new analytical method to evaluate casing integrity during

Research ArticleNew Analytical Method to Evaluate Casing Integrity duringHydraulic Fracturing of Shale Gas Wells

Jun Li1 Xueli Guo 12 Gonghui Liu13 Shuoqiong Liu2 and Yan Xi1

113e College of Petroleum Engineering China University of Petroleum Beijing China2CNPC Engineering Technology RampD Company Limited Beijing China3Beijing University of Technology Beijing China

Correspondence should be addressed to Xueli Guo clouder0713163com

Received 1 July 2018 Accepted 25 March 2019 Published 24 April 2019

Academic Editor Francesco Pellicano

Copyright copy 2019 Jun Li et al -is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An accurate analysis of casing stress distribution and its variation regularities present several challenges during hydraulicfracturing of shale gas wells In this paper a new analytical mechanical-thermal coupling method was provided to evaluate casingstress For this new method the casing cement sheath and formation (CCF) system was divided into three parts such as initialstress field wellbore disturbance field and thermal stress field to simulate the processes of drilling casing cementing andfracturing -e analytical results reached a good agreement with a numerical approach and were in-line with the actual boundarycondition of shale gas wells Based on this new model the parametric sensitivity analyses of casing stress such as mechanical andgeometry properties operation parameters and geostress were conducted during multifracturing Conclusions were drawn fromthe comparison between new and existing models -e results indicated that the existing model underestimated casing stressunder the conditions of the geostress heterogeneity index at the range of 05ndash225 the fracturing pressure larger than 25MPa anda formation with large elastic modulus or small Poissonrsquos ratio-e casing stress increased dramatically with the increase of in situstress nonuniformity degree-e stress decreased first and then increased with the increase of fracturing pressure -icker casinghigher fluid temperature and cement sheath with small modulus large Poissonrsquos ratio and thinner wall were effective to decreasethe casing stress-is newmethod was able to accurately predict casing stress which can become an alternative approach of casingintegrity evaluation for shale gas wells

1 Introduction

During the multistage fracturing process fracturing fluidsare pressed into a borehole with a high pump rate andpressure -e complex downhole environmentsmdashhighpressure and large temperature variationmdashincrease the riskof casing deformation -e volume fracturing techniqueeffectively reconstructs shale reservoirs however frequentand serious casing deformation failures occur [1] -erewere over 36 wells with casing deformation (including 112horizontal wells by 2017) during fracturing process in someshale gas plays Drilling tools were blocked and seriousdeformation sections were abandoned before fracturingoperation completion [2ndash4] -erefore casing integrity hasalways been themain issue in designing shale gas wells under

harsh conditions and accurately calculating casing stressbecomes primary to guarantee the casing safety A casingstress analysis presents several challenges regarding thedrilling completion and fracturing phases of shale gas wellsMany scholars have predicted the casing stress by simpli-fying the actual situations

Analytical solutions have been developed under dif-ferent conditions for casing-cement sheath-formationsystem In-plane and out-of-plane analyses for the stressfield around an internally pressurized cased cementedand remotely loaded circular hole were developed [5]Taking the in situ stresses and well trajectory into accountthe mechanical model of casing in the directional wellunder in situ stresses is established [6] Other scholars gavea comprehensively analytical solution of the stress

HindawiShock and VibrationVolume 2019 Article ID 4253241 19 pageshttpsdoiorg10115520194253241

distribution in a casing-cement-formation system under an-isotropic in situ stresses Fang et al [7] developed a multilayercemented casing system in the directional well under aniso-tropic formation to investigate the collapse resistance of casingunder nonuniform in situ stress and anisotropic formationBased on the stress function method a three-dimensionalmodel of the casing-cement sheath-formation (CCF) systemwas proposed subjected to linear crustal stress and then ananalytical solution of the model was obtained [8]

A finite element method (FEM) had been proposed toachieve a better understanding of the ultimate collapsestrength of casing [9] Using this method casing stress wasbetter analyzed subjected to external and nonuniformloading caused by void and pressure [10] -e results of aseries of finite element studies of the cemented casing undera variety of stress conditions for both burst and collapse werepresented demonstrating the inadequacy of accepted designequations under many cemented conditions [11] Apurpose-built finite element model was applied to simulatethe radial displacement of a casing string constrained withinan outer wellbore [12] Casing stress on the inner wall underthe condition of elliptic casing was calculated and analyzedto improve the designing and correction of casing strength[13] -e main controlling factor to the plastic limit load ofdefective casing was analyzed by the combination of hightemperature and high steam injection pressure [14]

