unloadingrheologicaltestandmodelresearchofhard...

Research ArticleUnloading Rheological Test and Model Research of HardRock under Complex Conditions

Longyun Zhang1 and Shangyang Yang 23

1Geotechnical and Structural Engineering Research Centre Shandong University Jinan 250061 China2Department of Physics and Mathematics Shandong Jiaotong University Jinan 250023 China3Key Laboratory for Health Monitoring and Control of Large Structures Shijiazhuang 050043 China

Correspondence should be addressed to Shangyang Yang 26916225qqcom

Received 4 December 2019 Revised 31 January 2020 Accepted 28 February 2020 Published 11 May 2020

Academic Editor Paweł Kłosowski

Copyright copy 2020 Longyun Zhang and Shangyang Yang 3is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

In order to study the unloading rheological properties of hard rock under complicated conditions the proposed Mengdigouhydropower station with a granite dam foundation was taken as the research object 3is paper studied the unloading rheologicalproperties of the granite by testing means and it obtained the unloading rheological deformation characteristics and deformationrules of the granite under complex stress conditions 3e granite unloading rheological characteristics of the whole stress-straincurve were analyzed and the nonlinear unloading rheological model (FOD-HKVP) was proposed based on the fractional-orderderivative When certain conditions are satisfied it is possible to degrade this model into the classical Nishihara model but itshould be noted that the FOD-HKVP model is a more reasonable way to determine the long-term strength of the granite and toconsider also the rock unloading rheological damage effect It introduced the fractional-order viscous body of combinationcomponents with a different fractional-order derivative to describe the nonlinear creep characteristics of the granite unloadingwhich provides an adequate description of steady-state creep and accelerated creep properties Following the analysis of test dataand parameter identification for the FOD-HKVPmodel the rationality of the model was validated Compared with the Nishiharamodel the fitting accuracy of FOD-HKVP model was more accurate

1 Introduction

3e engineering rock mass has a complex geologicalstructure and rheological properties are one of the mostimportant mechanical characteristics With the rapid de-velopment of the economy paired with rapid progress inscience and technology many large water resources andhydropower engineering facilities continuously extend deepunderground arriving at points where the geological rockmass is situated in a more complex environment For manylarge hydropower stations in China the buried depth of thefoundation engineering of underground caverns is largerand most of the dam foundation rock mass is comprised ofgranite marble and other hard masses 3e natural con-dition is that rock masses are in a three-dimensional high-stress state and so cavern excavation has the potential to lead

to rock mass unloading Importantly this results in changesin the mechanical properties of the rock thereby affectingthe stability and safety of the construction of the under-ground engineering As a result the evolution rules re-garding the creep mechanics behaviour of dam slopes underthe condition of unloading as well as research into theconstitutive model have important practical significance forthe long-term safety evaluation of high dam projects ulti-mately affecting the degree to which the normal operation ofhydropower stations can be guaranteed

Many experimental studies of rock creep characteristicshave been conducted by researchers both at home andabroad with themain reported result being that rock creep ischaracterised by several stages 3ese are the followingfirstly the initial creep stage secondly the steady-state creepstage and finally the accelerated creep stage With the

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 3576181 12 pageshttpsdoiorg10115520203576181

continuous development of rock mechanic tests and re-search the literature has gradually found that unloadingrock mass mechanics is more in accordance with the actualstate of engineering mechanics [1ndash4] In view of this re-searchers have proposed unloading rockmechanics which isprimarily concerned with the actual mechanical conditionsof engineering Extending this research based on unloadingrock mass mechanics many scholars have conducted rockunloading experiments in combination with practical en-gineering projects Some scholars have obtained theunloading creep characteristics of rocks under differentconditions mainly through the tests of rock specimens orrock materials and analyzed the creep deformation rulescreep rate change rules stress change rules creep damagecharacteristic and long-term strength of rocks in detailaccording to the test results For example in the studyconducted by Sorace [5] the researcher implemented thelong-term creep test of specimens to investigate the creepproperties of rock 3is involved several cycles of unloadingand loading at the maximum stress value for various stagesof steady-state creep followed by the proposal of a simplemathematical model to describe the test result Bazhin andMurashkin [6] studied the phenomenon of loading andunloading creep deformation as well as the stress relaxationcharacteristics of micropore areas of the material under thecondition of hydrostatic pressure Based on the research itwas possible to establish the nonlinear rheological modelwhich describes the materialrsquos loading and unloading plasticdeformation and creep deformation In the notable studyconducted by Zhao [7] the researcher examined the creepproperties of intact limestone specimens and fracturedspecimens by applying the multistage loading and unloadingtriaxial creep test 3e results indicated that fractured rockspecimens are associated with a longer creep time a largertransient creep and a larger steady-state creep rate In theabovementioned works the researchers mainly focused onthe creep characteristic of the rock under loading andunloading conditions while the creep characteristics ofrocks under unloading conditions were further studied bylater authors For example Jiang et al [8] applied theuniaxial compression and triaxial unloading capacity testwhich showed that unloading tests produce greater ex-pansion under identical deviatoric stress Additionally Liuet al [9] considered the excavation involved in unloadingrock destruction and this was followed by an investigationof marble unloading creep properties by implementingtriaxial unloading creep tests 3e findings indicated that therock unloading creep rate and creep deformation increasewith an increase in axial stress while they decrease with anincrease in confining pressure Deng et al [10] examined thegrading unloading rheological test in the context of sandymudstone and this was paired with the investigation of theunloading creep deformation characteristics and damagecharacteristics of mudstone under constant axial pressureunloading confining pressure and added axial pressureunloading confining pressure for two different paths Inaddition some researchers were dedicated to studying thebehavior of creep damage of rock under unloading condi-tions from microscopics for instance Wang et al [7]

conducted ample experiments to show the effect of initialcreep damage on unloading failure of rock from macro-mesoscopic perspective and verified the agreement betweenthe mesoscopic pore parameter and creep damage variable

