review article investigation progresses and applications...

Review ArticleInvestigation Progresses and Applications of FractionalDerivative Model in Geotechnical Engineering

Jinxing Lai1 Sheng Mao1 Junling Qiu1 Haobo Fan1 Qian Zhang2

Zhinan Hu2 and Jianxun Chen1

1School of Highway Changrsquoan University Xirsquoan 710064 China2School of Civil Engineering Shijiazhuang Tiedao University Shijiazhuang 050043 China

Correspondence should be addressed to Junling Qiu 870133597qqcom and Jianxun Chen chenjx1969chdeducn

Received 1 February 2016 Accepted 7 April 2016

Academic Editor Jose A T Machado

Copyright copy 2016 Jinxing Lai et al This 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

Over the past couple of decades as a new mathematical tool for addressing a number of tough problems fractional calculushas been gaining a continually increasing interest in diverse scientific fields including geotechnical engineering due primarilyto geotechnical rheology phenomenon Unlike the classical constitutive models in which simulation analysis gradually fails to meetthe reasonable accuracy of requirement the fractional derivative models have shown the merits of hereditary phenomena withlong memory Additionally it is traced that the fractional derivative model is one of the most effective and accurate approaches todescribe the rheology phenomenon In relation to this an overview aimed first at model structure and parameter determinationin combination with application cases based on fractional calculus was provided Furthermore this review paper shed light onthe practical application aspects of deformation analysis of circular tunnel rheological settlement of subgrade and relevant loessresearches subjected to the achievements acquired in geotechnical engineering Finally concluding remarks and important futureinvestigation directions were pointed out

1 Introduction

In practical engineering massive unstable failure is closelyrelated to the rheological characteristic of rock and soilRecently owing to guarantee of the long-term safety andstability of geotechnical engineering an increasing attentionhas been paid by relevant scholars to the investigations onthe rheology whose core content is constitutive model [12] Generally the classical constitutive model for rock andsoil can be classified into three distinct approaches namelyempirical and semiempirical models rheology theory basedmodels and visco-elastic-plastic component models Theempirical and semiempirical models which are based onexperiments were endowed with a small number of parame-ters On the other hand the rheology theory based modelswhich focus on the mechanical response thus reveal theproperties of rheology at the microscale The visco-elastic-plastic component model as a common approach which iscomposed of standard elements such as the Hooke springand the Newtonian dashpot has the advantage of flexible

description of different rheology deformation with visualconcept and clear physical significance [3ndash6] Neverthelessthe empirical and semiempirical models are restricted bythe duration of experiment whose results can only describethe stable stage of creep Meanwhile the visco-elastic-plasticcomponent models require more elements to establish con-stitutive equations so as to meet precision requirement [7]causing the problems of many parameters complicated con-stitutive relation and difficulty in determination of param-eters resulting in inconvenience for actual engineering Onthe whole because of its natural defects the classical modelshave difficulty in characterizing accurately the whole processcurve of creep of rock and soil Specifically at the primarystage of creep its theoretical values deviate much fromthe experimental results especially if the accelerated stageappears subsequently [8 9]

As a result more and more investigators employ frac-tional calculus to the establishment of constitutive modelswhen obtaining a model with a small number of param-eters and accurate description is the target point [10ndash12]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9183296 15 pageshttpdxdoiorg10115520169183296

2 Mathematical Problems in Engineering

In fact during last decades the field of fractional calcu-lus has attracted increasing interest of researchers in verydiverse scientific fields including mathematics chemistryengineering technology anomalous diffusion and evenmed-ical machinery and robot control [13ndash19] Indeed fractionalcalculus which is defined in the form of calculus with conciseequation has been found to be an accurate mathematicaltool for solving the difficulty during physical and mechanicalmodeling As for the rheology which is featured by timedependency the fractional derivative model can be a perfectapproach for describing rheology phenomenon in particularto the properties of hereditary phenomenawith longmemory[20 21] Over the years after devoting many continuousefforts toward the study on the application of fractionalcalculus by numerous researchers the fractional derivativemodel has gained much deep development Specific to theparameters which are the key point in the process ofapplication and to determine the feasibility of model it canbe obtained through inversion based on experiment datanumerical calculation and software fitting As a distinctapproach fractional derivative model is applied to describethe nonlinear viscoelastic property deformation propertyshear contraction and dilatation and damage growth duringaccelerated stage of creep in geotechnical engineering [9 2223] For more details on the creep phenomena fractionalderivative model is regarded as an effective approach thatcan not only well reflect the process of nonlinear gradualchange at primary stage but also present accurately at steadystage and accelerated stage under high stress However due toits pseudodifferential operator the storage memory propertyof fractional derivative model brings certain difficulty tonumerical calculation despite the fact that the rheologicalprocess curve is beautifully presented [24]

In this paper we systematically investigated currentresearch in the field of fractional derivative model structureand acquisition of parameters where the resulting modelshad been incorporated into creep analysis We began withan introduction to relevant concepts and developments offractional calculus providing an up-to-date review of keyconstitutive models which had been widely applied in thefield of geotechnical engineering Following this we compre-hensively reviewed the state of the art subjected to practicalapplications of fractional derivative model including threemajor aspects of deformation analysis of circular tunnelrheological settlement of the subgrade and relevant loessresearches

2 Model Structure Research

21 Presentation andDevelopment of Fractional Calculus Thefractional calculus has been verified that is a branch ofmathematics which investigates the property of the differen-tiation and integration of any order Its embryonic form canbe dated back to September 30 1695 when Leibniz repliedto the famous question ldquoWhat does the derivative 119889

119899

119891119889119909119899

for 119891(119909) mean if 119899 = 12rdquo proposed by Lrsquo Hospital [25]Due to its complicated calculation and undefined physicalsignificance the focus at early stage is mainly on researchin pure mathematics and for a long period of time the

research on fractional calculus has attracted little attentionfrom scholars in natural science and engineering technologyfield In 1936 Gemant firstly applied fractional derivativesto problems in elasticity [26] Until 1982 Mandelbrot [27]who was the founder of fractal theory pointed out thatthere was a self-similarity phenomenon between integer andfraction and many fractal dimensions existed in natureand many technological sciences Subsequently researchersfound relevant phenomenon or process of fractal geometrypower law phenomenon and memory process which couldestablish a close relationship with fractional calculus thusleading to an explosion of research activities on fractionalcalculus The contribution of Shestopal and Goss [28] wasfirstly applied to fractional derivative in Maxwell and Kelvinmodel for analysis of creep buckling Bagley and Torvik [29]conducted research on three-dimensional constitutive rela-tion of fractional derivative limitation of thermodynamicson model parameters and finite element method Throughutilizing the linear standard system of fractional derivativeZhang [30] established the constitutive relation of nitrilerubber and proved that it was equivalent with Rabotnovmodel Metzler and Nonnenmacher [31] took Zener modelbased on fractional order to explore the abnormal relaxationof polymeric materials Yin et al [32] firstly consideredfractional derivative element as soft-matter element whilebuilding the constitutive model of soil rheology phenomenarepresenting the material between an ideal fluid and an idealsolid

Up to now there are many different definitions of frac-tional calculus [33ndash35] among which basic ideology is tolook at a fractional derivative as the inverse operation ofa fractional integral It is usually chosen as the Riemann-Liouville form

119889minus120574

119891 (119905)

119889119905minus120574=1199050119863minus120574

119905

119891 (119905) = int

119905

1199050

(119905 minus 120585)120574minus1

Γ (120574)119891 (120585) 119889120585

119905 isin [1199050 119887]

(1)

where 119891(119905) is integrable function on the interval [1199050 119887] with

the order 120574 gt 0 119899minus 1 lt 120574 le 119899 (119899 is integer) and Γ(120574) is EulerrsquosGamma function which is defined by

Γ (120574) = int

infin

0

119890minus119905

119905120574minus1

119889119905 (2)

Correspondingly the Riemann-Liouville fractional de-rivative is given by

119889120574

119891 (119905)

119889119905120574=1199050119863120574

119905

119891 (119905) =119889119899

119889119905119899[1199050119863minus(119899minus120574)

119905

119891 (119905)] (3)

Laplace transformation formula of fractional differentialcan be expressed as

119871 [0119863minus120574

119905119891 (119905) 119901] = 119901

minus120574

119891 (119901) 120574 ge 0

119871 [0119863120574

119905119891 (119905) 119901] = 119901

120574

119891 (119901) 120574 le 0

(4)

where 119891(119905) can be integrated near 119905 = 0 with 0 le 120574 le 1 and119891(119901) is Laplace transformation of 119891(119905)

Mathematical Problems in Engineering 3

120590 120590

1205780

Figure 1 Abel dashpot with constant coefficient [32]

22 Research Achievements of Model Structure Just asfractional calculus is a generalization of the commonlyused integer-order differentiation and integration fractionalderivative model continues classical modelrsquos feature of beingconcise and easy to understand which not only can makeup the shortcomings of classical model but also has betterdescription effects Its essence is to replace a Newtoniandashpot in classical model with the fractional derivative Abeldashpot and thus obtain fractional derivative model throughseries-parallel connection with other several basic elementsIn geotechnical engineering the functional equations offractional model are mainly utilized to perform the processcurve of rheology phenomena with an attempt to adjustdeformation rate and control the stress (strain) [36 37]

