechanical properties intosix categories [42 43]e third category is called the plastic-elastic deformation feature in which the stress-strain curveis concave in the initial stage and then becomes a straight-line segment until destruction [44ndash46] Rocks with manypores and microfractures such as granite and sandstonereflect this deformation characteristic [47 48] In this studythe mechanical behavior and deformation characteristicsunder shear creep conditions were studied by fabricatingsimilar materials with the plastic-elastic deformation featureof rocks [49]

Experiments were performed on a RYL-600 computer-controlled rock shear rheometer which is usually used forrheological testing of materials such as rock and concreteand composed of three components the host the test andcontrol system and the computer system e rheometercan simultaneously apply vertical-axis compression testforce and horizontal-axis shear test force to a specimen andcontrol the loading by force or deformation In the shearcreep test the rock-like specimens were under constanttemperature and humidity conditions e loading rate ateach stage was 300Ns e specimens adopted the sametype of material mix ratio curing time and mold

specification (10 cm times 10 cm times 10 cm) rough the uni-axial and triaxial compression tests the mechanical pa-rameters of rock materials with different cement riversand and water ratio were obtained Finally the sandstonematerials with cement river sand and water ratio of 1 1 0406 were selected as research objects e mechanicalparameters of the rock-like specimens are shown in Table 1which is similar to the mechanical properties of sandstoneswith elastoplastic deformation characteristics Duringpouring the material was mixed according to the ratio andthe surface was smoothed after tamping in the steel mold tominimize adverse effects in the manufacturing process emold was taken apart after modeling for 24 hours andmaintained for 28 days All specimens are intact rock-likematerials

As the loading model shown in Figure 1 the specimenswere subjected to a fast shear test to compare the strengthcharacteristics of the rockrsquos fast shearing and shear creepconditions e test results are shown in Figure 2 At thebeginning of loading the slope of the stress-strain curveincreased with the loading stress and this process lasted for along time After the microcracks were compacted com-pletely the slope of the curve entered the linear elastic stagewith a constant size and then maintained this feature untilfailure Significant plastic elastic deformation characteristicswere exhibited e fast shear strengths of specimens 1ndash4were 41MPa 56MPa 645MPa and 733MPa respec-tively In the grading loading creep test normal stress wastaken as 5 10 15 and 20 of the uniaxial compressivestrength of a specimen e shear stress levels of specimen 1were 30 40 50 60 and 70 of the fast shear strengthe shear stress levels of specimen 2 were 30 40 50and 60 of the fast shear strength until destruction eshear stress levels of specimen 3 were 30 40 60 70and 80 of the fast shear strength e shear stress levels ofspecimen 4 were 30 50 60 70 and 80 of the fastshear strength e loading method is presented in Table 2and the process is shown in Figure 3

3 Test Results and Analysis

31 Mechanical Properties with Time e creep curves ofgraded shear loading under the same normal stress levelwere obtained through the creep shear test (Figure 4) In thecreep test curve obtained a straight line parallel to thelongitudinal axis was made to intersect with the creep curveunder the same normal stress and different shear forces andthe isochronal curve (Figure 5) was drawn by the stress anddisplacement values at each intersection point obtained Ascan be seen from Figure 5 the slope of the isochronous shearstress-shear displacement curve varies with stress Stress isthe factor that affects themechanical properties of rocks withplastic-elastic deformation characteristics which are dif-ferent from the characteristics of the microfracture com-paction stage completed immediately after loading in mostrocks To study the time deterioration effect on the rockcreep process the deformation modulus under differentcreep time conditions was calculated through the slope of thecurve section of the adjacent shear stress levels and the

2 Advances in Civil Engineering

0

1

2

3

4

5

6

7

8

Shea

r str

ess (

MPa

)

5 10 15 20 250Shear displacement (mm)

Figure 2 Shear stress-shear displacement relationship in the fast shear test under 312MPa normal stress

Table 1 e mechanical parameters of the rock-like specimens

Parameter σc (MPa) μ φ (deg) c (MPa)Specimen 156 021 60 312

Normal stress

Shea

r stre

ss

Shear platform

ConstraintConstraint

(a)

Horizontal shearing deviceControl system

Normal loading device

(b)

Figure 1 (a) Specimen loading model and (b) test equipment diagram

Table 2 Test grading loading method

Specimen number Normal stress (MPa) Shear strength (MPa)Shear stress (MPa)

1 2 3 4 51 078 41 123 164 205 246 2872 156 56 168 224 28 336 Broken3 234 645 1935 258 387 4515 5164 312 733 2199 3665 4398 5131 5864

Advances in Civil Engineering 3

Shear stress (MPa)123164204

246287

1416182022242628303234

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(a)


28336

10

15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(b)


4515516

15

20

25

30

35

40

45

50

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(c)


51315864

42

44

46

48

50

52

54

56

58

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(d)

Figure 4 Shear displacement-time curves under different normal stresses (a) 078MPa (b) 156MPa (c) 234MPa and (d) 312MPa

Δτ

Δτ

Δτ

Δτ

τ

t

Figure 3 Schematic diagram of grading loading process


relationship between deformation modulus and time wasplotted and is shown in Figure 6

e formula for calculating the deformation moduluscan be written as

E τ1 minus τ2

ctτ1

minus ctτ2

(1)

where τ1 and τ2 characterize the adjacent two-stage shear stressand ct

τ1 and ctτ2 indicate the shear displacement at different

moments under the respective shear stress conditionsFigure 6 indicates that for the rocks with plastic-elastic

deformation characteristics under the condition of lowshear stress the deformation modulus of rocks in creepbehavior increased with time which is contrary to the de-terioration of the mechanical properties of most rocks withtime [50 51] is result is due to the existence of manymicrocracks in plastic-elastic rocks [52 53] Low shear stresscannot easily compact the microcracks inside the rockcompletely in a short time and the microcracks that are not

compacted continue to be compacted in the following creepbehavior thus showing a hardening effect When the shearstress level was increased to a certain value the deformationmodulus weakened with time and showed aging deterio-ration is phenomenon occurred because the shear stressat this time exceeded the critical shear stress when themicrocrack was completely compacted which not onlystopped the hardening effect brought by microcrack com-paction but also developed the internal crack of the specimenwith time and produced significant aging degradation ef-fects When the shear stress continues to increase themechanical properties of the rock continued to deterioratewith time because the unfinished area of the rock when shearstress was applied to the upper stage continued to deteriorateunder the new first-order shear stress condition Meanwhilethe increase in shear stress led to the deterioration of the areathat is not easy to deteriorate under the condition of theshear stress of the upper level resulting in continuous in-ternal damage of the specimen

202 204 206 208 210 212 214

196198200202204206208210212214

Shea

r str

ess (

MPa

)