Finally rheological models represent fundamental the-oretical approaches that can be mobilised when analysingrock mass rheological mechanical properties Furthermorerheological modelling is a crucial tool that can be appliedwhen attempting to examine the implications of rock rhe-ological mechanics theory In recent years research in thedomain of rock rheological model has advanced consider-ably and the experience rheological model the originalcombination model and the nonlinear rheological me-chanics model among others have been widely usedAmong them the measure of using experimental data in-version on known rheological model parameters accom-panied by the identification of unknownmodels has becomeone of the more commonly used research methods

However the current rock mass rheological model theoryis still a difficult issue in the study of rock mass mechanicsMany major rock mass engineering projects have highlightedserious problems which underlie the existing research initia-tives addressing rock mass rheological model theory Manyscholars use damage mechanics thermodynamics fracturemechanics energy transfer and the theory of diffusion formodel analysis and this has underpinned the acquisition ofrelatively satisfactory results For example based on the theoryof fractional calculus Zhou et al [12] constructed a viscosityelastic-plastic constitutive model to describe the creep propertycharacteristics of halite Additionally based on soft rock ar-gillaceous siltstone constant axial compression and gradedand unloading confining pressure test results Wang et al [13]constructed the nonlinear damage rheological model Nedjarand Roy [14] established the three-dimensional mechanicalmodel on the basis of viscoelastic continuum damage me-chanics which enabled accurate descriptions of the long-termcreep behaviour of gypsum rock 3is was followed by theconstruction of a constitutivemodel that can describe the entirecreep process of granite under different temperature condi-tions In another noteworthy research project Chen andKonietzky [15] established a discrete element method relyingon a numerical model based on heterogeneous particles thepurpose of which was to describe the three-phase agingproperties of rock creep Furthermore Lu et al [16] proposed adual-scale model to simulate the creep aging damage defor-mation and fracture behaviour of heterogeneous brittle rockWu et al [17] established an improved Maxwell creep modelbased on the theory of fractional calculus and it was successfulin providing amore effective description of the creep propertiesof rock masses In addition based on an examination of therelationship between the macro- and the micromechanicsdefinitions of rock creep damage Li and Shao [18] established amicromechanics-based model 3is was then used to study thecreep properties of rocks under variable stress paths Byadopting the complex deformation mechanism of the creepmodel the valuable research conducted by Firme et al [19]provided an accurate description of the sticky elastoplasticcreep characteristics of soft rock and then applied the finiteelement [11] numerical simulation to the creep test Recently

2 Advances in Materials Science and Engineering

Tang et al [20] proposed a four-element creep model based onvariable fractional derivatives and continuous damage me-chanics which was used to simulate the creep behavior of rockpiecewise Hu et al [21] carried out creep test to figure out thecreep rule and mechanical property of fractured phyllite andmodified Burgerrsquos model to analyze the creep characteristics offractured phyllite Wang et al [22] carried out the triaxialunloading creep tests of layered rock specimen of Jinping IIhydropower station and acquired the creep deformation curvesunder different confining pressures 3rough analysing thecreep parameters a nonlinear creep damage model of rockunder unloading was proposed to reflect the nonlinear char-acteristics of rock creep

From what have been discussed above some of the modelswere established by the combination of elastic elements plasticelements or elastoplastic elements some were established bymodifying the classical mechanical models and others wereestablished by introducing new parameters Additionally somemodels with few parameters and good applicability weremostlyapplicable to soft rocks despite the wealth of literatureaddressing related topics it is important to recognize that theavailable evidence pertaining to the unloading rheologicalstudy of hard rock masses is insufficient One of the mainreasons for this relates to the limitations of testing equipmentand testing conditions especially regarding the unloadingrheological test for hard rock and its constitutive model3erefore with respect to the hard rock unloading rheologicaltest and the theoretical research further areas for explorationare numerous In view of this the purpose of the presentresearch was to apply the hard rock unloading rheological testto the proposed Mengdigou hydropower station thusinforming the present state of research regarding unloadingrock mass rheological studies Here it is particularly necessaryto introduce the mengdigou hydropower station which locatesin Sichuan China It is a concrete hyperbolic arch dam withthe maximum dam height of 240m light-colored granite ismainly developed in the dam foundation and the mechanicalproperties of granite have influenced greatly under the action ofexcavation and unloading during the long construction periodthat will last 81 years

According to the unloading rheological properties ofhard rock this paper capitalised on the theory of fractionalcalculus and it constructed the constitutive model to de-scribe the rheological deformation characteristics of hardrock unloading Noteworthily the model was verified 3efact that the publication of this result as well as relatedresults in the literature has promoted and enriched a newway of thinking about hard rock unloading rheologicalmechanics research should not be overlooked Furthermoreit has provided experimental and theoretical support for thewater resources and hydropower dam project all the whileaiding in the long-term stability of other comparable en-gineering initiatives

2 Hard Rock Unloading Rheological Testand Analysis

21 Unloading Rheological Test In order to study theunloading rheological characteristics of hard rock the

granite of Mengdigou hydropower station was taken as themain research object According to the actual engineeringsituation the unloading rheological test programme wasdesigned as keeping the axial stress (σ1) unchanged re-moving the confining pressure step by step 3en theunloading rheological tests of granite were carried out Allthe granite samples used in the tests were taken from thedam foundation of Mengdigou hydropower station and therock samples for testing were cut into ϕ50mmtimes 100mmstandard cylinder specimens 3e rock automatic triaxialservo rheometer (RLW-1000 kN) was used in this test whichwas independently researched and developed by ShandongUniversity 3e purpose of the instrument is to conduct theunloading rheological experimental study on the granitefrom the Mengdigou hydropower station

Based on the conventional triaxial test of hard rock wefound that the peak intensity of hard rock is higher while thebearing capacity is larger With the engineering practice ofhigher stress on a high dam slope it is necessary whenmaking unloading rheological test plans to take into con-sideration the larger initial confining pressure and thehigher-level axial stress 3ese are pertinent when con-ducting the unloading rheological experiment on the basis ofthe axial stress σ1 which remains unchanged and step-by-step unloading confining pressure 3e initial axial stresslevel is generally 40ndash50 of the peak intensity Furthermoreowing to the high strength of hard rock the creep defor-mation for time dependence is stronger and so in order toobserve more obvious creep deformation unloading rheo-logical length was set at each level for 50ndash70 h Additionallythis enabled the study to satisfy the accuracy requirements ofunloading rheological levels and the unloading step was setat 5ndash8 For the purpose of guaranteeing the smoothness ofthe test the classification of unloading at a constant dis-charge rate was 100Ns 3e test path is shown in Figure 1while the test process is outlined in Figure 2 where σ1represents the axial stress σ3 represents the confiningpressure and (σ1ndashσ3) represents the axial deviatoric stress