However the above analytical or FEM solutions areobtained by setting the casing and cement together instantlyat the beginning of the analysis -e strains induced by theinitial stress are included in the model which is not in-linewith the actual situation In fact initial stress has alreadyexisted in the formation before wellbore excavation -ewellbore stress redistributed after removing the rocks thatoriginally occupied the borehole volume and just affected thestress and displacement near wellbore zones In view of thismore sophisticated solutions have been developed withadditional parameters and appropriate assumptions re-garding the drilling completion and fracturing phases

Pattillo and Kristiansen [15] developed a finite modelstarting with the virgin reservoir which considered pre-wellbore depletion drilling the wellbore installation of casingand cement and subsequent draw down Behavior of variousconfigurations during subsequent draw down permitted themto be ranked according to the life expectancy of the resultingcompletion Gray et al [16] presented a staged-finite elementprocedure during the well construction which consideredsequentially the stress states and displacements at and near thewellbore But temperature flow and poroelasticity effectswere not included Mackay and Fontoura [17] carried anumerical analysis to determine the effect of salt creep andcontacts amongst the materials before during and afterdrilling the well focusing on the drilling of the wellbore andon the hardening of the cement Zhang et al [18] used thestaged-finite element modeling approach to simulate the wellconstruction processes and injection cycle using a ldquorealisticrdquobottom-hole state of stress to simulate the microannuligeneration by the tensile debonding of the cement-formationinterface Simone et al [19] developed an analytical solutionof single and double casing configurations to assess the

stresses during the drilling construction and productionphases But the nonuniform boundary stress was not con-sidered in this model Liu et al [20] presented an analyticalmethod for evaluating the stress field within a casing-cement-formation system of oilgas wells under anisotropic in situstresses in the rock formation and uniform pressure withinthe casing However the temperature was not considered inthis model

In this work to evaluate the thermal and mechanicalstresses of casing an analytical model of casing-cementsheath-formation system was established considering well-bore construction -e boundary stresses in the wellborecoordinate system were obtained through three-dimensionalrotations from principal in situ stresses-e casing stress wasobtained by dividing the model into three parts such asinitial stress field disturbance stress field and thermal stressfield -e continuous homogeneous isotropy and linearelasticity were taken into consideration Solutions werevalidated by a finite element method Sensitivity analyseswere conducted to estimate the influence of different factorson casing stress such as property of cement sheath andcasing fracturing pressure fluid temperature and initialgeostress Useful countermeasures were put forward todecrease casing stress during the fracturing operation

2 Method and Basic Conditions Analysis

21 Method Comparison

211 Existing Method For existing method the casingcement sheath and formation were set together at the be-ginning of analysis -en temperature boundary and loadswere added to the system to calculate wellbore stress dis-tribution as shown in Figure 1-is method did not considerthe process of wellbore construction and hydraulic frac-turing So the wellbore stress distribution could not accu-rately be calculated under the condition of formation straininduced by the initial in situ stress

212 New Method To exactly predict the wellbore stressdistribution a casing-cement sheath-formation model usinga new analysis method was provided -e analysis processwas divided into four steps (Figure 2) Before drilling theinitial stresses such as normal stress and shear stress wereloaded in the model and initial strains were produced -econstrained boundary conditions were assigned to the entiremodel which reached initial mechanical equilibrium Duringdrilling the rock was excavated causing stress concentrationaround the borehole -e drilling mud pressure Pm wasapplied to the internal face of the wellbore During casingand cementing casing and cement sheath elements wereadded simultaneously to the model and the cement hard-ening procedure was not considered Initial stress and strainin casing and cement sheath were ignored-e cement slurrypressure Pc was applied to the internal casing wall Duringfracturing a low temperature Tn and the fracturing pressurePf were assigned in the internal casing wall -e wellborestress field was obtained under the condition of thermal-mechanical coupling

2 Shock and Vibration

22 Stress Transformation -e stress state and coordinatetransformation system are shown in Figure 3 -e co-ordinate rotation processes from the principal in situ stresscoordinate system to the wellbore coordinate systemare shown as XHYhZy⟶ XprimeYprimeZprime ⟶ XPrimeYPrimeZPrime ⟶XPrimeprimeYPrimeprimeZPrimeprime ⟶ XYZ first rotating anticlockwise φ aroundthe Zv-axis and rotating clockwise β around the Zv-axissecond rotating anticlockwise α around the YPrime axis (afterthe second time rotation) and finally rotating anticlockwise90deg around the ZPrimeprime axis (after the third time rotation)