Shear displacement (mm)

Enlarge

01h05h1h

2h3h5h

8h12h

081012141618202224262830

Shea

r str

ess (

MPa

)

16 18 20 22 24 26 2814Shear displacement (mm)

(a)

16265

270

275

280

285

290

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(b)

290 295 300 305 310 315

44

46

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

35

40

45

50

55

Shea

r str

ess (

MPa

)

20 25 30 3515Shear displacement (mm)

01h05h1h

2h3h5h

8h12h

(c)

525 530 535 540 545 550 55548

49

50

51

52

53

54

Shea

r str

ess (

MPa

)


Enlarge

20

25

30

35

40

45

50

55

60

Shea

r str

ess (

MPa

)

42 44 46 48 50 52 54 56 5840Shear displacement (mm)

01h05h1h

2h3h5h

8h12h

(d)

Figure 5 Isochronal graphs under different normal stresses (a) 078MPa (b) 156MPa (c) 234MPa and (d) 312MPa


32 Determination of Long-Term Strength under DifferentNormal Stress Conditions According to the isochronouscurve method which is commonly used in rheologicalmechanics [54] the stress corresponding to the secondturning point of the isochronous curve after the fullycompacted linear elastic phase was used as the long-termstrength value of the rock In Figures 5(a)ndash5(d) the shearstress value corresponding to the second turning point is204MPa 28MPa 4515MPa and 5131MPa respec-tively us the long-term shear strength of the specimenunder the normal stress condition of 078MPa 156MPa234MPa and 312MPa is 204MPa 28MPa 4515MPaand 5131MPa respectively Correspondingly the ratioof long-term shear strength to fast shear strength is 0505 07 and 0702 respectively e phenomenonwherein the specimen had different ratios of long-termstrength to fast shear strength under different normalstress conditions could be caused by the numerous

microcracks in the rock with plastic-elastic deformationcharacteristics Under low normal stress the internaldamage of the rock accumulated continuously overtimeand with the increasing shear stress Damage occurredwhen the local damage reached the limit and lost thebearing capacity Under high normal stress the micro-cracks in the normal direction of the rock were com-pacted closely thereby making the hardening effectcaused by the compaction of microcracks more obviousthan that under low stress Meanwhile under the con-dition of dense microfractures the mechanical propertiesand internal continuity of the rock improved thusstrengthening the capability to transfer stress inside therock increasing the effective bearing area andstrengthening the capability to resist damage ereforethe ratio of long-term shear strength to fast shear strengthof the rock under high normal stress is greater than thatunder low normal stress

Shear stress (MPa)123ndash164164ndash205

205ndash246246ndash287

2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)

Figure 6 Deformationmodulus-time diagrams under different normal stresses (a) 078MPa (b) 156MPa (c) 234MPa and (d) 312MPa


4 Improved Nonlinear Shear Creep ModelBased on the Nishihara Model

e test results above illustrate that during the process ofcreep from the decay creep stage to the stable creep stage theinternal microcracks are continuously compacted whichcauses irreversible plastic deformation and a hardeningeffect erefore for rocks with an obvious hardening effectunder low shear stress conditions the traditional Nishiharamodel with linear parameters is no longer applicable emodel parameters must be described in a nonlinear mannerTo reflect the mechanical properties of the rock in the creepprocess accurately a time-varying factor was introduced toimprove the traditional Nishihara model and an improvedvariable-parameter nonlinear Nishihara shear creep modelof rock was established e rationality of the model wasverified by the creep test result of specimens

41 Description of the Characteristics of the Time-DependentParameter in the NishiharaModel e Nishihara model is arheological model composed of Hooke Kelvin and visco-plastic bodies in series as shown in Figure 7 e modelexpression is

c

τGM

+τ

GK

1 minus eminus GKηK( )t

1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

where GM is the Hooke body coefficient which is the in-stantaneous elastic modulus GK is the Kelvin model elasticcoefficient ηK is the Kelvin model viscosity coefficient η2 isthe viscous element coefficient in the viscoplastic body τs isthe long-term shear strength of the rock and ts is themoment when the accelerated creep stage begins

To describe the nonlinearity of parameters in theNishihara model under shear loading time-varying factor Dwas introduced to the Nishihara model by referring to themethod of establishing fractional-derivative models [55] todescribe the variation in element coefficients in the Nishi-hara model with time and a nonlinear shear creep modelwith variable parameters was constructed (Figure 8)

A large number of studies have shown that the internaldamagehardening variable is closely related to time as anexponential function [56] erefore on the basis of pre-vious studies the time-varying factor D is defined as

D μeminus αt

(3)

where D isin [0 1] μ and α are material parameters repre-senting the variation in element coefficients μ isin [minus infin 1] andα isin [0 +infin]

Compared with the nonlinear function in the fractional-derivative model which describes the aging characteristicsof the model parameters with only one parameter [55 57] inthis paper the function D used to describe the nonlinearcharacteristics of model parameters has two parameters

where μ represents the comparison of model parametersbefore and after creep occurs and α reflects the trend of themodel parameters over time It is not as simple and easy tocalculate as the function in fractional-derivate models but itcan describe the evolution characteristics of model pa-rameters during the creep process more comprehensively

In the Nishihara model the Kelvin body represents theattenuation and stable creep stages and the viscoplastic bodyrepresents the accelerated creep stage In the actual creepprocess all parameters change with time Time-varyingfactorDwas introduced to the Kelvin and viscoplastic bodiesto describe the relationship between the viscosity coefficientand time

e Kelvin model is composed of an elastic element anda viscous element in parallel and its constitutive equation is

τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)

After introducing time-varying factor D the Kelvinviscous element with variable parameters is expressed as

ηtK ηinfinK 1 minus D1( 1113857 ηinfinK 1 minus μ1e

minus α1t1113872 1113873 (7)

where ηtK is the actual viscosity coefficient that changes over

time when tinfin ηtK ηinfinK ηinfinK is the viscosity coefficient of

the final moment of rock stable creep μ1 is the materialparameter characterizing the damagehardening degree ofηK when μ1 is positive ηK hardens with time during rockcreep when μ1⟶ 1 ηK hardens into a plastomer at tinfinwhen μ1 0 ηK is constant which is the original Kelvinmodel when μ1 is negative ηK deteriorates with time duringrock creep when μ1⟶infin ηK softens into an ideal fluid at

τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp

Figure 8 Improved nonlinear shear creep model with variableparameters

GK

GM

η2ηK

τ ττs

γe γKel γvp

Figure 7 Nishihara rheological model


tinfin α1 is a parameter reflecting the trend of ηK over timein the creep process the larger α1 is the longer it takes for theηK to stabilize