3e conventional triaxial compression stress-straincurve of granite under different confining pressures is shownin Figure 3 3e cohesive force of granite was c 2130Mpawhile the internal friction angle was ϕ 5425deg

22UnloadingRheologicalTestAnalysis Based on the resultsof the unloading rheological test the rock unloading rhe-ological stress-strain curve was plotted Drawing on theconfining pressure of 40MPa as an example owing to thelimited length Figure 4 presents the unloading rheologicalstress-strain curve of the granite sample under the confiningpressure of 40MPa while Figures 5 and 6 illustrate themacroscopic and microscopic rock damage modes where ε1presents the axial strain of granite ε3 presents the lateralstrain of granite εv presents the volume strain of granite and(σ1ndashσ3) presents the deviatoric stress

By analysing the rock unloading creep curve charac-teristics of Figures 2ndash6 for the unloading rheological testdamage characteristics of the rock it is possible to draw thefollowing conclusions

Advances in Materials Science and Engineering 3

(1) Under various stress levels the unloading momentwill produce instantaneous elastic strain

(2) Rock flowing deformation exists when unloadingand there also exists a rheological threshold Whenthe stress level exceeds the rheological thresholdeven if the stress level remains the same aging de-formation will occur

(3) When the stress level is low rock creep strain overtime facilitates a slowdown in growth characterised

by slow creep properties On the contrary when thestress level is higher the rock creep strain shows asteady growth in speed over time and it is charac-terised by stable creep properties In addition whenthe stress level has reached the long-term strength ofthe rock rock creep strain will accelerate over timecharacterised by the accelerated creep property Fi-nally the rock will undergo dilatancy damage fa-cilitating a rapid increase in deformationcharacterised by brittle failure

(a) (b) (c)

(d) (e) (f )

Figure 2 Operation process for the tests (a) Specimen packaging (b) Specimen installation (c) Closed pressure chamber (d) Oil filled(e) Control system operation (f ) Specimen demolition

σ3

σ1ndashσ3

σ1

t

σ

Figure 1 Unloading path diagram


(4) 3emicroscopic damage forms of the rock show thatdamage occurred during the unloading rheologicalprocess

3e analysis of the characteristics of the hard rockunloading creep test curve indicates that the creep deformationof rock unloading involves instantaneous deformation slow

creep deformation steady-state creep deformation and ac-celeration creep deformation Each deformation stage presentsdistinctive characteristics When the stress level is lower thanlong-term strength hard rock displays typical viscoelasticcharacteristics however when the stress level is greater thanthe long-term strength and reaches the fracture stress level itdisplays typical sticky plastic characteristics

3 Modelling Problems That Must Be Solved

According to the unloading rheological test results theconstruction of a hard rock unloading creep model shouldinclude an elastic element viscous components and plasticcomponents 3is is because these components can describethe damage characteristics of hard rock unloading 3ereforethe following four problems need to be solved (1) reasonablydetermining the long-term strength of rock (2) identifyingdifferent degrees of damage at each unloading creep stage (3)ensuring that the steady-state creep stage unloading elementmodel reflects smooth deformation characteristics and (4)ensuring that the unloading accelerated creep phase com-ponent model clearly describes the unloading rheologicalcharacteristic of more apparent dilatancy

3e intention of this research is to draw on the followingmethods to solve these problems Firstly the creep rate in-tersection method is an effective measure that can be used toanalyze the long-term strength of hard rock unloading andalso to derive a reasonable long-term strength thresholdsecondly through the analysis of the characteristics of therock damage it will be appropriate to define damage variablesand establish the parameters damage through the evolutionequation and through the damage evolution of model pa-rameters at different stages of creep to reflect the damageeffect of rock during different unloading phases and finally tobuild the fractional unloading rheological element with thisprocess based on the change of the fractional-order deriva-tive-order time Regarding the last point this approach willnot only reflect the smooth deformation characteristics of theunloading steady-state creep stage but it will also ensuremoreapparent dilatancy for the unloading accelerated creep

31 Creep Rate Intersection Method Analysis regarding Long-Term Strength 3rough the analysis of the steady-statecreep properties it is possible to find that for sample in thesteady-state creep stage there must be a node in its axialcreep rate and the lateral creep rate curve which is thecritical point of rock creep damage 3is node correspondsto the strength which is the long-term strength of the rockAs outlined in Figure 7 this is referred to as the steady-statecreep rate intersection method

Based on the analysis of the test results from theunloading rheological test the rock sample became unstableafter volume expansion 3erefore the reliable values of thelong-term strength shall be at the volume strain compressionstage namely the εv gt 0 unloading creep stage In additionthe point of intersection for the unloading creep rate-stresscurve (CR-σ) becomes the first target of the ideal long-termstrength It should be noted that firstly the intersection

Figure 5 Macroscopic damage mode of granite

0

50

100

150

200

250

300

350

ndash12 ndash1 ndash08 ndash06 ndash04 ndash02 0 02 04 06 08 1

(σ1ndashσ 3

) MPa

ε10ndash4

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa

Lateral strain Axial strain

Figure 3 Triaxial compression stress (σ1ndashσ3)-strain (ε) curve forgranite under different confining pressures

0

10

20

30

40

50

60

70

80

90


(σ1ndashσ 3

) MPa

ε10ndash4

Axial strain ε1Lateral strain ε3

Volume strain εvUnloading starting point

ε1

εv

ε3

AB

C

D

O

E

Figure 4 Relationship between deviatoric stress (σ1ndashσ3) and strain (ε)


point meets the εv gt 0 unloading creep stage And secondlythe analysis shows that when stress level exceeds this stressvalue the rock enters an expanding and inflating stage3erefore it is usually reasonable to take this point as thelong-term strength threshold below which the rock canmaintain its long-term stability In the event that the stresslevel exceeds the critical value the rock will transition fromthe steady creep stage to the accelerated creep stage anddamage will be sustained rapidly