221 Fractional Derivative Abel Dashpot Research Thereare two kinds of most basic kernel operators Abel kerneland Rabotnov kernel The fractional derivative elementestablished based on Rabotnov kernel has the advantageof describing rheological curve with horizontal asymptotebut its equations are quite complicated resulting in hardlyacquired parameters For a comparison using Abel kernelto investigate creep phenomenon the basic ideology is toresearch the function forms after derivation of creep compli-ance for modified Abel dashpot Its relevant model has fewerparameters and stronger adaptability which is demonstratedby the increasing number of papers and special issues injournals [38ndash43]

Function equation of Abel kernel [36] is written asfollows

120601 (119905) =120582119905minus120574

Γ (1 minus 120574) (5)

(1) Abel Dashpot with Constant Coefficient The constitutiveequation of Abel dashpot with constant coefficient shown inFigure 1 is defined as follows

120590 (119905) = 120578120574119889120574

[120576 (119905)]

119889119905120574(0 le 120574 le 1) (6)

where 120578120574 is the viscosity coefficient In a special case of 120574 =

1 the constitutive equation (6) is usually applied to describethe Newton dashpot being equivalent to an ideal fluid with120578120574

= 120578 while a spring in the case of 120574 = 0 is equivalent to anideal solidwith 120578

0

= 119864Therefore Abel dashpotwith constantcoefficient bridges the gap between an ideal fluid and an idealsolid material response by interpolating between the Newtondashpot and a spring

120590 120590

1205781

120590s

Figure 2 Abel dashpot with variable coefficient [2]

Under the circumstance of constant stress in (6) namely120590(119905) = 120590 the element denotes the creep strain characteristicConducting the fractional integral calculation of (6) based onthe Riemann-Liouville operator we can gain

120576 (119905) =120590

120578120574

119905120574

Γ (1 + 120574)(0 le 120574 le 1) (7)

(2) Abel Dashpot withVariable Coefficient During the processof rheology in particular at the accelerated stage the crackingand damage growth at the microscale must be consideredhence the viscosity coefficient of rock or soil is no longerremaining unchanged which should be described by intro-ducing the damage variable [2 44 45] As a result theconstitutive relation of Abel dashpot with variable coefficientshown in Figure 2 deduces that

120590 (119905) = [120578120574

(1 minus 119863120596)]

119889120574

[120576 (119905)]

119889119905120574(0 lt 120574 le 1) (8)

where 119863120596is the damage variable defined by 0 le 119863

120596=

1 minus 119890minus120572119905

lt 1 and 120578120574

(1 minus 119863120596) = 120578120574

119890minus120572119905 indicates the variable

viscosity coefficient 120572 is related to properties of materials

222 Application of Fractional Derivative Model In terms ofthe issues in geotechnical engineering fractional derivativemodels are usually composed of basic element constantcoefficient Abel dashpot and variable coefficient Abel dash-pot in series and parallel which now have evolved fromsimple models such as fractional Maxwell and Kelvin mod-els to the fractional Nishihara model accounting for theefforts researchers devoted In contrast the simple modelsare connected in series or in parallel with the constantcoefficient Abel dashpot and spring element while the threeelementsrsquo solid model in series with variable coefficientdashpot leads to the fractionalNishiharamodel Additionallythree-dimensional constitutive equation is derived from one-dimensional constitutive equation of fractional derivativemodels The commonly used models are shown in Figure 3

Note that different rheological conditions require dif-ferent models In Figure 3 (a) (b) (c) and (d) belong toviscoelastic model while (e) belongs to viscoelastoplasticmodel In greater detail (a) (b) and (c) are all suitablefor describing creep that tends to stability finally and (c)shows the highest fitting precision For comparison (d) hasadvantage of describing the creep with decay and flow period(e) is regarded as special one that can approach the threestages with initial stage stable stage and accelerated stage

Taking the relevant investigations on soft soil as examplesshowing the evolution of model structure Sun and Zhang


120590120590

1205782E

(a) The Maxwell model

120590 120590

1205783

E

(b) The Kelvin model

120590120590

1205783

E1

E2

(c) The three elementsrsquo solid model (Kelvin-Voigt)

120590 1205901205782

1205783

E1

E2

(d) The Burgers model

120590120590

12057811205783

120590s

E1

E2

(e) The Nishihara model

Figure 3 The fractional derivative element model [36]

Experimental dataFractional derivative Kelvin modelClassical Kelvin model

200 400 600 800 1000 1200 1400 1600 18000Time (min)

10

15

20

25

30

35

J(t)

(10minus6

GPa

minus1)

Figure 4 Curves respectively simulated by fractional Kelvinmodeland Kelvin model (120590 = 375 kPa) [46]

[46] adopted the fractional derivative Kelvin model asshown in Figure 3(b) to simulate the creep laws of twostages of soft soil in Pearl River Delta in which equationform was simple and unified compared with classical modelAdditionally fewer parameters needed to be adjusted duringthe process of calculation In addition from the results shownin Figure 4 the classical Kelvinmodel brings a large deviationin comparison with the experimental data near the inflectionpoint of fitted curve and its offset increases with the timeIt can be predicted that accordingly in a very long time theclassical Kelvin model is seriously out of accordance with theexperimental data

Based on fractional calculus theory Li and Yue [47]proposed a five-element nonlinear rheological constitutivemodel which reserved a Newton dashpot as shown inFigure 5 for Hong Kong marine clay and Shanghai muckyclay under different stress levels When 120590

0lt 120590119904 the right side

120590120590

1205781205783

120590s

E1

E2

Figure 5 Fractional order derivative five-element model [47]

of the viscoplastic dashpot does not participate in the workthus the model is simplified as fractional derivative three-element solid model While 120590

0gt 120590119904 the viscoplastic dashpot

participates in the work the model can better characterizethe three stages of soft soil under high stress with high fittingaccuracy in which relevant coefficient of the whole curvecompared with experimental results reaches 09696 shownin Figure 6 In this regard the model has wide applicability inview simultaneously of a small number of parameters whichare easy to determine

To research creep properties of Zhanjiang soft clay Heet al [48] built the modified Burgers model and then theexpression of three-dimensional equation was derived basedon hereditary creep theory Consider

120576119894119895

=119904119894119895

2119866+

120590119898

3119896120575119894119895+

119904119894119895

2119866int

119905

0

120581 (119905) 119889119905 (9)

where 119904119894119895is the deviator stress tensor and119866 indicates the shear

modulus 120590119898is the confining pressure and 119896 expresses the

bulk modulus of deformation In a special case of 119894 = 119895120575119894119895

= 1 while 120575119894119895

= 0 in another case

3 Determination of Model Parameters

Simplicity and accuracy are two significant benchmarksto measure the applicability of constitutive model [39ndash41]Lewandowski and Chorązyczewski [42] pointed out that animportant problem connectedwith the fractional rheologicalmodels is an estimation of the model parameters fromexperimental data To some extent the key to accurately applythe fractional derivativemodel to practical engineering lies inthe precision of fitting parameters [42 43]


Table 1 Characteristics of the commonly used rheological model of fractional derivative

Fractional derivativemodel Creep compliance (119869) Number of

parameters

Kelvin

1

1199020

[1 minus 119864120574(119905)] 119905 ge 0

119864120574(119905) = 1 +

infin

sum

119899=1

(minus1)119899

(119905120578)120574119899

Γ(1 + 120574119899)

3

Three elementsrsquo solid(Kelvin-Voigt)

1

1198640

+1

120578120574

infin

sum

119899=0

(minus1198641120578120574

)119899

119905120574(1+119899)

120574 (1 + 119899) Γ [(1 + 119899) 120574]4

Burgers1

1198641

[1 minus (1 minus1198641+ 1198642

1198642

)

infin

sum

119899=0

(minus1)119899

(1198642

1205782

)

119899+1

119905120574(1+119899)

Γ [120574 (1 + 119899) + 1]] +

1

1205781

119905120574

Γ (119903 + 1)5

Nishihara

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] 120590 lt 120590

119904

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] +

1

12057821205742

1198901199051205742 120590 gt 120590

119904

6

Experimental valuesFractional derivative Nishihara model

100 200 300 400 500 600 700 8000Time (min)

120576(

)

1

2

3

4

5

6

7

8

Figure 6 Test data and fitting curve of five-element fractionalderivative model (120590 = 243 kPa) [47]

Literature [25] investigated that the creep equations ofdifferent models have unified form given by

120576 (119905) = 119869 (119905) 120590 (10)

where 119869(119905) refers to creep compliance which have differentformsTheir forms [49ndash52] and the number of parameters aresummarized in Table 1

31 Parameters Identification with Method of Numerical Anal-ysis Generally the numerical methods for determining theparameters of rheological model mainly include regressioninversion method least square method and decompositionmethod based on rheological curve Nevertheless all theseoptimized algorithms have some restricted conditions whendealing with relevant issues that is the optimized functionof constitutive relation should be differentiable Along withthe rapid development of numerical calculation and analysis