Equation (7) indicates that when t 0 the initial viscouselement coefficient η0K is

η0K ηinfinK 1 minus D1( 1113857 ηinfinK 1 minus μ1( 1113857 (8)

In accordance with (3)ndash(7) the constitutive equation ofnonlinear Kelvin model with parameter changes is obtainedas follows

τ τ1 + τ2 GK + ηtK _c

ηtK _c ηinfinK 1 minus μ1e

minus α1t1113872 1113873 _c τ minus GKc

(9)

Combined with the initial conditions t 0 and c 0 theKelvin nonlinear shear creep model equation with param-eter variation is obtained as follows

ckel τ

GK

1 minus eminus GKηinfinK( ) ln eα1tminus μ1( )minus ln 1minus μ1( )( )α1( )1113874 1113875 (10)

Similarly for the viscoplastic body introduced with time-varying factor D the viscous element with variable pa-rameters is expressed as

ηt2 ηinfin2 1 minus D2( 1113857 ηinfin2 1 minus μ2e

minus α2t1113872 1113873 (11)

where ηt2 is the actual viscosity coefficient that changes

over time when t infin ηt2 ηinfin2 ηinfin2 is the viscosity co-

efficient of the final moment of rock accelerated creepwhen μ2 gt 0 η2 hardens with time and the larger μ2represents the higher degree of hardening when μ2 lt 0 η2softens with time and the smaller μ2 represents the higherdegree of softness

e constitutive equation of viscoplastic body of Nish-ihara model is

τ minus τs η2 _cvp (12)

Combining (11) and (12) the constitutive equation of theviscous element with variable parameters is obtained asfollows

τ minus τs η2 1 minus D2( 1113857 _cvp η2 1 minus μ2eminus α2 tminus ts( )1113874 1113875 _cvp (13)

cvp τ minus τs

α2η2ln eα2 tminus ts( ) minus μ21113874 1113875 (14)

Combining (10) and (14) the constitutive model ofimproved nonlinear Nishihara shear creep model withvariable parameters can be obtained as follows

c(t)

τGM

+τ

GK

1 minus eminus GKηinfinK( ) ln eα1 tminus μ1( )minus ln 1minus μ1( )( )α1( )1113874 1113875 τ lt τs

τGM

+τ

GK

1 minus eminus GKηinfinK( ) ln eα1 tminus μ1( )minus ln 1minus μ1( )( )α1( )1113874 1113875

+τ minus τs

α2ηinfin2ln eα2 tminus ts( ) minus μ21113874 1113875 τ gt τs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

42 Model Verification To test the correctness of the im-proved nonlinear shear creep model established in thisstudy the model was used to fit the test curve with theexperimental data of specimens which was compared withthe fitting curve of the original Nishihara model to deter-mine the advantages and disadvantages of the improvedmodel e parameters of the improved model are given inTable 3 e fitting results are shown in Figure 9 and itindicates that the improved Nishihara model was closer tothe creep test curve than the original Nishihara model wasthereby reflecting the mechanical behavior of the specimencreep process and verifying the reasonableness of the model

43 Relationship between CreepModel Parameters and StressIn rock rheology constitutive model parameters have im-portant physical and mechanical significance Studying therelationship between model parameters and loading con-ditions (stress loading rate etc) can effectively reveal themechanical mechanism of rock rheology [58ndash60] In order tobetter understand the mechanical properties characterizedby D on the basis of Table 3 the relationship between μ1 α1and shear stress under the normal stress 156MPa is plottedin Figure 10 It could be seen from Figure 10 that μ1 is greaterthan 0 under each shear stress condition and the maximumvalue occurs when the shear stress is 168MPa and thendecreases with the increase of shear stress and finally sta-bilizes to 097 indicating that the nonlinear Kelvin viscositycoefficient has hardening under various shear stress con-ditions and the hardening degree is the highest under lowshear stress condition and the hardening degree tends to bestable after shear stress increases As shown in Figure 10 asthe shear stress increases α1 increases first then decays andthen continues to increase Combined with (7) it can be seenthat the time required for the viscosity coefficient in thenonlinear Kelvin body to reach stability is relatively longunder the condition of low shear stress and then fluctuateswith the increase of shear stress but all remain stable at acertain value Finally when the shear stress reaches the valuebeyond the long-term strength it decreases rapidly and thisphenomenon is caused by the compaction of rock-likespecimen under low shear stress conditionse relationshipbetween ηt

K and time in accordance with (7) and Table 4 isplotted in Figure 11 which can better show the changeprocess of ηt

K represented by μ1 and α1e relationship between Kelvin model elastic coefficient

GK and shear stress is plotted in Figure 12(b) and it showsthat when the range of the shear stress was from 168MPa to224MPa Kelvin model elastic coefficient GK increased withstress and this result indicates the obvious internal hard-ening effect of the specimen and the characteristics of theplastic-elastic deformation of the rock-like specimen thatnot only hardens under low-stress fast shear but also underlow-stress creep When the shear stress was loaded to28MPa the GK was attenuated indicating that under thisshear stress condition the internal microcracks of thespecimen began to develop and damage began to occurAfter the shear stress reached 336MPa the specimen


Table 3 Model parameters under various shear stresses conditions

Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK

(MPamiddoth)ηinfin2

(MPamiddoth) μ1 α1τs

(MPa) ts (h) μ2 α2Correlationcoefficient R2

078

123 08 1160 44866 mdash 0985 3861 mdash mdash mdash mdash 0998164 093 2435 174511 mdash 0977 0601 mdash mdash mdash mdash 0999205 101 2236 137981 mdash 0996 1096 mdash mdash mdash mdash 0997246 106 2159 105654 mdash 0993 1324 mdash mdash mdash mdash 0996287 108 1967 196583 mdash 0991 0703 mdash mdash mdash mdash 0999

156

168 106 1758 26098 mdash 0999 0435 mdash mdash mdash mdash 0999224 157 11667 121206 mdash 0977 0628 mdash mdash mdash mdash 099928 176 3927 98289 mdash 0971 0390 mdash mdash mdash mdash 0999336 074 710 39693 0087 0970 2032 28 02634 minus 1470 364 0996

234


312


Experimental dataImproved Nishihara modelOriginal Nishihara model

16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued


quickly entered the accelerated creep stage e viscosityparameter of the viscous element in the viscoplastic bodythat describes the accelerated creep stage was negative in-dicating that the internal damage of the specimen wascontinuously intensified and the creep rate was continuouslyaccelerated until failure

44 Long-Term Strength Determination Method Based onImproved Nonlinear Model It can be concluded from (10)that the rock-like specimen creep deformation is controlled by