In the principal stress coordinate system the principalhorizontal stress matrix is σ0 whose components are maxi-mum principal stress σH the minimum principal stress σhand the overburden pressure σv shown in equation (2) -estress matrix in the wellbore coordinate system is σAccording to the right-hand rule the direction cosine ma-trices rotating around X-axis Y-axis and Z-axis are shown inequation (1) [21] -e wellbore boundary stress is obtained bythe three-dimensional rotations from the principal in situstress coordinate system shown in equation (3)

Cxαx

1 0 0

0 cos αx sin αx

0 minussin αx cos αx

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cyαy

cos αy 0 minussin αy

0 1 0

sin αy 0 cos αy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Czαz

cos αz sin αz 0

minussin αz cos αz 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

σ0

minusσH 0 00 minusσh 00 0 minusσv

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

σ Cz90CyαCzβCzjσ0CT

zjCTzβC

TyαC

Tz90 (3)

Foramtion ForamtionDrilling fluid

Cement sheathFracturing fluid

ForamtionCasing


ForamtionCasing

Initial state Drilling Casing and completing Fracturing

Y

σy

σxX

τyx

τxy

Tf

TnPf

Y

σy

σxX

τyx

τxy

Tf

Tn

Pm

Y

σy

σxX

τyx

τxy

Tf

Tn

Pc

Y

σy

σxX

τyx

τxy

Tf

TnPf

Figure 2 Loading process of the new method Outer boundary temperature Tf inner boundary temperature Tn normal stress σx and σyshear stress τxy drilling mud pressure Pm cement slurry pressure Pc fracturing pressure Pf

Before loading

Y Y

σy

σxX

τyx

τyx

Tf

TnPfX

After loading

ForamtionCasing

Cement sheathFluid

Figure 1 Loading process of the existing method Outer boundary temperature Tf inner boundary temperature Tn normal stress σx andσy shear stress τxy

Shock and Vibration 3

where Cxαx Cyαy and Czαz are the coordinate rotationmatrices of the x y and z directions σ0 and σ are the stressmatrices in the principal stress coordinate and local wellborecoordinate systems and αx αy and αz are the rotation anglesin a counterclockwise direction when looking towards theorigin coordinate

23 BasicHypotheses A thermo-pressure coupling model ofcasing-cement sheath-formation (CCF) system was estab-lished (Figure 4) -e boundary stresses of σx σy and τxywere obtained by using equation (3) Compared to thelongitude of the well the radial dimension was very smalland a long cylindrical model was loaded by forces that wereperpendicular to the axial line and did not vary in lengthBoth the geometric form of the object and the external loadsexerted on the object did not change along the longitudinal(z-axis) direction and that the length of the object might betreated as an infinite one -ere were no additional re-strictions on the external loads -e wellbore stress wasobtained under this kind of stress state called the plane strainproblem [22 23]

For simplicity some assumptions have been made [24]

(1) Geometry the casing cement and borehole wereconcentric circles which were assumed to be per-fectly bonded to each other at each interface -eperfect bonding mathematically indicated that thecontinuity of radial stress and displacement wassatisfied at each interface

(2) -ermal effect the stress induced by wellboretemperature variation was assumed to be steady stateand the time effect was ignored

(3) Material to simplify the complex property of stronganisotropy and well-developed bedding planes ofshale formation [25] the formation was assumed tobe a linear elastic material with an infinite radius (R4⟶ infin) -e cement sheath was also a complexmaterial -e 3D images revealed the evolution of a

large connected pore network with characteristicwidths on the micrometer scale as hydration pro-ceeded [26] It was assumed to be an elastic materialneglecting the complex microstructure It is gener-ally accepted that the casing was an elastic-plasticmaterial and the casing yielding had nothing to dowith hydrostatic pressure In the model it was as-sumed that the deformation of the casing was withinthe elastic range

24 Stress Superposition -e boundary compression normalstresses (minusσx minusσy) were decomposed into uniform stress (p0)and deviator stress (s) expressed as equation (4) in theCartesian coordinate system -e other boundary stress wasshear stress (τxy)

minusσx 0

0 minusσy

⎡⎣ ⎤⎦ minusp0 0

0 minusp01113890 1113891 +

minuss 0

0 s1113890 1113891 (4)

where the uniform stress p0 (σx + σy)2 and the deviatorstress s (σx minus σy)2

According to the basic hypotheses in Section 22 thedeformation history of all phases in the CCF system wasindependent with each other -e principal of the linearsuperposition for the stress was applied as shown in thefollowing equation

σistress σprimeiUniformminusstress + σPrimeiDeviatorminusstress + σPrimeprimeiShearminusstress

+ σTiThermalminusstress

(5)