Variable 1 Variable 2 Variable m

Upper bound of the interval of variable valuesLower bound of the interval of variable valuesSelected path of number 1 ants Selected path of number 2 antsSelected path of number k ants

Selected grid nodeThe optimal point searched under the current grid

middot middot middot

Figure 7 Ant colony algorithm [53]

the estimation of the fractional derivative model has beenbrought to a whole new level

311 Ant Colony Algorithm Specific to the interval variablesof parameters in viscoelastic Voigt constitutive model thatwas the fractional derivative three elementsrsquo solid modelZhao [53] firstly established the solving models for parame-ters including the semianalytical finite element model basedon time domain homogenous field and the difference finiteelement model on the basis of regional heterogeneous fieldAnd then using optimized algorithm depended on ant colonytheory as shown in Figure 7 the single and combinedidentification of constitutive parameters all was realized


Calculated values under 120590 = 179MPaExperimental values under 120590 = 179MPaExperimental values under 120590 = 299MPaCalculated values under 120590 = 299MPaExperimental values under 120590 = 419MPaCalculated values under 120590 = 419MPa

120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20

(a) Fractional derivative Kelvin model

120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20


Figure 8 Calculation value of frozen soil and the uniaxial creep test value contrast [36]

Table 2 Particle swarm parameters [55]

Populations Close topopulations 119862

11198622

Maximumacceleration

80 15 30 30 15

312 Genetic Algorithm In order to address the uniaxialcreep of artificially frozen soil in the area ofHuainan-HuaibeiWang [36] adopted fractional derivative Kelvin model tosimulate the creep characteristics of the first and secondstages and then obtained the model parameters throughglobal optimization based on genetic algorithm As shownin Figure 8 fractional derivative Kelvin model has a higheraccuracy of simulation than the classical Kelvin model Incontrast to test data the former has the relevant coefficientof more than 099

313 Particle Swarm Optimization Algorithm Chen et al[54] applied particle swarm optimization for the purposeof identifying the creep parameters of generalized Kelvinmodel which was built with the experimental data on steadystage of artificially frozen soil under different loading stressthus material parameters of generalized Kelvin model wereobtained

Similarly Niu [55] built the fractional derivative Nishi-hara model intended to describe the deformation of frozensoil at accelerating rheology stage furthermore particleswarm optimization was applied to optimize the parame-ters of fractional derivative creep model The parametersusing particle swarm optimization algorithm were shown inTable 2

Special to the experiment conditions sandy clay had theloading values of 07120590

119904 respectively at the temperature of

minus5∘ minus10∘ and minus15∘The obtained parameters of the fractionalderivative Nishihara model were shown in Table 3 As aresult the relevant coefficient for fitting model parametersthrough particle swarm optimization algorithm was morethan 098 which highlighted the unique advantage of particleswarm optimization algorithm in application of optimizingthe fractional derivative Nishihara model for describingaccelerating rheology of frozen soil

314 Intelligence Algorithm Simulated annealing (SA) algo-rithm [56 57] was incorporated into standard particle swarmoptimization (PSO) algorithm by Xu et al [58] whichcould minimize the error caused by the limitations of PSOalgorithm The new method obtained was called SAPSOalgorithm as shown in Figure 9 Since PSO algorithm onlyaccepts the optimal solution point 119860 in Figure 9 will fallinto the local optimal solution on the left during the calcu-lating resulting in the global optimal solution being usuallyneglected Subjected to the SA algorithm point119860 can be onlyaccepted with a certain probability Therefore point 119861 as theglobal optimal solution has opportunity to be obtained bygetting out of the local optimal solution

After optimization rheological curve decomposition-particle swarm-simulated annealing algorithm is establishedto inverse the multiparameter constitutive model of rockrheology leading to effectively solving the initial valuedependence of nonlinear problem and gaining global optimalsolution simultaneously Additionally we can obtain fromFigure 10 that SAPSO algorithm shows rapid changes with


Table 3 Parameters of particle swarm fractional order derivative Nishihara model [55]

Soil Temperature Loadingcoefficient 120590 119864

11205782

120574 120590119904

Correlationcoefficient

Clay

minus5 03 105 006 mdash 01116 mdash 0986905 175 028 mdash 072 mdash 09990

minus10 03 15 024 mdash 00068 0994305 25 116 00198 mdash 09947


minus5 07 245 0000646 0000631 10494 237 09919minus10 07 35 011854 10638 000811 325 09873minus15 07 49 001117 000548 1139 474 09927

Optimal particle

Optimal particle

Local optimal solution Local optimal solution

Global optimal solution

AB

xt+1i

tk+1

xti tk

Figure 9 Sketches of SAPSO algorithm [58]

fitness value in the first 16 iterative steps and then reaches ahigh convergence precision

32 Parameter Solving Based on Software Based on MonteCarlo Gen et al [59] took the minimum error of experi-mental data as the criterion and then made the experimen-tal curves of nine samples fitted with standard fractionalderivative linear model thus determining the relaxation timeand coefficient of viscosity of Malan loess Meanwhile Sunand Zhang [46] and Li and Yue [47] by means of MATLABperformed effective identification for parameters aspects offractional derivative Kelvinmodel under the condition of lowstress of soft soil and fractional derivative five-element modelunder the condition of high stress which was featured bysimplicity and wide applicability

He [60] performed parameter fitting for fractional deriva-tive Maxwell model through 1stOpt mathematical softwareshown in Figure 11 As a result the changes law of rheologicalmodel parameters for 119876

2loess in cases of different water

content was gained as shown in Table 4

Table 4 Parameters fitting results of uniaxial rheological model of1198762loess in Yanrsquoan [60]

120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741

20 40 60 80 1000Iteration times (times)

Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10

Figure 10 Relationship between fitness and iteration times [58]

On the basis of parameter identification hence constitu-tive equation of steady creep model for Yanrsquoan119876

2loess under

the condition of uniaxial stress was obtained Consider

120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)

where 120590119904is determined by uniaxial creep experiment


Figure 11 Interface of 1stOpt [60]

33 Parameter Analysis

331 Parameter Sensitivity Analysis In order to get an ade-quate understanding of the effects of parameters 120574 and 120572 onthe time-dependent strain and stress Zhou et al [61] appliedprogrammable rheometer to carry out uniaxial rheological

experiments for salt rock sample of well Wangchu-1 of Jiang-han oilfield Subsequently based on the results of experimentfitting analysis of parameters 120574 and 120572 on rheological strainwas conducted and its effects were discussed simultaneouslyFigure 12(a) shows that rheological value and rheologicalspeed greatly depend on fractional derivative order and thehigher the fractional derivative order the higher the creepstrain which reflects the flexibility of fractional derivatemodel in description of rheology phenomena Moreover inthe special case of 120572 = 0 rheological procedures of saltrock never appear to be obvious accelerated stage as shownin Figure 12(b) namely the third stage which only appearsin the case of 120572 gt 0 What is more the accelerated stage ismore likely to occur as the values of 120572 increase The resultsobtained mutually verify the conclusions of Xu et al [58] Liuand Zhang [62] Yang and Fang [63] and He et al [64]

332 Special Case Introducing the Mittag-Leffler functionthe equations based on the fractional derivative Nishiharamodel deduce that

120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910

(119911) is Mittag-Leffler function with the case of11986411

(119911) = 119890119911 and 119864

12= (119890119911

minus 1)119911In a special case of 120574 = 1 and 120572 = 0 (12) is simplified as

120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)

Note that (13) refers to the constitutive relation of theclassical Nishihara model The classical Nishihara model isa special case of the fractional derivative Nishihara modelaccordingly [60]The conclusion is mutually verified with theresults of Yin et al [32] He et al [48] and Xu et al [58]

4 Engineering Application

41 Deformation Analysis of Circular Tunnel As the tra-ditional issue of underground engineering the stress ofsurrounding rock mass is redistributed due to tunnel exca-vation resulting in occasionally a large delayed deformationthat might lead to a delayed failure of structure and rockmass collapse especially for the tunnel in soft rock [6566] Therefore the deformation mechanism of surroundingrock should be investigated in the perspective of rheologybehavior since the property of associated time effects

Liu and Zhang [62] proposed to take advantage of frac-tional derivative Kelvin model to investigate the deformation

properties of horizontal circular tunnel [67 68] With theanalysis of the result it was shown that fractional derivativemodel could perform the relaxation properties of viscoelasticrock mass In addition different viscoelastic rock could beprecisely described through adjusting fractional order whichwas proved to the conclusions as mentioned previously Thedisplacement and strain components of circular tunnel basedon fractional Kelvin derivative model deduced that

119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)

Chen [69] took the effects of tunnel face into accountnamely the spatial effect as shown in Figure 13 at thesame time the region within quadruple tunnel radius fromtunnel face was defined as the near section contrarily as thefar section Using fractional derivative Maxwell model andclassic Burgers model respectively comparative analyses onthe displacement changed with time aspects after circulartunnel excavation and support were performed [70] with anattempt to research the deformation characteristics duringtwo stages of decay creep and stable creep of surroundingrock