GK when GK is small the mass of rock creep deformation islarge that is the larger GK is the smaller the creep defor-mation is GK characterizes the ability to resist creep defor-mation erefore in the improved nonlinear model theauthor proposes a method to determine the long-termstrength by using the curve of GK-shear stress is methoduses the nonlinear Nishihara model proposed in this paper tofit shear displacement-time curve to obtain the curve of GK minus

τ and then determines the shear stress of GK decay in thecurve as the long-term strength According to the parametersin Table 3 the GK minus τ curves of the specimen under different


5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)

Figure 9 Comparison of the test and calculated values of the creep model under normal stress (a) 078MPa (b) 156MPa (c) 234MPa and(d) 312MPa

μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1

Figure 10 e relationship between μ1 α1 and shear stress diagram

Table 4 Initial viscosity coefficient η0 under different shear stress conditions

Shear stress (MPa) 168 224 28 336η0 (MPamiddoth) 2356 6765 4684 0098


0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)

Figure 11 e relationship between ηtK and time diagram

10 12 14 16 18 20 22 24 26 28 30

Attenuation point (205 2236)

Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)

Attenuationpoint (28 3927)

0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)

Figure 12 e relationship between viscosity modulus GK-shear stress diagram under normal stress (a) 078MPa (b) 156MPa (c)234MPa and (d) 312MPa


normal stress levels were drawn (Figure 12) and were com-pared with the long-term shear strength determined by theisochronous curve method as shown in Table 5

Table 5 shows that the long-term shear strength deter-mined by these two methods is basically the same whichproves the effectiveness of the long-term strength deter-mination method proposed in this paper And comparedwith the isochronous curve method the modulus inflectionpoint method only needs to fit the experimental data todetermine the GK under each stress condition en thelong-term strength of the rock can be obtained according tothe attenuation of GK which has a simpler and faster op-eration process

5 Discussion and Conclusion

Research on the mechanical characteristics of plastic-elasticrock deformation under the conditions of fast shear andshear creep indicates that under low shear stress the me-chanical properties of rocks with plastic-elastic deformationare continuously strengthened during creep thereby con-tinuously enhancing the rockrsquos resistance to deformationWhen the shear stress level increases the microcracks insidethe rock are completely compacted At this time the me-chanical properties of the rock deteriorate with creep andaging Meanwhile the mechanical properties of rocks withplastic-elastic deformation under different normal stressconditions differ e rock bearing capacity and the damageresistance under high normal stress are better than thoseunder low normal stress erefore for rock mass engi-neering the rock with plastic-elastic deformation charac-teristics should be placed under low stress conditions andthe surrounding area should avoid high-intensity stress fieldto enhance the stability of the engineering and the project

In this work a time-varying factor D was introduced todescribe the viscous elements in a nonlinear manner and animproved nonlinear shear creep model based on theNishihara model was established After fitting the experi-mental data it is found that the model can reflect the shearcreep behavior of rock-like specimens effectively and themodel has rationality In addition under the normal stress of078MPa and 156MPa the ratio of long-term shearstrength to fast shear strength of rock samples with plastic-elastic deformation characteristics is 05 indicating that thecritical damage cumulative value required for rock reaches afailure state erefore in a project the long-term shearstrength at which the cracks inside the rock begin to developand expandmust be determined to prevent rock destructionAt present relevant scholars often monitor rock damageunder creep conditions by calculating the creep rate the

deformation modulus or the monitoring energy of theacoustic emission equipment However in practical engi-neering calculating the rate at each moment or providingspecial equipment is difficult By fitting the improvednonlinear Nishihara shear creep model to the surroundingrock displacement-time data monitored in the project thedamage can be discriminated according to the identifiedtrend of the viscous modulus Gk of the surrounding rock soas to provide timely support and avoid the structural in-stability of the project which is crucial in practical engi-neering Furthermore according to the fact that Gk of thenonlinear model can characterize the creep deformationresistance of rock-like specimen a method to determine thelong-term shear strength is proposed and compared withthe calculation results of isochronous curve method therationality and applicability of the method are verified

Notations

σc Uniaxial compressive strengthμ Poissonrsquos ratioφ Angle of internal frictionc Cohesive forceE e deformation modulusτ1 τ2 Adjacent two-stage shear stressctτ1 ct

τ2 e shear displacement at different momentsGM e Hooke body coefficientGK e Kelvin model elastic coefficientηK e Kelvin model viscosity coefficientη2 e viscous element coefficient in the viscoplastic

bodyτs e long-term shear strength of the rockts e moment when the accelerated creep stage

beginsμ α Material parameters representing the variation in

element coefficientsηinfinK e element coefficient after entering the stable

creep stageηt

K e real-time element coefficientμ1 α1 Material parameters representing the variation in

Kelvin model element coefficientsμ2 α2 Material parameters representing the variation in

viscoplastic body element coefficientsτs e long-term strength of the rock

Data Availability

All data generated or analyzed during this study are includedin this article

Table 5 Comparison of two methods for determining long-term shear strength

Specimen number Normal stress (MPa)Long-term shear stress (MPa)

Isochronous curve method Modulus inflection point method1 078 204 2042 156 28 283 234 4515 45154 312 5131 5131


Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (51774322 51774107 and 51774131)Hunan Provincial Natural Science Foundation of China(2018JJ2500) and Fundamental Research Funds for theCentral Universities of Central South University(2019zzts669) e authors wish to acknowledge thesesupports

References

[1] O Hamza and R Stace ldquoCreep properties of intact andfractured muddy siltstonerdquo International Journal of RockMechanics and Mining Sciences vol 106 pp 109ndash116 2018

[2] A Sainoki S Tabata H S Mitri D Fukuda and J-I KodamaldquoTime-dependent tunnel deformations in homogeneous andheterogeneous weak rock formationsrdquo Computers and Geo-technics vol 92 pp 186ndash200 2017

[3] P Cao Y Wen Y Wang H Yuan and B Yuan ldquoStudy onnonlinear damage creep constitutive model for high-stresssoft rockrdquo Environmental Earth Sciences vol 75 p 900 2016

[4] Y Zhao Y WangWWangWWan and J Tang ldquoModelingof non-linear rheological behavior of hard rock using triaxialrheological experimentrdquo International Journal of Rock Me-chanics and Mining Sciences vol 93 pp 66ndash75 2017

[5] C Jiang Z Zhang and J He ldquoNonlinear analysis of combinedloaded rigid piles in cohesionless soil sloperdquo Computers andGeotechnics vol 117 p 103225 2020

[6] K Hu F Qian H Li and Q Hu ldquoStudy on creep charac-teristics and constitutive model for thalam rock mass withfracture in tunnelrdquo Geotechnical amp Geological Engineeringvol 36 pp 827ndash834 2017