32 Analysis of Fracture Characteristics 3e ultrasonic flawdetection (UFD) test is regarded as one of the most effectiveways to study the evolution of internal structural rockdamage It relies on the longitudinal wave velocity of therock rheological ultrasonic test In this context it is possibleto define the damage variable [23] in the following way

D 1 minus1

1 + εv

vp

v0 (1)

where εv represents the volumetric strain vp represents thelongitudinal wave velocity and v0 represents the initial wavevelocity

Equation (1) can be further simplified to yield thefollowing

D 1 minus eminus αt

(2)

where α is the related coefficient of longitudinal wave ve-locity the initial wave velocity and the volumetric strain ofrock material Here we call it as damage parameter

Hence the damage evolution equation can be formu-lated as follows

ηβprime1 (t) (1 minus D)ηβ1 eminus αtηβ1

ηcprime1 (t) (1 minus D)ηc

1 eminus αtηc1

E0prime(t) (1 minus D)E0 eminus αtE0


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

33 Unloading Damage Rheology Element AnalysisGenerally complex viscoelastic materials represent themiddle ground between the ideal elastic and viscous ma-terials3is stems from the fact that they are characterised bymemory characteristics regarding stress and change ofstrain Noteworthily Hookersquos law of the elastic solid andNewtonrsquos law of the viscous fluid cannot accurately describethe characteristics In view of this the fractional-orderderivative can be used to describe the complicated physicaland mechanical process greatly overcoming the short-comings of ineffective inosculation between the theoreticaland experimental results [23ndash25]

34 8e Fractional-Order Viscous Body Hookersquos law indi-cates that the constitutive relation for ideal elastic solid isσ(t) sim d0ε(t)dt0 while Newtonrsquos law indicates that theconstitutive relation for the ideally viscous fluid isσ(t) sim d1ε(t)dt1 In view of this there is sufficient reason tothink that the constitutive relation σ(t) sim dβε(t)dtβ(0leβle 1) coincides with the mechanical property of the visco-elastic body between ideal elastic solids and a fluid with idealviscosity β takes a boundary value to describe the two viscousand elastic limit states and when β 1 the element repre-sents an ideal fluid however when β 0 the element rep-resents the ideal solid In addition when 0lt βlt 1 the elementrepresents some state of the object between the ideal fluid andsolid state which is referred to as a fractional viscous body3is is indicated in Figure 8

Axial

Lateral

Cree

p ra

te

σσ00

Figure 7 Schematic diagram of ldquointersection point of creep ratecurve methodrdquo

Figure 6 Microscopic damage mode of granite


3e constitutive relation for the fractional-order viscousbody components is as follows

σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)

where η is the fractional order viscous coefficient thephysical dimension of which is [stressmiddottime β] and β is thefractional-order derivative times of the fractional-orderviscous body

When the stress remains unchanged (namely whenσ(t) const) the fractional-order viscous body componentswill describe the rheological behaviour of material creepBased on RiemannndashLiouville fractional calculus operatortheory addressing the fractional integrals on the left- andright-hand sides of (4) can yield the creep equation offractional viscous body in the following way

ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)

Noteworthily when the strain remains the same(namely when ε(t) const) the fractional viscous body willdescribe the rheological behaviour of the materialrsquos stressrelaxation 3us drawing on (5) it is possible to derive thefollowing viscous stress relaxation equation

σ(t) ηεtminus β

Γ(1 minus β) 0lt βlt 1 (6)

As outlined in the above equation the fractional viscousbody for different rockmaterial properties by addressing thechange in the fractional-order derivative-order β value caneffectively reflect the nonlinear gradients of the materialcreep process and the gradient material stress relaxationprocess

4 Modelling and Solving the UnloadingNonlinear Rheological Constitutive ModelBased on the Fractional-Order Derivative

To solve the four key problems associated with constructingan unloading rheological model it is useful to draw on thefractional-order derivative hard rock unloading nonlinearrheological model (FOD-HKVP) as shown in Figure 9Based on the classical Nishihara model the key im-provements that must be made are as follows firstly thewhole process of the unloading rheological model mustconsider the damage of the rock material degradation ef-fect secondly the new long-term strength analysis methodshould be utilised which is more in line with the hard rockmass unloading rheological phenomenon and finally thetheory of fractional-order calculus should be introduced

Regarding the final point this is noteworthy because theintroduction of fractional-order calculus enables us tomore effectively reflect the nonlinear creep characteristicsof hard rock In particular this could reflect the dilation ofthe accelerating creep phase as well as the characteristics ofbrittle failure

41 Stress-Strain Relationship for the Injury Elastomer (H)According to Hookersquos law the creep equation for the injuryelastomer is as follows

εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)

where E0 is the elasticity modulus of H

42 Fractional Damage Viscoelastic Body (KV) Stress-StrainRelationship

σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee

minus αtεve(t) (8)

Namelydβ εve(t)1113858 1113859

dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)

Using the fractional-order differential equation to solve(9) will yield an expression for the creep equation of thefractional-order damage viscoelastic body as follows

εve(t) σηβ1

eαt

1113946t

0e

minus αττβminus 11113944

infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)

where ηβ1 is the viscosity coefficient of fractional damage for aviscoelastic body and where Eββ[(minus (E1η

β1))τβ]

1113936infink0((E1η

β1)kτβkΓ(kβ + β)) refers to the two-parameter

MittagndashLeffler function

43 8e Stress and Strain Relation of the Fractional DamageViscoplasticity Body (P)

σ ηc2e

minus αtdc εvp(t)1113960 1113961

dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)

Using (2)ndash(4) the Laplace transform and inversetransformation can be performed Consequently the creepequation of the viscous plastic body can be obtained

εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k

Γ(k + 1 + c) σ ge σs

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)

According to the series-parallel property of the modelelements the total strain of the FOD-HKVP model can beexpressed in the following way