10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1

Experimental data120574 = 05

120574 = 07

120574 = 09

(a) Parameter 120574

10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07

Experimental data120572 = 0013 hminus1

120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1

(b) Parameter 120572

Figure 12 Sensitivity of the creep strain to the fractional derivative order in the case [61]

2R

4R

Support (closed)

Spatial effect

Tunnel face

Excavation direction

Near section regionFar section region

Figure 13 Tunnel excavation

When we analyze the case of tunnel without supporttaking the displacement calculation of tunnel without side-wall as an example the creep displacement of its far sectionregion is investigated [71] that is neglecting the influenceof tunnel face with 120582 = 1 Thus the viscoelastic solutionfor sidewall displacement in far section region based onfractional derivative Maxwell model is obtained

119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)

The parameters of both Burgers model and fractionalderivative Maxwell model are firstly confirmed with particleswarm optimization and then the displacement values oftunnel sidewall in different periods are gained depicted inFigure 14 With the duration time of 1000 days Figure 14(a)denotes that the description of fractional derivative Maxwellmodel is essentially equivalent to classical model But when

the duration time comes to 10000 days the calculation resultsof two models are close to being consistent only in a certainperiod of time rather than the whole procedure shown inFigure 14(b) In greater detail it indicates that comparedwith the fractional derivative Maxwell model the calculationresults change greatly with time especially in later stage basedon Burgers model

Under the circumstance of supportmeasures as shown inFigure 15 the impacts of the interaction between tunnel faceand support and surrounding rock and support need to beconsidered simultaneously [72] It is assumed that time 119905

0has

passed when support structure is performed in the case of119905 lt 1199050 representing the state before the support structure is

applied while 119905 = 1199050represents the state of performing the

support structure Additionally the hypothesis is proposedthat the process of performing the support structure iscompleted instantaneously indicating that no displacementis produced during the installation of support Thus theviscoelastic solution for sidewall displacement in far sectionregion (120582 = 1) based on fractional derivative Maxwell modelis obtained

119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)

where the stiffness of supporting structure can be estimatedby the empirical formula proposed by Brady and Brown [73]

119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)

Subsequently the displacement values of tunnel sidewallwere depicted in Figure 16 with the same methods as men-tioned previously It can be known that the change trendsof sidewall displacement based on the fractional derivative


Burgers modelFractional derivative Maxwell model

200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100

(b) 0ndash10000 days

Figure 14 The creep displacement of tunnel sidewall without support [69]

Tunnel Support (closed)

a

bo

Figure 15 Schematic diagram of support

Maxwell model are basically consistent with Burgers modelin the range of 0sim1010 days Furthermore the deformationresults of two kinds of models tend to be convergent withthe same final value caused by the equation lim

119905rarrinfin119866(119905) =

lim119905rarrinfin

(1119869(119905)) = 0 Moreover through the comparativeanalysis of Figures 14 and 16 it is shown that the displacementvalue of tunnel sidewall decreases significantly and then tendsto be convergent after working of support structure

42 Rheological Settlement of Subgrade Mentioned in tra-ditional consolidation theory the soil skeleton is usuallyregarded as elastomer with pore water pressure [74] In rela-tion to this the additional stress formula of embankment isestablished with an assumption that foundation is consideredas an ideal linear elasticity [75 76] However in practicalroad engineering the state of embankment foundation ismore than extremely complicated with mechanical propertyof distinct viscoelasticity which mainly depends on structurecomposition temperature load frequency and load historyand so forth At the consolidation time scale processessuch as settlement of subgrade can be approached by creepdeformation subjected to a viscous rheology

On the basis of viscoelastic theory and in combinationwith the characterization of subgrade fill Wang [77] utilized

Fractional derivative Maxwell modelBurgers model

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)

Figure 16 The creep displacement of tunnel sidewall after support-ing [69]

the fractional derivative Kelvin-Voigt model and simplifiedthe rheological settlement problemof embankmentwith highfill to be a plane strain problem Meanwhile the EDJ-1 typedevice fromNanjing Soil Instrument Factory was selected asshown in Figure 17 devoted to undertake shear creep test oncompacted soil The stress-strain relationship was confirmedby the rheological constitutive model subsequently whichwas established through the curve cluster of consolidationcreep under step loading as shown in Figure 18

In greater detail taking the corresponding load stress1205900and 120590

0+ Δ120590 as example the curves of consolidation


Figure 17 The main test device [77]

o t

ɛ

800 kPa700kPa600kPa500kPa400kPa300kPa

200 kPa

100 kPa

Figure 18 Sketch of curve cluster [77]

creep were obtained from the experimental data treated bylinear interpolation Furthermore through subtracting cor-respondingly the curve values of the load stress 120590

0+Δ120590 from

1205900 the consolidation creep law at a certain depth resulted

from the fact that additional load Δ120590 could be calculatedthus parameters were determined through parameter iden-tification subsequently [78] In terms of practical subgradeaccording to the method as mentioned above consolidationcreep test was performed to each layer of stratified subgrade[79ndash81] following acquirement of the parameters of modelfor each layer Hence the total settlement of foundationwithin compressed layer was given by [77]

119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)

where 119867119894refers to the layer thickness of foundation and

its determination method is the same as the layer-wisesummation method

43 Relevant Loess Researches Through replacing Newtondashpot in classical model with Abel dashpot Gen et al[59] established fractional derivative standard linear modeland its analytical expression of shear stress relaxation was

120576(

)

Experimental valuesCalculated values

00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa

Figure 19 Test values and theoretic values for creep of loess sample(moisture content 8) [80]

obtained Furthermore the test for direct shear relaxationwasconducted under the conditions of different positive pressurefor Malan loess with different water contents resulting in theconclusion that the relaxation features of Malan loess couldbe represented effectively by the proposed model subjectedto fitting verification

Taking water damage variable 119863120596

of 1198762loess into

account Tang et al [82] obtained the equations of waterdamage evolution in terms of various creep parametersaccordinglyThrough introducing Abel dashpot with variablecoefficient the nonlinear creep model of 119876

2loess was

established thus its constitutive equation for creep behaviorwas given by [82]

120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1

1205742gt 1In addition to verify the correctness and accuracy of

the obtained constitutive equation a series of triaxial creeptests to 119876

2loess under different states of water content

was undertaken Meanwhile based on the test data theregression analysis for parameters estimation was performedusing 1stOpt mathematical software In the case of watercontent of 8 five group curves of model theoretical valuesand experimental values were depicted in Figure 19 repre-senting different procedure For a comparison it exhibits thatthe established model for 119876

2loess is able to characterize

effectively each stage of the instantaneous deformation decaycreep steady creep and accelerated creep thus having abright future for application


5 Summary and Perspectives

For the purpose of describing the rheology phenomena ingeotechnical engineering this paper has provided an up-to-date review of key constitutive models in combination withmodel structure and acquisition of parameters Comparedwith classical constitutive model the fractional derivativemodels have shown the extruding advantage of more conciseequation fewer parameters and higher precision which alsohave been improved amended and perfected constantlyCurrently with the significant improvement of numericalmethods the parameter of the fractional derivative modelthat is rather difficult to determine in the pass time has notbeen the fetters to hinder the research thus the applicationof fractional derivative models in geotechnical engineer-ing has been successful with gradually mature technologyWe therefore comprehensively reviewed the state-of-the-art of fractional derivative model for practical applicationincluding three major aspects of deformation analysis ofcircular tunnel rheological settlement of the subgrade andrelevant loess researches To some extent the rheologicalmodels based on fractional calculus are developing towardprecision theorization design of three dimensions and soforth and have established a better research frameworkHowever through the application case analysis and summarymentioned above we can obtain the following conclusionsand perspectives

(1) As a powerfulmathematical tool the fractional deriva-tivemodels have advantage of describing effectively thewholeprocedure of rheology phenomenon and are endowed witha small number of parameters Nevertheless along with theconstant refinement of analysis and design in geotechnicalengineering what is crucial and then is proposed duringthe process of application is a higher demand on parametersprecision Therefore how to obtain more precise parametersbased on experimental data is the target that investigatorsassiduously seek Meanwhile theoretical research ought tobe deepened and tends to clear the physical significance andfind out its change laws of some parameters such as 120574 and120578

(2) The application case analysis shows that the descrip-tion of the fractional derivative model has a preferableapplicability by adjusting the fractional derivative order offractional derivative dashpot In other words the fractionalderivative model is one of the most effective ways to describethe rheology which ismore flexible for the dynamic deforma-tion Butwe need to know that fractional derivativemodel hasmany restrictions in the actual application of engineering forinstance the tunnel under study is assumed to be horizontallycircularTherefore actual experiences should be accumulatedin the processing of engineering and on this basis theapplicability of fractional derivative model ought to keep onimproving Specifically monitoring in situ test of tunnelingshould be strengthened and then the predictive value forcorresponding deformation will be gained in combinationwith the research on accelerated stage of creep Moreoverin order to ensure the safety of construction site and avoidthe occurrence of accident in the long run the fractionalderivative model should be applied to predict the failure

phenomena such as collapse and large deformation of tunnelengineering

(3) According to the complicated problems of geotech-nical engineering the results obtained based on classicalmodel due to its limitation deviate largely from the actualfield condition Obviously the current finite element softwareprograms which are based on classical constitutive modelare powerless in numerical simulation Furthermore owingto the bad potential of operation and cost-effectiveness labo-ratory simulation experiment has difficulty in satisfying engi-neering demands Considering the complicated mechanicsproperties of practical engineering it is necessary to deepenthe investigation on three-dimensional fractional deriva-tive model Moreover it can be envisaged that fractionalderivative model would be more suitable for the analysisof engineering problems in the coming years especially ifthe existing numerical software is equipped with fractionalderivative models by conducting secondary development