[7] Q Zhang M Shen W Ding and C Clark ldquoShearing creepproperties of cements with different irregularities on twosurfacesrdquo Journal of Geophysics and Engineering vol 9 no 2pp 210ndash217 2012

[8] S N Moghadam K Nazokkar R J Chalaturnyk andH Mirzabozorg ldquoParametric assessment of salt cavern per-formance using a creep model describing dilatancy andfailurerdquo International Journal of Rock Mechanics and MiningSciences vol 79 pp 250ndash267 2015

[9] Y Zhao L Zhang W Wang et al ldquoCreep behavior of intactand cracked limestone under multi-level loading andunloading cyclesrdquo Rock Mechanics amp Rock Engineeringvol 50 no 6 pp 1409ndash1424 2017

[10] Z-L He Z-D Zhu X-H Ni and Z-J Li ldquoShear creep testsand creep constitutive model of marble with structural planerdquoEuropean Journal of Environmental and Civil Engineeringvol 23 no 11 pp 1275ndash1293 2017

[11] L Yang and Z-D Li ldquoNonlinear variation parameters creepmodel of rock and parametric inversionrdquo Geotechnical andGeological Engineering vol 36 no 5 pp 2985ndash2993 2018

[12] H Mansouri and R Ajalloeian ldquoMechanical behavior of saltrock under uniaxial compression and creep testsrdquo Interna-tional Journal of Rock Mechanics and Mining Sciencesvol 110 pp 19ndash27 2018

[13] A Fahimifar M Karami and A Fahimifar ldquoModifications toan elasto-visco-plastic constitutive model for prediction of

creep deformation of rock samplesrdquo Soils and Foundationsvol 55 no 6 pp 1364ndash1371 2015

[14] H Ozsen I Ozkan and C Sensogut ldquoMeasurement andmathematical modelling of the creep behaviour of Tuzkoyrock saltrdquo International Journal of Rock Mechanics amp MiningSciences vol 66 pp 128ndash135 2014

[15] A Singh C Kumar L G Kannan K S Rao andR Ayothiraman ldquoEstimation of creep parameters of rock saltfrom uniaxial compression testsrdquo International Journal ofRock Mechanics and Mining Sciences vol 107 pp 243ndash2482018

[16] C C Xia L Jin and R Guo ldquoNonlinear theoretical rheo-logical model for rock a review and some problemsrdquo ChineseJournal of Rock Mechanics and Engineering vol 30 no 3pp 453ndash463 2011

[17] L Liu G-M Wang J-H Chen and S Yang ldquoCreep ex-periment and rheological model of deep saturated rockrdquoTransactions of Nonferrous Metals Society of China vol 23no 2 pp 478ndash483 2013

[18] F Deleruyelle T A Bui HWong N Dufour D K Tran andX S Zhang ldquoAnalytical study of the post-closure behaviour ofa deep tunnel in a porous creeping rock massrdquo ComptesRendus Mecanique vol 344 no 9 pp 649ndash660 2016

[19] C Y Zhang H Lin C M Qiu T T Jiang and J H Zhangldquoe effect of cross-section shape on deformation damageand failure of rock-like materials under uniaxial compressionfrom both a macro and micro viewpointrdquo InternationalJournal of Damage Mechanics vol 20 pp 1ndash20 2020

[20] X H Xu S P Ma M F Xia F J Ke and Y L Bai ldquoDamageevaluation and damage localization of rockrdquo Ieoretical andApplied Fracture Mechanics vol 42 no 2 pp 131ndash138 2004

[21] S P Jia L W Zhang B S Wu H D Yu and J X Shu ldquoAcoupled hydro-mechanical creep damage model for clayeyrock and its application to nuclear waste repositoryrdquo Tun-nelling and Underground Space Technology vol 74 pp 230ndash246 2018

[22] S N Moghadamaab ldquoModeling time-dependent behavior ofgas caverns in rock salt considering creep dilatancy andfailurerdquo Tunnelling amp Underground Space Technology vol 33pp 171ndash185 2013

[23] J F Shao Q Z Zhu and K Su ldquoModeling of creep in rockmaterials in terms of material degradationrdquo Computers andGeotechnics vol 30 no 7 pp 549ndash555 2003

[24] H Lin X Ding R Yong W Xu and S Du ldquoEffect of non-persistent joints distribution on shear behaviorrdquo ComptesRendus Mecanique vol 347 no 6 pp 477ndash489 2019

[25] Y Shen Y Wang Y Yang Q Sun T Luo and H ZhangldquoInfluence of surface roughness and hydrophilicity onbonding strength of concrete-rock interfacerdquo Constructionand Building Materials vol 213 pp 156ndash166 2019

[26] D Lei H Lin Y Chen R Cao and Z Wen ldquoEffect of cyclicfreezing-thawing on the shear mechanical characteristics ofnonpersistent jointsrdquo Advances in Materials Science andEngineering vol 2019 Article ID 9867681 14 pages 2019

[27] C Zhang C Pu R Cao T Jiang and G Huang ldquoe stabilityand roof-support optimization of roadways passing throughunfavorable geological bodies using advanced detection andmonitoring methods among others in the Sanmenxia BauxiteMine in Chinarsquos Henan Provincerdquo Bulletin of Engineering Ge-ology and the Environment vol 78 no 7 pp 5087ndash5099 2019

[28] J Jin P Cao Y Chen C Pu D Mao and X Fan ldquoInfluenceof single flaw on the failure process and energy mechanics ofrock-like materialrdquo Computers and Geotechnics vol 86pp 150ndash162 2017


[29] Y Chen and H Lin ldquoConsistency analysis of Hoek-Brownand equivalent Mohr-coulomb parameters in calculatingslope safety factorrdquo Bulletin of Engineering Geology and theEnvironment vol 78 no 6 pp 4349ndash4361 2019

[30] H Lin Y Zhu J Yang and Z Wen ldquoAnchor stress anddeformation of the bolted joint under shearingrdquo Advances inCivil Engineering vol 2020 Article ID 3696489 10 pages2020

[31] X-Y Yu T Xu M Heap G-L Zhou and P Baud ldquoNu-merical approach to creep of rock based on the numericalmanifold methodrdquo International Journal of Geomechanicsvol 18 no 11 Article ID 04018153 2018

[32] H W Zhou C P Wang B B Han and Z Q Duan ldquoA creepconstitutive model for salt rock based on fractional deriva-tivesrdquo International Journal of Rock Mechanics and MiningSciences vol 48 no 1 pp 116ndash121 2011