-e stress distribution of the thermal-pressure cou-pling model around a wellbore was decomposed into fourparts as shown in Figure 5 -e stress field of the first partwas induced by the uniform stress and inner casingpressure -e second one was induced by the deviatorstress and the third one was induced by the shear stress-ermal stress of the fourth part was induced by thetemperature variation

Wellbore trajectory

σv

σh

σH

YPrime

Yprimeh (East)

Xprimeh (North)

YPrime Y XXPrime

XPrime

ZPrime Z ZvZ Z

X

ZPrime

Y

Oβ

α

φ

Xh

XH

Principle in-situ stress

Figure 3 -e coordinate rotation processes (a) stress state of a horizontal well and (b) coordinate transformation system Principal in situstress coordinate system XHYhZy geodetic coordinate system XprimeYprimeZprime coordinate system XPrimeYPrimeZPrime after the second time rotationcoordinate system XPrimeprimeYPrimeprimeZPrimeprime after the third time rotation local wellbore coordinate system XYZ wellbore deviation angle α wellboreazimuth angle β the maximum horizontal stress azimuth angle φ


It was convenient to convert the Cartesian coordinatesystem into the polar coordinate system to calculate wellborestress -e normal boundary stresses in the polar coordinatesystem were expressed as follows under the conditions of theinfinite outer boundary radius of R4

σprime30r

1113868111386811138681113868rinfin minusp0 minus s cos(2θ)

τprime30rθ

11138681113868111386811138681113868rinfin s sin(2θ)

⎧⎪⎨

⎪⎩(6)

where σprime30r and τprime30rθ are the initial normal and shear stresses inthe formation respectively

Since temperature and stress were coupled the stressdistribution around a cased wellbore induced by tempera-ture variation was hard to solve in the closed form Howeverthe steady-state condition made the temperature and stressdecouple and the problem analytically solvable [27]

3 Stress Distribution around Wellbore

31 Stress Induced byUniformStress Under the condition ofthe uniform internal pressure and external stress the stress

and displacement in a thin wall cylinder were obtained byusing the following equations shown in Figure 6

uprimei

r 12Gi

1minus 2μi( 1113857Aiprimer + Ciprime1r

1113876 1113877qi

+12Gi

1minus 2μi( 1113857Biprimer + Ciprime1r

1113876 1113877qi+1 minus rεprimei0r

(7)

σprimeir Aiprime minusCiprime 1r2

1113874 1113875qi + Biprime minusCiprime 1r2

1113874 1113875qi+1

σprimeiθ Aiprime + Ciprime 1r2

1113874 1113875qi + Biprime + Ciprime 1r2

1113874 1113875qi+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(8)

where σprimeir σprimeiθ and uprimei

r are the radial stress tangential stressand radial displacement respectively σprimei0r σprimei0θ and uprime

i0r are

the initial radial stress tangential stress and displacement Eiis the material elastic modulus μi is the material Poissonrsquosratio Gi Ei((1 + μi)2) is the material shear modulus qiqi+1 were the interfacial pressure positive in the radial in-crease direction i 1 2 3 represented the casing cement

Y

Y

R4

R3 R2 R1

σy

σx

TnPf σx

σy

X

τyx

τyx

τxy

Tf

θ

Z

X

ForamtionCasing

Cement sheathFluid

Figure 4 CCF composite assembly Formation boundary temperature Tf internal casing temperature Tn internal casing pressure Piradius Ri i 1 2 3 4 present the radii of the internal casing wall outer casing wall internal wellbore and formation boundary respectivelythe counterclockwise angle from the x-direction to the calculated point θ

Shear stress Thermal stressUniform stress Deviator stress

Y

R4

R3 R2 R1

Pf

p0

p0

X R3 R2 R1R3 R3

R2 R2R1 R1

R4 R4 R4

Tn

Tfndashs

s

Y Y Y

X X

τyx

τxy

X


ForamtionCasing


ForamtionCasing


ForamtionCasing


ForamtionCasing

Figure 5 Stress decompositions Inner casing pressure Pi thermal stress σT


sheath and formation respectively Aiprime R2

i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2

i ) Ciprime AiprimeR2

i+1 are the constants Ri (i 12 3 4) is the radii of internal casing wall external casingwall external cement sheath wall and formation boundaryrespectively

311 Formation Stress Before drilling the borehole theinitial geostress field already existed in the formation Whenthe rock was removed from the borehole the wellbore stressfield redistributed to produce a disturbance field which onlyaffected the near-wellbore zones [28] So the model wasdecomposed into two parts such as the original field and thedisturbance field -e original field had initial stress anddisplacement-e disturbance field was induced by drilling awellbore and mud pressure In view of this the actual stressfield of F1 in the formation induced by the uniform stressand internal pressure was decomposed into three parts asshown in Figure 7 -ey were the original stress field of A1the excavation disturbance field of B1 induced by drilling ofa wellbore and the interface disturbance field of C1 inducedby the fluid column pressure