η β

Figure 8 3e fractional-order derivative (FOD) viscous bodymodel


ε(t) εe(t) + εve(t) + εvp(t) (14) By substituting (3) (7) and (10) into (14) it is possible toderive the constitutive equation of the nonlinear rheologicalmodel (FOD-HKVP model) based on the fractional order

ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ σ lt σs

ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

It is not difficult to see that when α 0 β 1 and c 1the model degenerates as follows

ε(t) σE0

+σE1

1 minus eminus E1η1( )t

1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Noteworthily (16) is entirely consistent with the classicalNishihara model 3erefore it is clear that the classicalNishihara model is a particular instance of the FOD-HKVPmodel when α 0 β 1 and c 1

5 Model Verification

It should first be noted that the parameters E0 E1 ηβ1 ηc

2 and

α are contained in the hard rock unloading nonlinearrheological model Equation (15) namely the creep equa-tion of FOD-HKVP model was used to preliminarily fit thecreep test curve of hard rock unloading and during theattenuation creep and steady-state creep stages α is ap-proximately 000001 hminus 1 Contrastingly during the accel-erating creep stage α amounts to around 000005 hminus 1 3isphenomenon is consistent with the experimental conclusionthat the degree of deterioration in the accelerated creep stageis greater than that of attenuation creep and steady-statecreep 3erefore in order to simplify the calculation α takesthe value of 000001 hminus 1 when the FOD-HKVP model is

distinguishing (note that the stress level is lower than thelong-term strength) while it alternatively takes the value of000005 hminus 1 (note that the stress level is higher than the long-term strength) Furthermore the parameters E0 E1 ηβ1 η

c

2

β and c can be identified using the objective functionmethod [13]

To verify the validity of the unloading rheological model(FOD-HKVP) it is useful to make use of the compiledcalculation procedure to carry out the finite element analysisand then compare with the experimental data

3e numerical model size is ϕ50mmtimes 100mm standardcylinder specimens a total of 8000 units and 8421 nodes3emodel is constrained in the Z-direction at the bottom asshown in Figure 10

According to the results of unloading rheological testthe creep rate node method was used to determine the long-term strength of rock which is 626MPa And the damageparameter α 000001 hminus 1 when σlt626MPa whileα 000005 hminus 1 when ge626MPa and Poissonrsquos ratio wasassumed to be a fixed value which is 02 3e axial creepparameter σ identification results of the granite under theinitial unloading confining pressure of 40MPa are shown inTable 1 and the model theory curve and test curve areoutlined in Figure 11

Table 1 shows the model parameter identification re-sults and under identical unloading confiding pressureconditions with varying stress levels the modelrsquos param-eters are different Furthermore each level of stress ischaracterised by a distinct set of parameter values In viewof this we now encounter a form of nonlinear rheology

FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)

η1β (1 ndash D) β η2

γ (1 ndash D) γ

σσ

Figure 9 Nonlinear unloading rheological constitutive model based on the fractional-order derivative


Analysis of the axial rheological parameter change ruleindicates the following

(1) When the deviatoric stress amounts to 60MPaparameter volatility is significant 3e reason for thisis because partial damage occurred within the rockthereby meaning that it does not arise in the cor-relation analysis In addition to this with an increasein the stress level the elasticity variable E(t) di-minishes gradually thus fully demonstrating thedeterioration effect of hard rock in the unloadingcreep process

(2) 3e viscous variables ηβ1and ηc

2 in two viscous

bodies decrease gradually in line with rock attenu-ation creep steady-state creep and accelerated creep3is is indicative of the fact that the viscous variableis characterised by its deterioration effect in thecontext of the unloading rheological process

Table 1 Model parameter identification results

Deviatoric stress (MPa) E0 (MPa) E1 (MPa) ηβ1 (MPamiddoth) β ηc2 (MPamiddoth) c R2

50 36011 114359 554376 0525 mdash mdash 0997060 38383 34874 250921 0150 mdash mdash 0997370 35913 227796 4848376 0605 1056001 0598 0997375 33651 198630 2221598 0566 865258 0610 09979775 30245 67181 782149 0445 462178 0470 09996

100mm

50mm

Figure 10 Mesh sketch of calculation model

01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)

Test value (the axial)FOD-HKVP model

σ1ndashσ3 =50MPa

σ1ndashσ3 = 60MPaσ1ndashσ3 = 70MPa

σ1ndashσ3 = 75MPa

σ1ndashσ3 = 775MPa

σ1ndashσ3 = 80MPa

Figure 11 Experimental data and the fitting curves for the FOD-HKVP model


013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4

Test valueFitted value of FOD-HKVP modelFitted value of Nishihara model

(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02

021σ1ndashσ3 = 70MPa

0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)

Figure 12 Comparison diagram of fitting results with FOD-HKVP model and the classical Nishihara model (a) σ1ndashσ3 50MPa(b) σ1ndashσ3 60MPa (c) σ1ndashσ3 70MPa (d) σ1ndashσ3 75MPa (e) σ1ndashσ3 775MPa


(3) 3e fractional derivatives β and c in two viscousbodies are different In particular when the stresslevel exceeds the long-term strength the fractionalderivative decreases with an increase in the stresslevel and in the context of the steady-state creepstage β is greater than c However following thetransition from the steady-state creep stage into theaccelerated creep stage c becomes greater than β Intheory this is because the two types of viscous bodiesdescribe different stages of the creep process and theclassification of viscous bodies in the fractional-or-der viscous plastic bodies accounts for theaccelerated creep stage As such its derivative orderc is greater than the derivative order of the viscouscoefficient for the viscoelastic body β

Using the parameter identification results for FOD-HKVP model verification Figure 11 compares the theo-retical curve against the test curve Based on the modelvalidation results given in the figure it is not difficult to seethat the theoretical curve closely aligns with the test curvethus showing that the FOD-HKVP model reflects theunloading creep property of each phase in terms of the hardrock unloading rheological characteristics As such theFOD-HKVP model test and theoretical analysis are a closematch

6 Comparison with the Nishihara Model

We know from the previous analysis that the classicalNishihara model is a particular instance of the FOD-HKVPmodel So in order to further verify the applicability of FOD-HKVP model typical test values of each creep stage of rocksamples were selected and then fitted with FOD-HKVPmodel and the classical Nishihara model respectively andthe fitting results of the two models were compared andanalyzed Figure 12 shows the fitted results