[33] M-D Wei F Dai J-W Zhou Y Liu and J Luo ldquoA furtherimproved maximum tangential stress criterion for assessingmode I fracture of rocks considering non-singular stress termsof the Williams expansionrdquo Rock Mechanics and Rock En-gineering vol 51 no 11 pp 3471ndash3488 2018

[34] G Chen T Chen Y Chen R Huang and M Liu ldquoA newmethod of predicting the prestress variations in anchoredcables with excavation unloading destructionrdquo EngineeringGeology vol 241 pp 109ndash120 2018

[35] Y Chen and C C Li ldquoPerformance of fully encapsulatedrebar bolts and D-Bolts under combined pull-and-shearloadingrdquo Tunnelling and Underground Space Technologyvol 45 pp 99ndash106 2015

[36] Y Zhao Y Wang W Wang L Tang Q Liu and G ChengldquoModeling of rheological fracture behavior of rock crackssubjected to hydraulic pressure and far field stressesrdquo Ieo-retical and Applied Fracture Mechanics vol 101 pp 59ndash662019

[37] Y Zhao L Zhang W Wang C Pu W Wan and J TangldquoCracking and stress-strain behavior of rock-like materialcontaining two flaws under uniaxial compressionrdquo RockMechanics and Rock Engineering vol 49 no 7 pp 2665ndash2687 2016

[38] H Chen X Fan H Lai Y Xie and Z He ldquoExperimental andnumerical study of granite blocks containing two side flawsand a tunnel-shaped openingrdquo Ieoretical and AppliedFracture Mechanics vol 104 Article ID 102394 2019

[39] Y Wang H Zhang H Lin Y Zhao and Y Liu ldquoFracturebehaviour of central-flawed rock plate under uniaxial com-pressionrdquo Ieoretical and Applied Fracture Mechanicsvol 106 Article ID 102503 2020

[40] F Dai M D Wei N W Xu Y Ma and D S Yang ldquoNu-merical assessment of the progressive rock fracture mecha-nism of cracked chevron notched Brazilian disc specimensrdquoRock Mechanics and Rock Engineering vol 48 no 2pp 463ndash479 2015

[41] H-B Du F Dai Y Xu Z Yan and M-D Wei ldquoMechanicalresponses and failure mechanism of hydrostatically pressur-ized rocks under combined compression-shear impactingrdquoInternational Journal of Mechanical Sciences vol 165 ArticleID 105219 2020

[42] C C Xia and Z Q Sun Engineering Rock Joint MechanicsTongji Press Shanghai China 2002

[43] S Xie H Lin Y Wang et al ldquoA statistical damage consti-tutive model considering whole joint shear deformationrdquoInternational Journal of DamageMechanics vol 29 Article ID105678951990077 2020

[44] X Fan H Lin H Lai R Cao and J Liu ldquoNumerical analysisof the compressive and shear failure behavior of rock con-taining multi-intermittent jointsrdquo Comptes RendusMecanique vol 347 no 1 pp 33ndash48 2019

[45] H Lin H Yang Y Wang Y Zhao and R Cao ldquoDetermi-nation of the stress field and crack initiation angle of an openflaw tip under uniaxial compressionrdquoIeoretical and AppliedFracture Mechanics vol 104 Article ID 102358 2019

[46] Y Wang H Lin Y Zhao X Li P Guo and Y Liu ldquoAnalysisof fracturing characteristics of unconfined rock plate underedge-on impact loadingrdquo European Journal of Environmentaland Civil Engineering vol 23 pp 1ndash16 2019

[47] Y-H Huang S-Q Yang P G Ranjith and J Zhao ldquoStrengthfailure behavior and crack evolution mechanism of granitecontaining pre-existing non-coplanar holes experimentalstudy and particle flow modelingrdquo Computers and Geo-technics vol 88 pp 182ndash198 2017

[48] S-Q Yang X-R Liu and H-W Jing ldquoExperimental in-vestigation on fracture coalescence behavior of red sandstonecontaining two unparallel fissures under uniaxial compres-sionrdquo International Journal of Rock Mechanics and MiningSciences vol 63 pp 82ndash92 2013

[49] E M Gell S MWalley and C H Braithwaite ldquoReview of thevalidity of the use of artificial specimens for characterizing themechanical properties of rocksrdquo Rock Mechanics and RockEngineering vol 52 no 9 pp 2949ndash2961 2019

[50] G Wang L Zhang Y Zhang and G Ding ldquoExperimentalinvestigations of the creep-damage-rupture behaviour of rocksaltrdquo International Journal of Rock Mechanics and MiningSciences vol 66 pp 181ndash187 2014

[51] CMuller T Fruhwirt D Haase R Schlegel andH KonietzkyldquoModeling deformation and damage of rock salt using thediscrete element methodrdquo International Journal of Rock Me-chanics and Mining Sciences vol 103 pp 230ndash241 2018

[52] H Lin S Xie R Yong Y Chen and S Du ldquoAn empiricalstatistical constitutive relationship for rock joint shearingconsidering scale effectrdquo Comptes Rendus Mecaniquevol 347 no 8 pp 561ndash575 2019

[53] R-H Cao P Cao H Lin X Fan C Zhang and T LiuldquoCrack initiation propagation and failure characteristics ofjointed rock or rock-like specimens a reviewrdquo Advances inCivil Engineering vol 2019 Article ID 6975751 31 pages2019

[54] S Zhang W Liu and H Lv ldquoCreep energy damage model ofrock graded loadingrdquo Results in Physics vol 12 pp 1119ndash1125 2019

[55] H Khajehsaeid ldquoA comparison between fractional-order andinteger-order differential finite deformation viscoelasticmodels effects of filler content and loading rate on materialparametersrdquo International Journal of Applied Mechanicsvol 10 no 9 Article ID 1850099 2018

[56] R B Wang W Y Xu W Wang and J C Zhang ldquoAnonlinear creep damagemodel for brittle rocks based on time-dependent damagerdquo European Journal of Environmental andCivil Engineering vol 17 no sup1 pp s111ndashs125 2013

[57] C Zopf S E Hoque and M Kaliske ldquoComparison of ap-proaches to model viscoelasticity based on fractional timederivativesrdquo Computational Materials Science vol 98pp 287ndash296 2015

[58] A S Khan O Lopez-Pamies and R Kazmi ldquoermo-me-chanical large deformation response and constitutive mod-eling of viscoelastic polymers over a wide range of strain ratesand temperaturesrdquo International Journal of Plasticity vol 22no 4 pp 581ndash601 2006


[59] H Khajehsaeid S Reese J Arghavani and R NaghdabadildquoStrain and stress concentrations in elastomers at finite de-formations effects of strain-induced crystallization fillerreinforcement and deformation raterdquo Acta Mechanicavol 227 no 7 pp 1969ndash1982 2016