In the polar coordinate system the initial conditions ofA1 were σ30r σ30θ minusp0 εprime30r p02G3(1minus 2μ3) andboundary stress conditions of B1 andC1 were σprime3r |rR3 B1 p0and σprime3r |rR3 C1 minusp3prime Substituting the initial and boundaryconditions into equations (7) and (8) the displacement andstress in formation were obtained as shown below BecauseR4 approached to infinity A3prime 0 and C3prime R2

3 wereobtained

uprime3

r (r) 1

2G3

R23

rp3prime minusp0( 1113857

σprime3r (r) minusp0 minusR23

r2p3prime minusp0( 1113857

σprime3θ (r) minusp0 +R23


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)

where uprime3r was the radial displacement in formation and uprime3rand σprime3θ were the radial and tangential stresses in formationrespectively

312 Casing and Cement Sheath Stress -e pressures atcasing-cement sheath interface and cement sheath-formation interface were p2prime and p3prime respectively (Fig-ure 8) -e initial stresses of the casing and the cementsheath were σprimei0r σprimei0θ 0 and εprimei0r 0 -e boundary stressconditions were σprime2r |rR3

minusp3prime and σprime2r |rR2 minusp2prime

σprime1r |rR1 minuspi

Substituting these initial and boundary conditionsin equations (7) and (8) the displacement and stress incasing and cement sheath were obtained as follows Sub-scripts 1 and 2 represent the casing and cement sheathrespectively

uprime1r 1

2G11minus 2μ1( 1113857A1primer + C1prime

1r

1113876 1113877pi

+1

2G11minus 2μ1( 1113857B1primer + C1prime

1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

σprime1r A1prime minusC1prime1r2

1113874 1113875pi + B1prime minusC1prime1r2

1113874 1113875 minusp2prime( 1113857

σprime1θ A1prime + C1prime1r2

1113874 1113875p2prime + B1prime + C1prime1r2

1113874 1113875 minusp2prime( 1113857


1113874 1113875p2prime + B2prime minusC2prime1r2

1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

According to the hypotheses that cement sheath-formation interface and casing-cement sheath interfacewere perfectly bonded to each other the interfacial dis-placement continuity conditions were expressed in thefollowing equation

uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)

Substituting equation (10) into equation (12) the binaryequations were obtained as

AAp2prime minusBBp3prime CCpi

DDp2prime minusEEp3prime FFp0

⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1

Figure 6 Stress induced by the uniform stress


where

AA 1

2G11minus 2μ1( 1113857B1primeR2 + C1

1R2

1113890 1113891

+1

2G21minus 2μ2( 1113857A2primeR2 + C2prime

1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1

2G11minus 2μ1( 1113857B1primeR3+C1

1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

-e interfacial pressures p2prime and p3prime could be calculated byusing equation (14) Substituting them into equation (11)the stresses induced by uniform stress were obtainedsubsequently

32 Stress Induced by Deviator Stress -e deviator stressboundary conditions are shown in Figure 9 To calculate thestress distribution induced by deviator stress the stressfunction was defined as

ϕ APrimei r4

+ BPrimei r2

+ CPrimei +DPrimeir2

1113888 1113889cos(2θ) (15)

-e stress and strain under the condition of nonuniformstress are

σPrimeir minus 2BPrimei +4CPrimeir2

+6DPrimeir4

1113888 1113889cos 2θ

σPrimeiθ 12APrimei r2 + 2BPrimei +6DPrimeir4

1113888 1113889cos 2θ

τPrimeirθ 6APrimei r2 + 2BPrimei minus2CPrimeir2minus6DPrimeir4

1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimeir minus1 + μi

Ei

12υiAPrimei r

2+ 2BPrimei + 1minus μi( 1113857

4CPrimeir2

+6DPrimeir4

1113890 1113891

middot cos 2θminus εPrimei0r

εPrimeiθ 1 + μi

Ei

12 1minus μi( 1113857APrimei r2

+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891

middot cos 2θminus εPrimei0θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0

Figure 7 Wellbore stress components under the condition of uniform stress

Casing

pi p0pprime2 pprime3

Cement sheath Formation

Figure 8 Interface pressures induced by uniform stress pcos2θ scos2θ

RiRi+1

Figure 9 Stress induced by deviator stress Outer stress s cos 2θinterface pressure p cos 2θ


where σPrimeir and σPrimeiθ are the radial and tangential stressesεPrimei0r and εPrimei0θ are the initial radial and tangential strains andAPrimei BPrimei CPrimei andDPrimei were the constants i 1 2 3 representedthe casing cement sheath and formation