(4) In the field of geotechnical engineering there wereaccumulated numerous constitutive models (including clas-sical models and fractional models) with diversified formsHence it is essential to conclude and summarize the exist-ing identification technologies and then constitutive modeldatabase of geotechnical materials should be established incombination with various identification algorithms devotedto the convenient choices and efficient applications withclear classification of constitutive models For instance thefractional derivative Maxwell model should be labeled asone that can only describe the rheology whose initial valuetends toward zero which is not applicable to the case ofcompacted soil Finally taking our results into considerationa constitutive model based on fractional derivative whichis concise and is able to describe rheology phenomenonaccuratelymay be incorporated intomodel database andmaybe the best choice in the future

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is financially supported by the Brainstorm Projecton Social Development of Shaanxi Provincial Science andTechnology Department (Grant No 2016SF-412) and theSpecial Fund for Basic Scientific Research of Central Collegesof Changrsquoan University (Grant no 31082116011) and the KeyIndustrial Research Project of Shaanxi Provincial Scienceand Technology Department (Grant no 2015GY185) and theIntegrated Innovation Project of Shaanxi Provincial Scienceand Technology Department (Grant no 2015KTZDGY01-05-02)

References

[1] D Yin W Zhang C Cheng and Y Li ldquoFractional time-dependent Bingham model for muddy clayrdquo Journal of Non-Newtonian Fluid Mechanics vol 187-188 pp 32ndash35 2012


[2] H W Zhou C P Wang B B Han and Z Q Duan ldquoA creepconstitutive model for salt rock based on fractional derivativesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 48 no 1 pp 116ndash121 2011

[3] Y P Yao B Y Zhang and J G Zhu ldquoBehaviors constitutivemodels and numerical simulation of soilsrdquo Tumu GongchengXuebaoChina Civil Engineering Journal vol 45 no 3 pp 127ndash150 2012

[4] Z-C Wang and L-P Qiao ldquoA review and discussion on creepbehavior of soil and its modelsrdquo Rock and Soil Mechanics vol32 no 8 pp 2251ndash2260 2011

[5] D M Cruden ldquoThe form of the creep law for rock under uni-axial compressionrdquo International Journal of RockMechanics andMining Sciences ampGeomechanics Abstracts vol 8 no 2 pp 105ndash126 1971

[6] M Oeser and T Pellinien ldquoComputational framework forcommon visco-elastic models in engineering based on thetheory of rheologyrdquoComputers andGeotechnics vol 42 pp 145ndash156 2012

[7] H Zhang Z Y Wang Y Zheng P J Duan and S Ding ldquoStudyon tri-axial creep experiment and constitutive relation ofdifferent rock saltrdquo Safety Science vol 50 no 4 pp 801ndash8052012

[8] A Heibig and L I Palade ldquoWell posedness of a linearized frac-tional derivative fluid modelrdquo Journal of Mathematical Analysisand Applications vol 380 no 1 pp 188ndash203 2011

[9] G B Davis M Kohandel S Sivaloganathan and G TentildquoThe constitutive properties of the brain paraenchyma Part2 Fractional derivative approachrdquo Medical Engineering andPhysics vol 28 no 5 pp 455ndash459 2006

[10] H-A A Rafael R-O M Pedro M del Toro Roberto andC-B Jorge ldquoSimulation of creep phenomenon in clay soilsusing rheology and fractional differential equationsrdquo IngenierıaInvestigacion y Tecnologıa vol 15 no 4 pp 561ndash574 2014

[11] F Barpi and S Valente ldquoA fractional order rate approach formodeling concrete structures subjected to creep and fracturerdquoInternational Journal of Solids and Structures vol 41 no 9-10pp 2607ndash2621 2004

[12] T Pritz ldquoFive-parameter fractional derivative model for poly-meric damping materialsrdquo Journal of Sound and Vibration vol265 no 5 pp 935ndash952 2003

[13] P Yang Y C Lam and K-Q Zhu ldquoConstitutive equation withfractional derivatives for the generalized UCMmodelrdquo Journalof Non-Newtonian Fluid Mechanics vol 165 no 3-4 pp 88ndash972010

[14] V A Vyawahare and P S V Nataraj ldquoFractional-order mod-eling of neutron transport in a nuclear reactorrdquo Applied andMathematial Modelling vol 37 no 23 pp 9747ndash9767 2013

[15] R Droghei and E Salusti ldquoA comparison of a fractional deriva-tive model with an empirical model for non-linear shock wavesin swelling shalesrdquo Journal of PetroleumScience andEngineeringvol 125 pp 181ndash188 2015

[16] Z Xu and W Chen ldquoA fractional-order model on new exper-iments of linear viscoelastic creep of Hami Melonrdquo Computersand Mathematics with Applications vol 66 no 5 pp 677ndash6812013

[17] B Li and W Xie ldquoAdaptive fractional differential approach andits application to medical image enhancementrdquo Computers andElectrical Engineering vol 45 pp 324ndash335 2015

[18] B J West ldquoFractional calculus in bioengineeringrdquo Journal ofStatistical Physics vol 126 no 6 pp 1285ndash1286 2007

[19] C Celauro C Fecarotti A Pirrotta and A C Collop ldquoExperi-mental validation of a fractional model for creeprecovery test-ing of asphalt mixturesrdquo Construction and Building Materialsvol 36 pp 458ndash466 2012

[20] A Schmidt and L Gaul ldquoOn the numerical evaluation of frac-tional derivatives in multi-degree-of-freedom systemsrdquo SignalProcessing vol 86 no 10 pp 2592ndash2601 2006

[21] L Deseri M D Paola and M Zingales ldquoFree energy andstates of fractional-order hereditarinessrdquo International Journalof Solids and Structures vol 51 no 18 pp 3156ndash3167 2014

[22] J H Kang F Z Zhou C Liu and Y Liu ldquoA fractional non-linear creep model for coal considering damage effect andexperimental validationrdquo International Journal of Non-LinearMechanics vol 76 pp 20ndash28 2015

[23] M-M He N Li Y-S Chen and C-H Zhu ldquoDynamic defor-mation behavior of rock based on fractional order calculusrdquoChinese Journal of Geotechnical Engineering vol 37 no s1 pp178ndash184 2015

[24] C C Zhang H H Zhu B Shi et al ldquoTheoretical investigationof interaction between a rectangular plate and fractional vis-coelastic foundationrdquo Journal of Rock Mechanics and Geotech-nical Engineering vol 45 pp 324ndash335 2015

[25] W Chen H G Sun and X C Li Mechanical and EngineeringProblems of Fractional Derivative Model Science Press BeijingChina 2012

[26] A Gemant ldquoA method of analyzing experimental resultsobtained from elasto-viscous bodiesrdquo Journal of Applied Physicsvol 7 no 8 pp 311ndash317 1936

[27] B BMandelbrotTheFractal Geometry of Nature Freeman SanFrancisco Calif USA 1982

[28] V O Shestopal and P C J Goss ldquoThe estimation of columncreep buckling durability from the initial stages of creeprdquo ActaMechanica vol 52 no 3-4 pp 269ndash275 1984

[29] R L Bagley and P J Torvik ldquoOn the fractional calculus modelof viscoelastic behaviorrdquo Journal of Rheology vol 30 no 1 pp133ndash155 1986

[30] W M Zhang ldquoA new rheological model theory with fractionalorder derivativesrdquo Natural Science Journal of Xiangtan Univer-sity vol 23 no 1 pp 31ndash36 2001

[31] R Metzler and T F Nonnenmacher ldquoFractional relaxationprocesses and fractional rheological models for the descriptionof a class of viscoelastic materialsrdquo International Journal ofPlasticity vol 19 no 7 pp 941ndash959 2003

[32] D S Yin J J Ren C L He and W Chen ldquoNew rheologicalmodel element for geomaterialsrdquo Chinese Journal of RockMechanics and Engineering vol 26 no 9 pp 1899ndash1903 2007

[33] K B Oldham and J Spanier The Fractional Calculs AcademicPress New York NY USA 1974

[34] T FNonnenmacher andRMetzler ldquoOn the Riemann-Liouvillefractional calculus and some recent applicationsrdquo Fractals vol3 no 3 pp 557ndash566 1995

[35] G Rudolf and M Francesco Essentials of Fractional CalculusMaPhySto Center Aarhus Denmark 2000

[36] X Wang Genetic fractional order derivative of artificial frozensoil creep constitutive model research [PhD thesis] AnhuiUniversity of Science and Technology 2013

[37] F Wu J-F Liu Z-D Wu Y Bian and Z-W Zhou ldquoFractionalnonlinear creep constitutive model of salt rockrdquo Rock and SoilMechanics vol 35 no s8 pp 162ndash167 2014