[60] H Khajehsaeid ldquoModeling nonlinear viscoelastic behavior ofelastomers using a micromechanically motivated rate-de-pendent approach for relaxation times involved in integral-based modelsrdquo Constitutive Models for Rubber IX CRC PressBoca Raton FL USA pp 181ndash188 2015


underground chamber as it changes with time Yang and Li[11] replaced the viscous element of the Nishihara modelwith a new element by using fractional calculus andestablished a nonlinear Nishihara model that describes rockrheology Shao et al [23] established a full stress-strain creepmodel from the perspective of microstructure evolution byconsidering various factors ese models effectively de-scribe the nonlinear characteristics of model parametersduring rock creep but focus on the creep mechanicalproperties of rocks under uniaxial and triaxial compressionResearch on rock characteristics under direct shear creepconditions is relatively scarce e primary failure mode ofrocks under natural conditions is shear failure [24ndash30]erefore the shear creep failure behavior of rocks should bestudied e classical Nishihara model can describe themechanical properties of rocks in decay and stable creepstages [11 31 32] but its parameters are constant and cannotreflect the nonlinear characteristics of creep To reveal themechanism of rock mechanical behavior under shear creepconditions and establish a corresponding shear creep modelthis study investigated the shear rheological mechanicalcharacteristics of rocks with plastic-elastic deformation byperforming a shear test on a similar sandstone material Atime-varying factor was introduced to the Nishihara modelto describe the functional relationship between the viscositycoefficient and time On the basis of this factor an improvednonlinear Nishihara shear creep model that can reveal theunsteady and nonlinear characteristics of parameters wasestablished e modelrsquos rationality was verified through acomparison with experimental values

2 Specimen Preparation and Test Methods

Natural rock masses contain many micro- and macrocrackswhich lead to different physical and mechanical charac-teristics [33ndash41] Miller divided the stress-strain curves ofrocks with different physical and mechanical properties intosix categories [42 43]e third category is called the plastic-elastic deformation feature in which the stress-strain curveis concave in the initial stage and then becomes a straight-line segment until destruction [44ndash46] Rocks with manypores and microfractures such as granite and sandstonereflect this deformation characteristic [47 48] In this studythe mechanical behavior and deformation characteristicsunder shear creep conditions were studied by fabricatingsimilar materials with the plastic-elastic deformation featureof rocks [49]

Experiments were performed on a RYL-600 computer-controlled rock shear rheometer which is usually used forrheological testing of materials such as rock and concreteand composed of three components the host the test andcontrol system and the computer system e rheometercan simultaneously apply vertical-axis compression testforce and horizontal-axis shear test force to a specimen andcontrol the loading by force or deformation In the shearcreep test the rock-like specimens were under constanttemperature and humidity conditions e loading rate ateach stage was 300Ns e specimens adopted the sametype of material mix ratio curing time and mold

specification (10 cm times 10 cm times 10 cm) rough the uni-axial and triaxial compression tests the mechanical pa-rameters of rock materials with different cement riversand and water ratio were obtained Finally the sandstonematerials with cement river sand and water ratio of 1 1 0406 were selected as research objects e mechanicalparameters of the rock-like specimens are shown in Table 1which is similar to the mechanical properties of sandstoneswith elastoplastic deformation characteristics Duringpouring the material was mixed according to the ratio andthe surface was smoothed after tamping in the steel mold tominimize adverse effects in the manufacturing process emold was taken apart after modeling for 24 hours andmaintained for 28 days All specimens are intact rock-likematerials

As the loading model shown in Figure 1 the specimenswere subjected to a fast shear test to compare the strengthcharacteristics of the rockrsquos fast shearing and shear creepconditions e test results are shown in Figure 2 At thebeginning of loading the slope of the stress-strain curveincreased with the loading stress and this process lasted for along time After the microcracks were compacted com-pletely the slope of the curve entered the linear elastic stagewith a constant size and then maintained this feature untilfailure Significant plastic elastic deformation characteristicswere exhibited e fast shear strengths of specimens 1ndash4were 41MPa 56MPa 645MPa and 733MPa respec-tively In the grading loading creep test normal stress wastaken as 5 10 15 and 20 of the uniaxial compressivestrength of a specimen e shear stress levels of specimen 1were 30 40 50 60 and 70 of the fast shear strengthe shear stress levels of specimen 2 were 30 40 50and 60 of the fast shear strength until destruction eshear stress levels of specimen 3 were 30 40 60 70and 80 of the fast shear strength e shear stress levels ofspecimen 4 were 30 50 60 70 and 80 of the fastshear strength e loading method is presented in Table 2and the process is shown in Figure 3

3 Test Results and Analysis

31 Mechanical Properties with Time e creep curves ofgraded shear loading under the same normal stress levelwere obtained through the creep shear test (Figure 4) In thecreep test curve obtained a straight line parallel to thelongitudinal axis was made to intersect with the creep curveunder the same normal stress and different shear forces andthe isochronal curve (Figure 5) was drawn by the stress anddisplacement values at each intersection point obtained Ascan be seen from Figure 5 the slope of the isochronous shearstress-shear displacement curve varies with stress Stress isthe factor that affects themechanical properties of rocks withplastic-elastic deformation characteristics which are dif-ferent from the characteristics of the microfracture com-paction stage completed immediately after loading in mostrocks To study the time deterioration effect on the rockcreep process the deformation modulus under differentcreep time conditions was calculated through the slope of thecurve section of the adjacent shear stress levels and the


0

1

2

3

4

5

6

7

8

Shea

r str

ess (

MPa

)





Normal stress

Shea

r stre

ss

Shear platform


(a)



(b)




1 2 3 4 51 078 41 123 164 205 246 2872 156 56 168 224 28 336 Broken3 234 645 1935 258 387 4515 5164 312 733 2199 3665 4398 5131 5864



246287

1416182022242628303234

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(a)


28336

10

15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(b)


4515516

15

20

25

30

35

40

45

50

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(c)


51315864

42

44

46

48

50

52

54

56

58

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(d)


Δτ

Δτ

Δτ

Δτ

τ

t





E τ1 minus τ2

ctτ1

minus ctτ2

(1)






202 204 206 208 210 212 214

196198200202204206208210212214

Shea

r str

ess (

MPa

)


Enlarge

01h05h1h

2h3h5h

8h12h

081012141618202224262830

Shea

r str

ess (

MPa

)


(a)

16265

270

275

280

285

290

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(b)

290 295 300 305 310 315

44

46

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

35

40

45

50

55

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(c)

525 530 535 540 545 550 55548

49

50

51

52

53

54

Shea

r str

ess (

MPa

)


Enlarge

20

25

30

35

40

45

50

55

60

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(d)