From the geometric equations

εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)

-e radial displacement uPrimei

r and tangential displacementuPrime

i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r

3+ 2BPrimei rminus 1minus μi( 1113857

4CPrimeirminus2DPrimeir3

1113890 1113891

middot cos 2θminus rεPrimei0r

uPrimei

θ 1 + μi

2Ei

4 3minus 2μi( 1113857APrimei r3

+ 4BPrimei rminus 1minus 2μi( 11138574CPrimei

r+4DPrimeir3

1113890 1113891

middot sin 2θ + r 1113946 εPrimei0r minus εPrimei0θ1113874 1113875dθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei

1minus μi( 1113857σPrimei0r minus μiσPrimei0θ1113876 1113877

εPrimei0θ 1 + μi

Ei

1minus μi( 1113857σPrimei0θ minus μiσPrimei0

r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

where σPrimei0r and σPrimei0θ were the initial radial and tangentialstresses

321 Formation Stress Similar to that of uniform stress theactual stress field F2 in the strata induced by the non-uniform stress was decomposed into three parts theoriginal stress field A2 the disturbance field B2 induced bythe wellbore excavation and the interface pressure C2 in-duced by the interface pressure (Figure 10)

In the polar coordinate system initial stresses wereσPrime30r minuss cos(2θ) σPrime30θ s cos(2θ) and τPrime30rθ s sin(2θ)initial strains were εPrime30r minus(1 + μ3)E3 middot s cos(2θ) andεPrime30θ (1 + μ3)E3 middot s cos(2θ) and the boundary stresses wereσPrime3r |rinfin minuss cos(2θ) σPrime3θ s cos(2θ) and τPrime3rθ |rinfin

s sin(2θ) Substituting the initial and boundary conditionsinto (14) and (15) it was obtained that APrime3 0 andBPrime3 S2-e displacements and stresses in formation were expressed asshown in the following equations

uPrime3r minus1

G3minus 1minus μ3( 1113857

2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1

G3minus 1minus 2μ3( 1113857

CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)

σPrime3r minus s +4CPrime3r2

+6DPrime3r4

1113888 1113889cos 2θ

σPrime3θ s +6DPrime3r4

1113888 1113889cos 2θ

τPrime3rθ sminus2CPrime3r2minus6DPrime3r4

1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)

322 Casing and Cement Sheath Stress For casing andcement sheath in the polar coordinate system initial stresseswere σPrimei0r σPrimei0θ 0 and initial strains were εPrimei0r εPrimei0θ 0Substituting the initial and boundary conditions intoequations (14) and (15) the displacements and stresses wereobtained as follows

uPrimei

r minus1Gi

2μiAPrimei r

3+ BPrimei rminus 1minus μi( 1113857

2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi

3minus 2μi( 1113857APrimei r3

+ BPrimei rminus 1minus 2μi( 1113857CPrimeir

+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

-e interfacial displacement and stress continuity andboundary conditions were expressed in the followingequation

σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)


Substituting equations (20)ndash(23) into the followingequation equations were obtained as

minus2BPrime1 minus4CPrime1

R21minus6DPrime1

R41

1113888 1113889cos 2θ 0

6R21APrime1 + 2BPrime1 minus

2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)

minus 2BPrime1 +4CPrime1

R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0

6APrime1R22 + 2BPrime1 minus

2CPrime1

R22minus6DPrime1

R42

1113888 1113889

minus 6APrime2R22 + 2BPrime2 minus

2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)

minus BPrime2 +2CPrime2

R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss

3APrime2R23 + BPrime2 minus

CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B

Prime1 minus 1minus μ1( 1113857

2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1

3minus 2μ1( 1113857R32APrime1 + R2B

Prime1 minus 1minus 2μ1( 1113857

1R2

CPrime1 +1

R32DPrime11113890 1113891minus

1G2

middot 3minus 2μ2( 1113857R32APrime2 + R2B


1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where the constants of APrime1 BPrime1 CPrime1 DPrime1 APrime2 BPrime2 CPrime2 DPrime2 CPrime3

andDPrime3 were calculated by the total 10 equations in equa-tions (24)ndash(28) -en wellbore stress distribution inducedby deviator stress was obtained by substituting these 10constants and APrime3 andBPrime3 into equations (19)ndash(22)