[38] R F Meng D S Yin C Zhou and H Wu ldquoFractional descrip-tion of time-dependent mechanical property evolution in


materials with strain softening behaviorrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 40 no 1 pp 398ndash406 2016

[39] Q Li K H Wang and K H Xie ldquoRecognition of models andback analysis of parameters for artificial frozen soilsrdquo Rock andSoil Mechanics vol 23 no 11 pp 1895ndash1899 2004

[40] C Zopf S E Hoque and M Kaliske ldquoComparison ofapproaches to model viscoelasticity based on fractional timederivativesrdquo Computational Materials Science vol 98 pp 287ndash296 2015

[41] B-R Chen X-J Zhao X-T Feng H-B Zhao and S-Y WangldquoTime-dependent damage constitutive model for the marblein the Jinping II hydropower station in Chinardquo Bulletin ofEngineering Geology and the Environment vol 73 no 2 pp499ndash515 2014

[42] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[43] T Beda ldquoAn approach for hyperelastic model-building andparameters estimation a review of constitutive modelsrdquo Euro-pean Polymer Journal vol 50 no 1 pp 97ndash108 2014

[44] Y J Song and S Y Lei ldquoMechanical model of rock nonlinearcreep damage based on fractional calculusrdquo Chinese Journal ofUnderground Space and Engineering vol 9 no 1 pp 91ndash1222013

[45] M Caputo and M Fabrizio ldquoDamage and fatigue described bya fractional derivativemodelrdquo Journal of Computational Physicsvol 293 pp 400ndash408 2015

[46] H-Z Sun and W Zhang ldquoAnalysis of soft soil with viscoelasticfractional derivative Kelvin modelrdquo Yantu LixueRock and SoilMechanics vol 28 no 9 pp 1983ndash1986 2007

[47] R D Li and J C Yue ldquoNonlinear rheological constitute of softsoil based on fractional order derivative theoryrdquo Journal of BasicScience and Engineering vol 22 no 5 pp 856ndash864 2014

[48] L-J He L-W Kong W-J Wu X-W Zhang and Y CaildquoA description of creep model for soft soil with fractionalderivativerdquo Rock and Soil Mechanics vol 32 no 2 pp 239ndash2492011

[49] R C Koeller ldquoApplications of fractional calculus to the theoryof viscoelasticityrdquo Journal of AppliedMechanics vol 51 no 2 pp299ndash307 1984

[50] J Ding H Zhou D Liu Q Chen and J Liu ldquoResearch onfractional derivative three elements model of salt rockrdquo ChineseJournal of Rock Mechanics and Engineering vol 33 no 4 pp672ndash678 2014

[51] Z M Yao Y Zhou Y Xu et al ldquoGenetic algorithms fractionalorder derivative burgers creep model for artificial frozen soilrdquoIndustrial Construction vol 43 no 11 pp 73ndash76 2013

[52] Y-G Kang and X-E Zhang ldquoNonstationary parameter frac-tional burgers model of rock creeprdquo Rock and Soil Mechanicsvol 32 no 11 pp 3237ndash3248 2011

[53] X Zhao Identification of fractional viscoelastic constitutiveparameters [PhD thesis] Dalian University of Technology2012

[54] J-H Chen Z-M Yao Y Xu and H-L Wang ldquoParticle swarmfractional order derivative model of artificial frozen soil creeppropertiesrdquo Journal of the China Coal Society vol 38 no 10 pp1763ndash1768 2013

[55] L S Niu Study of laboratory prepared frozen soil property andfractional order derivative Nishiharamodel [PhD thesis] AnhuiUniversity of Science and Technology Huainan China 2014

[56] Y R Liu and H M Tang Rockmass Mechanics ChemicalIndustry Press Beijing China 2008

[57] J Liu Experimental investigation and theoretic analysis on themechanical properies of layered rock salt [PhD thesis] Instituteof Rock and Soil Mechanics Chinese Academy of Sciences2006

[58] G-W Xu C He X-Y Hu and S-MWang ldquoA modified Nishi-hara model based on fractional calculus theory and its param-eter intelligent identificationrdquo Rock and Soil Mechanics vol 36no 2 pp 132ndash147 2015

[59] X F Gen B X Li and T D Miao ldquoStudy on shear stressrelaxation properties of Malan loessrdquo Northwestern Seismologi-cal Journal vol 32 no 1 pp 36ndash41 2010

[60] Q F He Study on the mechanical and rheological propertiesof Yanrsquoan Q2 Loess [PhD thesis] Changrsquoan University XirsquoanChina 2008

[61] H W Zhou C P Wang Z Q Duan M Zhang and J F LiuldquoTime-based fractional derivative approach to creep constitu-tive model of salt rockrdquo Scientia Sinica Physica Mechanica ampAstronomica vol 42 no 3 pp 310ndash318 2012

[62] L-C Liu andW Zhang ldquoDeformation properties of horizontalround adits in viscoelastic rocks with fractional Kelvin modelrdquoRock and Soil Mechanics vol 26 no 2 pp 287ndash289 2005

[63] T Z Yang and B Fang ldquoAsymptotic analysis of an axiallyviscoelastic string constituted by a fractional differentiationlawrdquo International Journal of Non-Linear Mechanics vol 49 pp170ndash174 2013

[64] Z L He Z D Zhu N Wu Z Wang and S Cheng ldquoStudy ontime-dependent behavior of granite and the creep model basedon fractional derivative approach considering temperaturerdquoMathematical Problems in Engineering vol 2016 Article ID8572040 10 pages 2016

[65] H-P Lai S-Y Wang and Y-L Xie ldquoExperimental research ontemperature field and structure performance under differentlining water contents in road tunnel firerdquo Tunnelling andUnderground Space Technology vol 43 pp 327ndash335 2014

[66] J X Lai J L Qiu Z H Feng J Chen andH Fan ldquoPrediction ofsoil deformation in tunnelling using artificial neural networksrdquoComputational Intelligence and Neuroscience vol 2016 ArticleID 6708183 16 pages 2016

[67] C Cao T Ren and C Cook ldquoCalculation of the effect ofPoissonrsquos ratio in laboratory push and pull testing of resin-encapsulated boltsrdquo International Journal of RockMechanics andMining Sciences vol 64 pp 175ndash180 2013

[68] D S Yin X M Duan and X J Zhou ldquoFractional time-dependent deformation component models for characterizingviscoelastic Poissonrsquos ratiordquo European Journal of MechanicsmdashASolids vol 42 pp 422ndash429 2013

[69] L Chen A study on rheological characteristics of soft rocktunnel with a deep overburden [PhD thesis] Southwest JiaotongUniversity Chengdu China 2014

[70] J X Lai J L Qiu J X Chen H Fan and K Wang ldquoNewtechnology and experimental study on snow-melting heatedpavement system in tunnel portalrdquo Advances in MaterialsScience and Engineering vol 2015 Article ID 706536 11 pages2015

[71] J Lai C Guo J Qiu and H Fan ldquoStatic analytical approachof moderately thick cylindrical ribbed shells based on first-order shear deformation theoryrdquo Mathematical Problems inEngineering vol 2015 Article ID 274091 14 pages 2015


[72] J Lai H Fan J Chen J Qiu and K Wang ldquoBlasting vibrationmonitoring of undercrossing railway tunnel using wirelesssensor networkrdquo International Journal of Distributed SensorNetworks vol 2015 Article ID 703980 7 pages 2015

[73] B Brady and E BrownRockMechanic for UndergroundMiningGeorge Allen amp Unwin London UK 1st edition 1985

[74] K-H Xie M-M Lu A-F Hu and G-H Chen ldquoA generaltheoretical solution for the consolidation of a composite foun-dationrdquo Computers and Geotechnics vol 36 no 1-2 pp 24ndash302009

[75] J X Lai H Q Liu J L Qiu and J Chen ldquoSettlement analysis ofsaturated tailings dam treated by CFG pile composite founda-tionrdquo Advances in Materials Science and Engineering vol 2016Article ID 7383762 10 pages 2016

[76] M-H Zhao X-J Zhan and X-J Zou ldquoNumerical simulationof pile installation and subsequent consolidation in clayrdquoJournal of Hunan University Natural Sciences vol 40 no 2 pp1ndash8 2013

[77] Z C Wang Research on constitutive model and its calculationmethod of high-filled embankment rheological settlement [PhDthesis] Xiangtan University Xiangtan China 2011

[78] Y Xie and X Yang ldquoCharacteristics of a new-type geocellflexible retaining wallrdquo Journal of Materials in Civil Engineeringvol 21 no 4 pp 171ndash175 2009

[79] H Zhang Z Wang Y Zheng P Duan and S Ding ldquoStudy ontri-axial creep experiment and constitutive relation of differentrock saltrdquo Safety Science vol 50 no 4 pp 801ndash805 2012

[80] F Song JM Zhang andG R Cao ldquoExperimental investigationof asymptotic state for anisotropic sandrdquo Acta Geotechica vol10 no 5 pp 571ndash585 2015