As it can be seen from the fitting curve in Figure 12 thefitting result of the Nishiharamodel deviates greatly from thetest value while the FOD-HKVPmodel can better match thetest data and can reflect the whole unloading creep process ofgranite Granite belongs to hard rock material at the initialconfining pressure (40MPa) unloading conditions when thestress level exceeds its long-term strength value (626MPa)the plastic deformation effects significantly particularlyunder the higher horizontal stress the granite rock turnsinto the viscoplastic deformation Compared with the statusof viscoelastic deformation with lower stress the Nishiharamodel can better describe the relaxation properties of rockwhich is reflected in Figure 12 that is when the stress level ismore than 70MPa the Nishihara model presents a betterfitting status However compared with the FOD-HKVPmodel the fitting degree is still defective It can be seen fromthe comparison between the fitting value and the experi-mental value of the two models that the FOD-HKVP modeloccupies the advantage 3is is because the fractional vis-cosity coefficient introduced in the FOD-HKVP model canbetter reflect the viscoelastic deformation at low stress leveland the viscoplastic deformation at high stress level

7 Conclusion

3e purpose of this research has been to study the unloadingrheological properties of hard rock and draw on the theoryof fractional-order calculus as well as damage theory toconstruct a nonlinear viscous elastic and plastic damagerheological model which is called FOD-HKVP modelherein In addition a comparative analysis was made be-tween the FOD-HKVP model and the classical Nishiharamodel 3e conclusions are as follows

(1) Relying on the background project of mengjigouhydropower station the unloading rheological test ofthe granite under confining pressure of 40MPa wascompleted according to the test programme

(2) Based on the unloading rheological damage char-acteristics of granite and the analysis of rheologicalstress-strain curve the macroscopic and microscopicfailure modes the research sought to illuminate theentire unloading creep process of hard rock

(3) 3e FOD-HKVP model was built by using the curvevariation characteristics of unloading creep test ofgranite and experimental verification proceeded onthe basis of the unloading rheological model It isworth acknowledging that the model identificationresult is reasonable and in particular it is consistentwith the unloading creep rule of granite and theresults indicated a close match between the FOD-HKVP model test and theoretical analysis

(4) 3e FOD-HKVP model was compared with theNishihara model the results indicated that theFOD-HKVP model can effectively describe thethree unloading creep stages for granite When thestress level is lower below the long-term strengththe Nishihara model cannot simulate the visco-elastic deformation of granite while the FOD-HKVP model presents a perfect fitting curve Whenthe stress level is higher above the long-termstrength the Nishihara model fitting has beenimproved to reflect the viscoplastic creep charac-teristics of granite but the accuracy is not goodwhile the FOD-HKVP model still presents a perfectfitting curve 3at is to say the FOD-HKVP modelhas obvious advantages in the whole creep processwhich relate primarily to the timely adjustment ofthe viscous coefficient and its fractional derivativeorder as well as its description of the nonlinearchange process that extends from the steady-statecreep stage to the accelerated creep stage Note-worthily the model provides an accurate reflectionof unloading creep damage characteristics for thehard rock at various stages

Data Availability

Some of the experimental test data used to support thefindings of this study are included within the article 3eother data used to support the findings of this study areavailable from the corresponding author upon request


Conflicts of Interest

3e authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

3is work was supported by the National Natural ScienceFoundation of China (no 51809156) and the Key Laboratoryof Large Structure Health Monitoring and Control Shi-jiazhuang (no KLLSHMC1914)

References

[1] Y Wu and L Jiding ldquoUnloading properties of marblerdquo Rockand Soil Mechanics vol 5 no 1 pp 30ndash36 1984

[2] T Li and W Lansheng ldquoAn experimental study on the de-formation and failure features of a basalt under unloadingconditionrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 12 no 4 pp 321ndash327 1993

[3] J Shen L Wang and Q Wang ldquoDeformation and fracturefeatures of unloaded rock massrdquo Chinese Journal of RockMechanics and Engineering vol 22 no 12 pp 2028ndash20312003

[4] J S O Lau and N A Chandler ldquoInnovative laboratorytestingrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1427ndash1445 2004

[5] S Sorace ldquoEffects of initial creep conditions and temporaryunloading on the long-term response of stonesrdquo Materialsand Structures vol 31 no 212 pp 555ndash562 1998

[6] A A Bazhin and E V Murashkin ldquoCreep and stress re-laxation in the vicinity of a micropore under the conditions ofhydrostatic loading and unloadingrdquo Doklady Physics vol 57no 8 pp 294ndash296 2012

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

[8] D-Y Jiang J-Y Fan and J Chen ldquoStudy of dilatancycharacteristics of salt rock under unloading action of con-fining pressurerdquo Rock and Soil Mechanics vol 34 no 7pp 1881ndash1886 2013

[9] J Liu L Wang and J-L Pei ldquoExperimental study on creepdeformation and long-term strength of unloading-fracturedmarblerdquo European Journal of Environmental and Civil En-gineering vol 19 pp 97ndash107 2015

[10] H-F DengM-L Zhou and L I Jian ldquoExperimental researchon unloading triaxial rheological mechanical properties ofsandy mudstonerdquo Rock and Soil Mechanics vol 37 no 2pp 315ndash322 2016

[11] YWang Q Qiao and J Li ldquo3e effect of initial creep damageon unloading failure properties of sandstone from macro-mesoscopic perspectiverdquo Periodica Polytechnica-Civil Engi-neering vol 63 no 4 pp 1004ndash1015 2016

[12] 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

[13] Y Wang J Li and H Deng ldquoInvestigation on unloadingtriaxial rheological mechanical properties of soft rock and itsconstitutive modelrdquo Rock and Soil Mechanics vol 33 no 1pp 3338ndash3344 2012

[14] B Nedjar and R L Roy ldquoAn approach to the modeling ofviscoelastic damage Application to the long-term creep of

gypsum rock materialsrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 37 no 9pp 1066ndash1078 2013

[15] W Chen and H Konietzky ldquoSimulation of heterogeneitycreep damage and lifetime for loaded brittle rocksrdquo Tecto-nophysics vol 633 pp 164ndash175 2014