33 Stress Induced by Shear Stress -e stress induced byshear stress was uPrimeprime

ir uPrimeprime

i

θ σPrimeprimeir σPrimeprimeiθ and τPrimeprimeirθ i 1 2 3 repre-sented the casing cement sheath and formation re-spectively (Figure 11) -e angle of Ω between σx and x-direction was calculated by using equation (29) -en theprincipal stresses were obtained as follows [29]

Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy

σminusπ4 minusτxy1113896 (30)

It could be found that the stress distribution induced byshear stress was similar with that by deviator stress whencounterclockwise rotating the angle of π4 -e stresses anddisplacements were obtained by substituting the referencevariable θ θprime(minusπ4) into the stress induced by deviatorstress discussed in Section 32

34 Stress Induced by Temperature Variation -e thermalfield was obtained by using the steady temperature distri-bution model to calculate the thermal stress When frac-turing fluids were pumped into a wellbore with a high pump

A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1

Figure 10 Formation stress components under the nonuniform stress condition


rate they were always in the turbulent state -e heattransfer coefficient between casing and fluid was calculatedusing the Marshall model [30] shown in the followingequation

h Stkm

D 00107

km

D

ρaDeff 4QπD2( 1113857

K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)

where h is the heat transfer coefficient (Wmiddotmminus2middotdegCminus1) St is theStanton number Pr is the Prandtl number Reg is theReynolds number μwapp is the fluid apparent viscosity D isthe inner diameter (m) Deff is the equivalent diameter (m)ρa is the fluid density (kgmiddotmminus3) n is the liquidity index K isthe consistency coefficient (Pamiddotsn) v is the fluid velocity Q isthe fracturing pump rate (m3middotminminus1) km is the coefficient ofheat conductivity (Wmiddotmminus1middotdegCminus1) and Cm is the fluid specificheat capacity (Jmiddotkgminus1middotdegCminus1)

-e temperature distribution among casing cementsheath and formation is shown in Figure 12 In the cylin-drical coordinate system of CCF the differential equationof steady heat conduction of the cylinder is expressed as [31]

d2T

dr2+1r

dT

dr 0 (32)

Temperature field distribution solutions were obtainedaccording to integral and boundary conditions kidTdr

hn(Ti minusTn) T|rRi Ti T|rRi+1

Ti+1 shown in the follow-ing equation

Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn

ln Ri+1Ri( 1113857 + i22( )minus(5i2) + 3( )kiRih

BTi

Tn lnRi+1 minusTi+1 lnRi + i22( 1113857minus(5i2) + 3( 1113857 kiRih( 1113857Ti+1

ln Ri+1Ri( 1113857 + i22( )minus(5i2) + 3( ) kiRih( 1113857

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)

where Ti is the temperature (degC) Ti is the temperature at theinterface (degC) Tn is the fluid temperature (degC) ki is thematerial thermal conductivity (Wmiddotmminus1middotdegCminus1) Ri is the radius

(m) and ATi andBT

i were the constants i 1 2 3 representedcasing cement sheath and formation respectively

-e heat flow density continuity conditions wereexpressed as

ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)

-e temperatures at interfaces of casing-cement sheathand cement sheath-formation system were defined as T2 andT3 and were calculated by using the following equation

1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)

Interfacial temperature of Ti was obtained by solvingequation (36) -e steady-state temperature field around thewellbore could be calculated by substituting Ti into equa-tions (33) and (34) According to thermal elastic mechanicsconstitutive equations for a plane strain problem wereexpressed as

εTr

1 + μi

Ei

1minus μi( 1113857σTr minus μiσ

Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei

1minus μi( 1113857σTθ minus μiσ

Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)

-e actual thermal stress field F3 in the strata inducedby the temperature changes was decomposed into two

Shear stress field

R3 R3

R4

R2 R2R1 R1

R4

Y Yσπ4 = ndashτxy

σπ4 = τxy

XX

τyx

τxy

Stress transformation

Figure 11 Stress distribution induced by shear stress

CasingCement sheathFormation

R4 R3 R2 R1

T4 T3 T2 T1

Figure 12 -e distribution of interface temperatures


parts the original stress field A3 and the disturbance fieldB3 induced by the temperature variation shown inFigure 13

-e initial stresses were σTi0r σTi0

θ 0 and the initialstrains were εTi0

r εTi0θ 0 -e stresses and displacements

induced by thermal variations were expressed as

uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi

2 are the constants σTir and σTi

θ are the radialand tangential stresses (Pa) uTi

r is the radial displacement(m) ΔTi is the temperature changes (degC) pi is the interfacepressure (Pa) and αi is the material thermal expansioncoefficient i 1 2 3 represented casing cement sheath andformation respectively