[81] H N Wang Z P You S W Goh P Hao and X M HuangldquoLaboratory evaluation on the high temperature rheologicalproperties of rubber asphalt a preliminary studyrdquo CanadianJournal of Civil Engineering vol 39 no 10 pp 1125ndash1135 2012

[82] H Tang Z Duan F-S Zhao F Song and X-N Li ldquoCreepdamage model of Q2 loess in consideration of moisture contentbased on fractional calculusrdquo Journal of Yangtze River ScientificResearch Institute vol 32 no 8 pp 78ndash83 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In fact during last decades the field of fractional calcu-lus has attracted increasing interest of researchers in verydiverse scientific fields including mathematics chemistryengineering technology anomalous diffusion and evenmed-ical machinery and robot control [13ndash19] Indeed fractionalcalculus which is defined in the form of calculus with conciseequation has been found to be an accurate mathematicaltool for solving the difficulty during physical and mechanicalmodeling As for the rheology which is featured by timedependency the fractional derivative model can be a perfectapproach for describing rheology phenomenon in particularto the properties of hereditary phenomenawith longmemory[20 21] Over the years after devoting many continuousefforts toward the study on the application of fractionalcalculus by numerous researchers the fractional derivativemodel has gained much deep development Specific to theparameters which are the key point in the process ofapplication and to determine the feasibility of model it canbe obtained through inversion based on experiment datanumerical calculation and software fitting As a distinctapproach fractional derivative model is applied to describethe nonlinear viscoelastic property deformation propertyshear contraction and dilatation and damage growth duringaccelerated stage of creep in geotechnical engineering [9 2223] For more details on the creep phenomena fractionalderivative model is regarded as an effective approach thatcan not only well reflect the process of nonlinear gradualchange at primary stage but also present accurately at steadystage and accelerated stage under high stress However due toits pseudodifferential operator the storage memory propertyof fractional derivative model brings certain difficulty tonumerical calculation despite the fact that the rheologicalprocess curve is beautifully presented [24]

In this paper we systematically investigated currentresearch in the field of fractional derivative model structureand acquisition of parameters where the resulting modelshad been incorporated into creep analysis We began withan introduction to relevant concepts and developments offractional calculus providing an up-to-date review of keyconstitutive models which had been widely applied in thefield of geotechnical engineering Following this we compre-hensively reviewed the state of the art subjected to practicalapplications of fractional derivative model including threemajor aspects of deformation analysis of circular tunnelrheological settlement of the subgrade and relevant loessresearches

2 Model Structure Research

21 Presentation andDevelopment of Fractional Calculus Thefractional calculus has been verified that is a branch ofmathematics which investigates the property of the differen-tiation and integration of any order Its embryonic form canbe dated back to September 30 1695 when Leibniz repliedto the famous question ldquoWhat does the derivative 119889

119899

119891119889119909119899

for 119891(119909) mean if 119899 = 12rdquo proposed by Lrsquo Hospital [25]Due to its complicated calculation and undefined physicalsignificance the focus at early stage is mainly on researchin pure mathematics and for a long period of time the

research on fractional calculus has attracted little attentionfrom scholars in natural science and engineering technologyfield In 1936 Gemant firstly applied fractional derivativesto problems in elasticity [26] Until 1982 Mandelbrot [27]who was the founder of fractal theory pointed out thatthere was a self-similarity phenomenon between integer andfraction and many fractal dimensions existed in natureand many technological sciences Subsequently researchersfound relevant phenomenon or process of fractal geometrypower law phenomenon and memory process which couldestablish a close relationship with fractional calculus thusleading to an explosion of research activities on fractionalcalculus The contribution of Shestopal and Goss [28] wasfirstly applied to fractional derivative in Maxwell and Kelvinmodel for analysis of creep buckling Bagley and Torvik [29]conducted research on three-dimensional constitutive rela-tion of fractional derivative limitation of thermodynamicson model parameters and finite element method Throughutilizing the linear standard system of fractional derivativeZhang [30] established the constitutive relation of nitrilerubber and proved that it was equivalent with Rabotnovmodel Metzler and Nonnenmacher [31] took Zener modelbased on fractional order to explore the abnormal relaxationof polymeric materials Yin et al [32] firstly consideredfractional derivative element as soft-matter element whilebuilding the constitutive model of soil rheology phenomenarepresenting the material between an ideal fluid and an idealsolid

Up to now there are many different definitions of frac-tional calculus [33ndash35] among which basic ideology is tolook at a fractional derivative as the inverse operation ofa fractional integral It is usually chosen as the Riemann-Liouville form

119889minus120574

119891 (119905)

119889119905minus120574=1199050119863minus120574

119905

119891 (119905) = int

119905

1199050

(119905 minus 120585)120574minus1

Γ (120574)119891 (120585) 119889120585

119905 isin [1199050 119887]

(1)

where 119891(119905) is integrable function on the interval [1199050 119887] with

the order 120574 gt 0 119899minus 1 lt 120574 le 119899 (119899 is integer) and Γ(120574) is EulerrsquosGamma function which is defined by

Γ (120574) = int

infin

0

119890minus119905

119905120574minus1

119889119905 (2)

Correspondingly the Riemann-Liouville fractional de-rivative is given by

119889120574

119891 (119905)

119889119905120574=1199050119863120574

119905

119891 (119905) =119889119899

119889119905119899[1199050119863minus(119899minus120574)

119905

119891 (119905)] (3)

Laplace transformation formula of fractional differentialcan be expressed as

119871 [0119863minus120574

119905119891 (119905) 119901] = 119901

minus120574

119891 (119901) 120574 ge 0

119871 [0119863120574

119905119891 (119905) 119901] = 119901

120574

119891 (119901) 120574 le 0

(4)

where 119891(119905) can be integrated near 119905 = 0 with 0 le 120574 le 1 and119891(119901) is Laplace transformation of 119891(119905)


120590 120590

1205780





120601 (119905) =120582119905minus120574

Γ (1 minus 120574) (5)


120590 (119905) = 120578120574119889120574

[120576 (119905)]

119889119905120574(0 le 120574 le 1) (6)




0


120590 120590

1205781

120590s



120576 (119905) =120590

120578120574

119905120574

Γ (1 + 120574)(0 le 120574 le 1) (7)


120590 (119905) = [120578120574

(1 minus 119863120596)]

119889120574

[120576 (119905)]

119889119905120574(0 lt 120574 le 1) (8)


120596=

1 minus 119890minus120572119905

lt 1 and 120578120574

(1 minus 119863120596) = 120578120574







120590120590

1205782E


120590 120590

1205783

E


120590120590

1205783

E1

E2


120590 1205901205782

1205783

E1

E2


120590120590

12057811205783

120590s

E1

E2




200 400 600 800 1000 1200 1400 1600 18000Time (min)

10

15

20

25

30

35

J(t)

(10minus6

GPa

minus1)





120590120590

1205781205783

120590s

E1

E2






120576119894119895

=119904119894119895

2119866+

120590119898

3119896120575119894119895+

119904119894119895

2119866int

119905

0

120581 (119905) 119889119905 (9)




= 1 while 120575119894119895

= 0 in another case






parameters

Kelvin

1

1199020

[1 minus 119864120574(119905)] 119905 ge 0

119864120574(119905) = 1 +

infin

sum

119899=1

(minus1)119899

(119905120578)120574119899

Γ(1 + 120574119899)

3


1

1198640

+1

120578120574

infin

sum

119899=0

(minus1198641120578120574

)119899

119905120574(1+119899)

120574 (1 + 119899) Γ [(1 + 119899) 120574]4

Burgers1

1198641

[1 minus (1 minus1198641+ 1198642

1198642

)

infin

sum

119899=0

(minus1)119899

(1198642

1205782

)

119899+1

119905120574(1+119899)

Γ [120574 (1 + 119899) + 1]] +

1

1205781

119905120574

Γ (119903 + 1)5

Nishihara

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] 120590 lt 120590

119904

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] +

1

12057821205742

1198901199051205742 120590 gt 120590

119904

6


100 200 300 400 500 600 700 8000Time (min)

120576(

)

1

2

3

4

5

6

7

8



120576 (119905) = 119869 (119905) 120590 (10)












120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20


120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20





11198622

Maximumacceleration

80 15 30 30 15












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590 120590

1205780





120601 (119905) =120582119905minus120574

Γ (1 minus 120574) (5)


120590 (119905) = 120578120574119889120574

[120576 (119905)]

119889119905120574(0 le 120574 le 1) (6)




0


120590 120590

1205781

120590s



120576 (119905) =120590

120578120574

119905120574

Γ (1 + 120574)(0 le 120574 le 1) (7)


120590 (119905) = [120578120574

(1 minus 119863120596)]

119889120574

[120576 (119905)]

119889119905120574(0 lt 120574 le 1) (8)


120596=

1 minus 119890minus120572119905

lt 1 and 120578120574

(1 minus 119863120596) = 120578120574







120590120590

1205782E


120590 120590

1205783

E


120590120590

1205783

E1

E2


120590 1205901205782

1205783

E1

E2


120590120590

12057811205783

120590s

E1

E2




200 400 600 800 1000 1200 1400 1600 18000Time (min)

10

15

20

25

30

35

J(t)