[16] Y Lu D Elsworth and L Wang ldquoA dual-scale approach tomodel time-dependent deformation creep and fracturing ofbrittle rocksrdquo Computers and Geotechnics vol 60 pp 61ndash762014

[17] F Wu J F Liu and J Wang ldquoAn improved Maxwell creepmodel for rock based on variable-order fractional derivativesrdquoEnvironmental Earth Sciences vol 73 no 11 pp 6965ndash69712015

[18] X Li and Z Shao ldquoInvestigation of macroscopic brittle creepfailure caused by microcrack growth under step loading andunloading in rocksrdquo Rock Mechanics and Rock Engineeringvol 49 no 7 pp 2581ndash2593 2016

[19] P A L P Firme D Roehl and C Romanel ldquoAn assessmentof the creep behaviour of Brazilian salt rocks using the multi-mechanism deformation modelrdquo Acta Geotechnica vol 11no 6 pp 1445ndash1463 2016

[20] H Tang D Wang R Huang X Pei and W Chen ldquoA newrock creep model based on variable-order fractional deriva-tives and continuum damage mechanicsrdquo Bulletin of Engi-neering Geology and the Environment vol 77 no 1pp 375ndash383 2018

[21] 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 and Geological Engineeringvol 36 pp 827ndash834 2018

[22] R Wang Y Jiang C Yang F Huang and C Wang ldquoAnonlinear creep damage model of layered rock underunloading conditionrdquoMathematical Problems in Engineeringvol 2018 Article ID 8294390 8 pages 2018

[23] K Balasubramaniam J S Valluri and R V Prakash ldquoCreepdamage characterization using a low amplitude nonlinearultrasonic techniquerdquo Materials Characterization vol 62no 3 pp 275ndash286 2011

[24] Z Hou ldquoMechanical and hydraulic behavior of rock salt in theexcavation disturbed zone around underground facilitiesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 40 no 5 pp 725ndash738 2003

[25] Z He Z Zhu and M Zhu ldquoAn unsteady creep constitutivemodel based on fractional order derivativesrdquo Rock and SoilMechanics vol 37 no 3 pp 737ndash744 2016


continuous development of rock mechanic tests and re-search the literature has gradually found that unloadingrock mass mechanics is more in accordance with the actualstate of engineering mechanics [1ndash4] In view of this re-searchers have proposed unloading rockmechanics which isprimarily concerned with the actual mechanical conditionsof engineering Extending this research based on unloadingrock mass mechanics many scholars have conducted rockunloading experiments in combination with practical en-gineering projects Some scholars have obtained theunloading creep characteristics of rocks under differentconditions mainly through the tests of rock specimens orrock materials and analyzed the creep deformation rulescreep rate change rules stress change rules creep damagecharacteristic and long-term strength of rocks in detailaccording to the test results For example in the studyconducted by Sorace [5] the researcher implemented thelong-term creep test of specimens to investigate the creepproperties of rock 3is involved several cycles of unloadingand loading at the maximum stress value for various stagesof steady-state creep followed by the proposal of a simplemathematical model to describe the test result Bazhin andMurashkin [6] studied the phenomenon of loading andunloading creep deformation as well as the stress relaxationcharacteristics of micropore areas of the material under thecondition of hydrostatic pressure Based on the research itwas possible to establish the nonlinear rheological modelwhich describes the materialrsquos loading and unloading plasticdeformation and creep deformation In the notable studyconducted by Zhao [7] the researcher examined the creepproperties of intact limestone specimens and fracturedspecimens by applying the multistage loading and unloadingtriaxial creep test 3e results indicated that fractured rockspecimens are associated with a longer creep time a largertransient creep and a larger steady-state creep rate In theabovementioned works the researchers mainly focused onthe creep characteristic of the rock under loading andunloading conditions while the creep characteristics ofrocks under unloading conditions were further studied bylater authors For example Jiang et al [8] applied theuniaxial compression and triaxial unloading capacity testwhich showed that unloading tests produce greater ex-pansion under identical deviatoric stress Additionally Liuet al [9] considered the excavation involved in unloadingrock destruction and this was followed by an investigationof marble unloading creep properties by implementingtriaxial unloading creep tests 3e findings indicated that therock unloading creep rate and creep deformation increasewith an increase in axial stress while they decrease with anincrease in confining pressure Deng et al [10] examined thegrading unloading rheological test in the context of sandymudstone and this was paired with the investigation of theunloading creep deformation characteristics and damagecharacteristics of mudstone under constant axial pressureunloading confining pressure and added axial pressureunloading confining pressure for two different paths Inaddition some researchers were dedicated to studying thebehavior of creep damage of rock under unloading condi-tions from microscopics for instance Wang et al [7]

conducted ample experiments to show the effect of initialcreep damage on unloading failure of rock from macro-mesoscopic perspective and verified the agreement betweenthe mesoscopic pore parameter and creep damage variable

Finally rheological models represent fundamental the-oretical approaches that can be mobilised when analysingrock mass rheological mechanical properties Furthermorerheological modelling is a crucial tool that can be appliedwhen attempting to examine the implications of rock rhe-ological mechanics theory In recent years research in thedomain of rock rheological model has advanced consider-ably and the experience rheological model the originalcombination model and the nonlinear rheological me-chanics model among others have been widely usedAmong them the measure of using experimental data in-version on known rheological model parameters accom-panied by the identification of unknownmodels has becomeone of the more commonly used research methods