2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)






c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References

































































0

1

2

3

4

5

6

7

8

Shea

r str

ess (

MPa

)





Normal stress

Shea

r stre

ss

Shear platform


(a)



(b)




1 2 3 4 51 078 41 123 164 205 246 2872 156 56 168 224 28 336 Broken3 234 645 1935 258 387 4515 5164 312 733 2199 3665 4398 5131 5864



246287

1416182022242628303234

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(a)


28336

10

15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(b)


4515516

15

20

25

30

35

40

45

50

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(c)


51315864

42

44

46

48

50

52

54

56

58

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(d)


Δτ

Δτ

Δτ

Δτ

τ

t





E τ1 minus τ2

ctτ1

minus ctτ2

(1)






202 204 206 208 210 212 214

196198200202204206208210212214

Shea

r str

ess (

MPa

)


Enlarge

01h05h1h

2h3h5h

8h12h

081012141618202224262830

Shea

r str

ess (

MPa

)


(a)

16265

270

275

280

285

290

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(b)

290 295 300 305 310 315

44

46

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

35

40

45

50

55

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(c)

525 530 535 540 545 550 55548

49

50

51

52

53

54

Shea

r str

ess (

MPa

)


Enlarge

20

25

30

35

40

45

50

55

60

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(d)







2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)






c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References


































































246287

1416182022242628303234

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(a)


28336

10

15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(b)


4515516

15

20

25

30

35

40

45

50

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(c)


51315864

42

44

46

48

50

52

54

56

58

Shea

r disp

lace

men

t (m

m)

2 4 6 8 10 12 14 16 18 20 22 240Time (h)

(d)


Δτ

Δτ

Δτ

Δτ

τ

t





E τ1 minus τ2

ctτ1

minus ctτ2

(1)






202 204 206 208 210 212 214

196198200202204206208210212214

Shea

r str

ess (

MPa

)


Enlarge

01h05h1h

2h3h5h

8h12h

081012141618202224262830

Shea

r str

ess (

MPa

)


(a)

16265

270

275

280

285

290

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(b)

290 295 300 305 310 315

44

46

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

35

40

45

50

55

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(c)

525 530 535 540 545 550 55548

49

50

51

52

53

54

Shea

r str

ess (

MPa

)


Enlarge

20

25

30

35

40

45

50

55

60

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(d)







2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)






c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References



































































E τ1 minus τ2

ctτ1

minus ctτ2

(1)






202 204 206 208 210 212 214

196198200202204206208210212214

Shea

r str

ess (

MPa

)


Enlarge

01h05h1h

2h3h5h

8h12h

081012141618202224262830

Shea

r str

ess (

MPa

)


(a)

16265

270

275

280

285

290

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(b)

290 295 300 305 310 315

44

46

Shea

r str

ess (

MPa

)


Enlarge

15

20

25

30

35

40

45

50

55

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(c)

525 530 535 540 545 550 55548

49

50

51

52

53

54

Shea

r str

ess (

MPa

)


Enlarge

20

25

30

35

40

45

50

55

60

Shea

r str

ess (

MPa

)


01h05h1h

2h3h5h

8h12h

(d)







2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)






c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References





































































2 4 6 8 10 120Time (h)

12

14

16

18

20

22

Def

orm

atio

n m

odul

us (M

Pa)

(a)


30

31

32

33

34

35

36

37

38

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(b)



2 4 6 8 10 120Time (h)

1012141618202224262830

Def

orm

atio

n m

odul

us (M

Pa)

(c)


4398ndash51315131ndash5864

15

20

25

30

35

40

45

50

Def

orm

atio

n m

odul

us (M

Pa)

2 4 6 8 10 120Time (h)

(d)






c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References




































































c

τGM

+τ

GK


1113874 1113875 τ lt τs

τGM

+τ

GK


1113874 1113875 +τ minus τs

η2t τ gt τs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)




D μeminus αt

(3)






τ τ1 + τ2 GKc + ηK _c (4)

c c1 c2 τ1GK

τ2ηK

t (5)

c(t) τ

GK


1113874 1113875 (6)



minus α1t1113872 1113873 (7)




τ ττs

GM

GK

ηK D1η2 D2

γe γKel γvp


GK

GM

η2ηK

τ ττs

γe γKel γvp










(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References








































































(9)


ckel τ

GK




minus α2t1113872 1113873 (11)








cvp τ minus τs



c(t)

τGM

+τ

GK


τGM

+τ

GK


+τ minus τs


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)








Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References


































































Normal stress(MPa)

Shear stress(MPa)

GM

(MPa)GK

(MPa)ηinfinK




078


156


234


312



16

18

20

22

24

26

28

30

32

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(a)


15

20

25

30

35

40

45

50

55

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(b)

Figure 9 Continued







5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References






































































5 10 15 20 250Time (h)

15

20

25

30

35

40

45

Shea

r disp

lace

men

t (m

m)

(c)


424446485052545658606264

Shea

r disp

lace

men

t (m

m)

5 10 15 20 250Time (h)

(d)


μ1α1

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

0970

0975

0980

0985

0990

μ1

0

5

10

15

20

α1





0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References

































































0 2 4 6 8 10 12 14 16 18 20 22 240

200

400

600

800

1000

1200

1400ηt K

(MPa

middoth)

Time (h)

Shear stress (MPa)

224168 28

336

Enlarge

(a)

000 005 010 015 020 025 03000

05

10

15

20

25

30

35

40

ηt K (M

Pamiddoth

)

Time (h)

336MPa

(b)


10 12 14 16 18 20 22 24 26 28 30


Shear stress (MPa)

10

12

14

16

18

20

22

24

26

Visc

osity

mod

ulus

Gk (

MPa

)

(a)


0

20

40

60

80

100

120

140

Visc

osity

mod

ulus

Gk (

MPa

)

18 20 22 24 26 28 30 32 3416Shear stress (MPa)

(b)

15 20 25 30 35 40 45 50 55


Shear stress (MPa)

5

10

15

20

25

30

35

40

45

Visc

osity

mod

ulus

Gk (

MPa

)

(c)

21

22

23

24

25

26

Attenuationpoint

(5131 2173)

Visc

osity

mod

ulus

Gk (

MPa

)

25 30 35 40 45 50 55 6020Shear stress (MPa)

(d)









Notations






creep stageηt




Data Availability








Acknowledgments


References







































































Notations






creep stageηt




Data Availability








Acknowledgments


References



































































Acknowledgments


References






































































































Documents

Improved Nonlinear Nishihara Shear Creep Model with ...downloads.hindawi.com/journals/ace/2020/7302141.pdf · 0 1 2 3 4 5 6 7 8 Shear stress (MPa) 0 5 10152025 Shear displacement