(10minus6

GPa

minus1)





120590120590

1205781205783

120590s

E1

E2






120576119894119895

=119904119894119895

2119866+

120590119898

3119896120575119894119895+

119904119894119895

2119866int

119905

0

120581 (119905) 119889119905 (9)




= 1 while 120575119894119895

= 0 in another case






parameters

Kelvin

1

1199020

[1 minus 119864120574(119905)] 119905 ge 0

119864120574(119905) = 1 +

infin

sum

119899=1

(minus1)119899

(119905120578)120574119899

Γ(1 + 120574119899)

3


1

1198640

+1

120578120574

infin

sum

119899=0

(minus1198641120578120574

)119899

119905120574(1+119899)

120574 (1 + 119899) Γ [(1 + 119899) 120574]4

Burgers1

1198641

[1 minus (1 minus1198641+ 1198642

1198642

)

infin

sum

119899=0

(minus1)119899

(1198642

1205782

)

119899+1

119905120574(1+119899)

Γ [120574 (1 + 119899) + 1]] +

1

1205781

119905120574

Γ (119903 + 1)5

Nishihara

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] 120590 lt 120590

119904

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] +

1

12057821205742

1198901199051205742 120590 gt 120590

119904

6


100 200 300 400 500 600 700 8000Time (min)

120576(

)

1

2

3

4

5

6

7

8



120576 (119905) = 119869 (119905) 120590 (10)












120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20


120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20





11198622

Maximumacceleration

80 15 30 30 15












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590120590

1205782E


120590 120590

1205783

E


120590120590

1205783

E1

E2


120590 1205901205782

1205783

E1

E2


120590120590

12057811205783

120590s

E1

E2




200 400 600 800 1000 1200 1400 1600 18000Time (min)

10

15

20

25

30

35

J(t)

(10minus6

GPa

minus1)





120590120590

1205781205783

120590s

E1

E2






120576119894119895

=119904119894119895

2119866+

120590119898

3119896120575119894119895+

119904119894119895

2119866int

119905

0

120581 (119905) 119889119905 (9)




= 1 while 120575119894119895

= 0 in another case






parameters

Kelvin

1

1199020

[1 minus 119864120574(119905)] 119905 ge 0

119864120574(119905) = 1 +

infin

sum

119899=1

(minus1)119899

(119905120578)120574119899

Γ(1 + 120574119899)

3


1

1198640

+1

120578120574

infin

sum

119899=0

(minus1198641120578120574

)119899

119905120574(1+119899)

120574 (1 + 119899) Γ [(1 + 119899) 120574]4

Burgers1

1198641

[1 minus (1 minus1198641+ 1198642

1198642

)

infin

sum

119899=0

(minus1)119899

(1198642

1205782

)

119899+1

119905120574(1+119899)

Γ [120574 (1 + 119899) + 1]] +

1

1205781

119905120574

Γ (119903 + 1)5

Nishihara

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] 120590 lt 120590

119904

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] +

1

12057821205742

1198901199051205742 120590 gt 120590

119904

6


100 200 300 400 500 600 700 8000Time (min)

120576(

)

1

2

3

4

5

6

7

8



120576 (119905) = 119869 (119905) 120590 (10)












120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20


120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20





11198622

Maximumacceleration

80 15 30 30 15












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












parameters

Kelvin

1

1199020

[1 minus 119864120574(119905)] 119905 ge 0

119864120574(119905) = 1 +

infin

sum

119899=1

(minus1)119899

(119905120578)120574119899

Γ(1 + 120574119899)

3


1

1198640

+1

120578120574

infin

sum

119899=0

(minus1198641120578120574

)119899

119905120574(1+119899)

120574 (1 + 119899) Γ [(1 + 119899) 120574]4

Burgers1

1198641

[1 minus (1 minus1198641+ 1198642

1198642

)

infin

sum

119899=0

(minus1)119899

(1198642

1205782

)

119899+1

119905120574(1+119899)

Γ [120574 (1 + 119899) + 1]] +

1

1205781

119905120574

Γ (119903 + 1)5

Nishihara

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] 120590 lt 120590

119904

1

1198640

+1

1198641

[1 minus

infin

sum

119899=0

(minus1)119899(11990511986411205781)120574119899

Γ (1 + 120574119899)] +

1

12057821205742

1198901199051205742 120590 gt 120590

119904

6


100 200 300 400 500 600 700 8000Time (min)

120576(

)

1

2

3

4

5

6

7

8



120576 (119905) = 119869 (119905) 120590 (10)












120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20


120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20





11198622

Maximumacceleration

80 15 30 30 15












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120576(

)

2 4 6 80Time (h)

0

4

8

12

16

20


120576(

)


2 4 6 80Time (h)

0

2

4

6

8

10

12

14

16

18

20





11198622

Maximumacceleration

80 15 30 30 15












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11205782

120574 120590119904


Clay





Optimal particle

Optimal particle



AB

xt+1i

tk+1

xti tk








120596 () 1198640(105 kPa) 120578 (105 kPasdotmin) 120574

5 3029 57093120590minus05813 00109120590

minus03369

7 1361 11068120590minus03529 00337120590

minus01172

10 1109 16234120590minus03711 00496120590

minus00964

12 0787 31993120590minus01921 00265120590

minus02053

15 0577 3166120590minus03786 00122120590

minus03588

18 0419 49773120590minus03832 0026120590

minus01839

20 0311 25315120590minus02093 00424120590

minus01495

22 0229 12174120590minus01585 00238120590

minus02741


Fitn

essF

(x)

(120583120576)

00

02

04

06

08

10



2loess under


120576 (119905) = 119869se (119905) 120590 = (1

1198640

+1

120578

119905120574

intinfin

0

119890minus119905119905120574119889119905)120590 120590 lt 120590

119904 (11)








120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120576 (119905) =

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) 120590 le 120590119904

120590

1198640

+120590

1198641

minus120590

1198641

1198641205741

(minus1198641

120578120574

1

119905120574

) +120590 minus 120590119904

120578120574

2

119905120574

11986411+120574

(120572119905) 120590 ge 120590119904

(12)

where 119864119909119910


(119911) = 119890119911 and 119864

12= (119890119911


120576 (119905)

=

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) 120590 le 120590

119904

120590

1198640

+120590

1198641

(1 minus 119890minus(11986411205781)119905) +

120590 minus 120590119904

1205782

119905 120590 ge 120590119904

(13)






119906 (119905) =1198862

1205900

21198661199031 minus 119864

119886[minus (

119905

120591)

120574

]

120576119903(119905) = minus

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

120576120579(119905) =

1198862

1205900

211986611990321 minus 119864

119886[minus (

119905

120591)

120574

]

(14)



10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

15

20

25

30

35

40St

rain

()

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120572 = 001 hminus1


120574 = 07

120574 = 09


10

15

20

25

30

35

40

Stra

in (

)

50 100 150 200 250 300 3500Time (h)

120590 = 18MPa 120574 = 07


120572 = 001 hminus1

120572 = 0007 hminus1

120572 = 0hminus1



2R

4R

Support (closed)

Spatial effect

Tunnel face





119880119903(119905) =

12059001198872

2119903[

1

119866+

1

120578

119905120574

Γ (1 + 120574)] (0 le 120574 le 1) (15)




0has




119880119887(119905) =

1205900119887

119870119904+ 2119866lowast (119905)

[1 +119870119904

2119866] (16)


119870119904=

1198641015840

(1198872

minus 1198862

)

(1 + ]1015840) [(1 minus 2]1015840) 1198872 + 1198862] (17)




200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











200 400 600 800 1000 12000Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

5

10

15

20

25

30

(a) 0ndash1000 days


1 100 1000 1000010Time (days)

The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

10

20

30

40

50

60

70

80

90

100




a

bo



119905rarrinfin119866(119905) =

lim119905rarrinfin





The d

ispla

cem

ent o

f tun

nel s

idew

allU

b(c

m)

0

2

4

6

8

10

12

14

100 10000 1000000 1E8 1E101Time (days)







o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











o t

ɛ


200 kPa

100 kPa



0+Δ120590 from



119878 =

119899

sum

119894=1

Δ119878119894= 119902

119899

sum

119894=1

Δ120578119894(119909 119911)

sdot 1

1198640119894

+1

1198641119894

1 minus 119864120574119894 1

[minus120572120574119894

119894(

119905

120591)

120574119894

]119867119894

(18)




120576(

)


00

05

10

15

20

25

30

35

40

100 200 300 400 500 600 700 8000Time (h)

120590 = 518kPa

120590 = 439kPa

120590 = 362kPa120590 = 268kPa

120590 = 175kPa






2loess was


120576 (119905 120596)

=

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741) 120590 le 120590

119904

120590

119864lowast119867

+120590

120578lowast1

1199051205741

Γ (1 + 1205741)+

120590 minus 120590119904

1205782

1199051205742

Γ (1 + 1205742) 120590 ge 120590

119904

(19)

where 119864lowast

119867= 119864119867[minus119863120576119867(120596)

] 120578lowast1

= 1205781[1 minus 119863

120576119867(120596)] 0 lt 120574

1lt 1















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Competing Interests


Acknowledgments


References






























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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