However the current rock mass rheological model theoryis still a difficult issue in the study of rock mass mechanicsMany major rock mass engineering projects have highlightedserious problems which underlie the existing research initia-tives addressing rock mass rheological model theory Manyscholars use damage mechanics thermodynamics fracturemechanics energy transfer and the theory of diffusion formodel analysis and this has underpinned the acquisition ofrelatively satisfactory results For example based on the theoryof fractional calculus Zhou et al [12] constructed a viscosityelastic-plastic constitutive model to describe the creep propertycharacteristics of halite Additionally based on soft rock ar-gillaceous siltstone constant axial compression and gradedand unloading confining pressure test results Wang et al [13]constructed the nonlinear damage rheological model Nedjarand Roy [14] established the three-dimensional mechanicalmodel on the basis of viscoelastic continuum damage me-chanics which enabled accurate descriptions of the long-termcreep behaviour of gypsum rock 3is was followed by theconstruction of a constitutivemodel that can describe the entirecreep process of granite under different temperature condi-tions In another noteworthy research project Chen andKonietzky [15] established a discrete element method relyingon a numerical model based on heterogeneous particles thepurpose of which was to describe the three-phase agingproperties of rock creep Furthermore Lu et al [16] proposed adual-scale model to simulate the creep aging damage defor-mation and fracture behaviour of heterogeneous brittle rockWu et al [17] established an improved Maxwell creep modelbased on the theory of fractional calculus and it was successfulin providing amore effective description of the creep propertiesof rock masses In addition based on an examination of therelationship between the macro- and the micromechanicsdefinitions of rock creep damage Li and Shao [18] established amicromechanics-based model 3is was then used to study thecreep properties of rocks under variable stress paths Byadopting the complex deformation mechanism of the creepmodel the valuable research conducted by Firme et al [19]provided an accurate description of the sticky elastoplasticcreep characteristics of soft rock and then applied the finiteelement [11] numerical simulation to the creep test Recently

















(a) (b) (c)

(d) (e) (f )


σ3

σ1ndashσ3

σ1

t

σ












0

50

100

150

200

250

300

350


(σ1ndashσ 3

) MPa

ε10ndash4

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa



0

10

20

30

40

50

60

70

80

90


(σ1ndashσ 3

) MPa

ε10ndash4



ε1

εv

ε3

AB

C

D

O

E





D 1 minus1

1 + εv

vp

v0 (1)




(2)





1 eminus αtηc1



⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)



Axial

Lateral

Cree

p ra

te

σσ00





σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)



ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)


σ(t) ηεtminus β







εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)



σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee



dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)


εve(t) σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)


β1))τβ]





σ ηc2e


dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)


εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)


η β




ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References











































(a) (b) (c)

(d) (e) (f )


σ3

σ1ndashσ3

σ1

t

σ












0

50

100

150

200

250

300

350


(σ1ndashσ 3

) MPa

ε10ndash4

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa



0

10

20

30

40

50

60

70

80

90


(σ1ndashσ 3

) MPa

ε10ndash4



ε1

εv

ε3

AB

C

D

O

E





D 1 minus1

1 + εv

vp

v0 (1)




(2)





1 eminus αtηc1



⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)



Axial

Lateral

Cree

p ra

te

σσ00





σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)



ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)


σ(t) ηεtminus β







εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)



σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee



dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)


εve(t) σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)


β1))τβ]





σ ηc2e


dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)


εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)


η β




ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References





































0

50

100

150

200

250

300

350


(σ1ndashσ 3

) MPa

ε10ndash4

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa



0

10

20

30

40

50

60

70

80

90


(σ1ndashσ 3

) MPa

ε10ndash4



ε1

εv

ε3

AB

C

D

O

E





D 1 minus1

1 + εv

vp

v0 (1)




(2)





1 eminus αtηc1



⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)



Axial

Lateral

Cree

p ra

te

σσ00





σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)



ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)


σ(t) ηεtminus β







εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)



σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee



dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)


εve(t) σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)


β1))τβ]





σ ηc2e


dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)


εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)


η β




ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References






























D 1 minus1

1 + εv

vp

v0 (1)




(2)





1 eminus αtηc1



⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)



Axial

Lateral

Cree

p ra

te

σσ00





σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)



ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)


σ(t) ηεtminus β







εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)



σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee



dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)


εve(t) σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)


β1))τβ]





σ ηc2e


dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)


εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)


η β




ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References





























σ(t) ηdβε(t)

dtβ 0lt βlt 1 (4)



ε(t) ση

tβ

Γ(1 + β) 0lt βlt 1 (5)


σ(t) ηεtminus β







εe(t) σ

E0(1 minus D)

σE0e

minus αt (7)



σ ηβ1eminus αtd

β εve(t)1113858 1113859

dtβ+ Ee



dtβ

σηβ1

eαt

minusE

ηβ1εve(t) (9)


εve(t) σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ (10)


β1))τβ]





σ ηc2e


dtc+ σs

(11)

Namely

dc εvp(t)1113960 1113961

dtcσ minus σs

ηc2

eαt

(12)


εvp

0 σ lt σs

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(13)


η β




ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References





























ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk


ε(t) σ

E0eminus αt

+σηβ1

eαt

1113946t

0e


infin

k0

E1ηβ11113872 1113873

kτβk

Γ(kβ + β)dτ +

σ minus σs

ηc2

tc

1113944

infin

k0

(αt)k


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


ε(t) σE0

+σE1


1113874 1113875 σ lt σs

ε(t) σE0

+σE1


1113874 1113875 +σ minus σs

η2t σ ge σs

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)




2 and



c

2






FOD-H FOD-KV FOD-P

σs

εeεve εvp

E1 (1 ndash D)

E0 (1 ndash D)


γ (1 ndash D) γ

σσ






2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References































2 in two viscous





100mm

50mm


01

015

02

025

03

035

04

0 50 100 150 200 250 300 400350

ε10

ndash4

Time (h)


σ1ndashσ3 =50MPa


σ1ndashσ3 = 75MPa


σ1ndashσ3 = 80MPa



013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References




























013

014

015

0 20 40 60 80 100Time (h)

σ1ndashσ3 = 50MPa

ε10

ndash4


(a)

015

016

017

018

0 20 40 60 80

ε10

ndash4

Time (h)

σ1ndashσ3 = 60MPa


(b)

ε10

ndash4 02


0190 20 40 60 80

Time (h)


(c)

ε10

ndash4

022

023

025

024

0 20 40 60 80Time (h)


σ1ndashσ3 = 75MPa

(d)

ε10

ndash4

026

027

028

029

0 20 40 60 80Time (h)



(e)








7 Conclusion






Data Availability





Acknowledgments


References

































7 Conclusion






Data Availability





Acknowledgments


References






























Acknowledgments


References




























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