-e temperatures were known and the boundary wasfree at internal casing and external formation So radialstress at inner and outer boundaries equals to zero andradial displacement at the outer boundary equals to zero aswell -e boundary and interfacial displacement continuityconditions were expressed as

uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)

Substituting equations (39) and (40) into the followingequation the equations were obtained as

CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)

-e constants of CT11 CT1

2 CT21 CT2

2 CT31 andCT3

2 wereobtained by equations (42)ndash(44) -e wellbore stress wasobtained by substituting these constants into equation (40)

-e total stresses were obtained using the followingequation

σir σprimeir + σPrimeir + σPrimeprimeir + σTi

r

σiθ σprimeiθ + σPrimeiθ + σPrimeprimeiθ + σTi

θ

σiz μi σi

r + σiθ( 1113857

τirθ τPrimeirθ + τPrimeprimeirθ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)

where σir is the radial stress σ

iθ is the tangential stress σ

iz is

the axial stress and τirθ is the shear stress

35 Estimation ofWellbore Integrity It is generally acceptedthat the yield of isotropic material such as casing has nothingto do with hydrostatic pressure while hydrostatic pressure isnot considered in vonMises yield criterion So this criterionwas adopted to determine the casing failure

f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)

where J2 is the second stress partial tensor k is the criticalvalue of failure and σij is the stress components i j 1 2 3represented the three directions of the system respectively

For uniaxial tensionJ2

1113968 σ

3

radic the von Mises stress

could be expressed as follows in the polar coordinate


σMises

12

σr minus σθ( 11138572

+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation

From 2009 to 2017 PetroChina has drilled 141 fracturingwells (including 112 horizontal wells) in the Changning-Weiyuan National Shale Gas Demonstration Area -egeometrical dimensions of the CCF model were a wellborediameter of 85 in casing diameter of 55 in and casingthickness of 917mm According to the Saint-Venantprinciple a formation boundary dimension should befive to six times larger than that of the wellbore geometryto avoid the influence of boundary effect on wellborestress In view of this the model geometry was2000 times 2000mm while the corresponding wellbore di-ameter was 2159mm -e direction of horizontal in situstress was N120degE -e well deviation angle was 90deg andthe wellbore azimuth was N30degE indicating that thehorizontal trajectory was along the minimum in situ stressdirection -e internal casing pressure was calculated fromthe pump pressure plus the downhole hydrostatic fluidpressure -e external boundary stress was obtained fromthe geostress data of the shale reservoir -e thermal andmechanical properties of different materials are presentedin Table 1 -e casing stress and displacement were cal-culated and analyzed considering thermal-pressurecoupling

-e applied maximum horizontal stress σH was 82MPathe minimum horizontal stress σh was 55MPa the verticalstress σv was 57MPa the inner casing pressure Pi was75MPa the boundary temperature T4 was 100degC the fluidtemperature Ta was 20degC and the convective heat transfercoefficient was obtained by using equation (20) (1890Wmiddotmminus2middotdegCminus1) at the pump rate of 20m3min

-e finite element analysis method was adopted tovalidate the results of the analytical models A steady-statethermal analysis followed by a static structural analysiswas conducted to calculate the stress considering thermal-pressure coupling -e solutions of radial stress cir-cumferential stress and Mises stress are compared inFigure 14

-e analytical solutions of radial stress circumferentialstress and Mises stress were in good agreement with theresults obtained by a finite element method which indicates

the validity of the analytical method -e maximum de-viation between analytical and finite element results was14ndash139 indicating that the analytical model could pro-vide an accurate calculation of stress distribution for theCCF system

From Figures 14(a) and 14(b) the radial stress in-creased with the increase of radius in casing and cementsheath but decreased in the formation -e absolute valueof radial stress calculated by the new model was smallerthan that of the existing model-is was mainly because thenew model excluded the strain induced by the initial stressFrom Figures 14(c) and 14(d) the circumferential stressdecreased with the increase of radius in the casing andcement sheath and increased slowly to a constant value inthe formation -e interfacial stress at the internal casingwall was larger than that at the external casing wall -esolutions calculated by the new model were larger thanthose by the existing model From Figure 14(e) casingMises stress obtained by the newmodel was larger than thatof the existing model It could be explained that circum-ferential stress was larger than radial stress and had a maininfluence on Mises stress

-e radial displacements along the 0deg direction calcu-lated by the new model and existing model under the sameconditions were shown in Figure 15 -ere was an obviousdifference for two models especially at the outer boundary-e displacements of new model approached zero when theouter boundary was infinite which reached an agreementwith the actual boundary condition However the dis-placements obtained by the existing model increased linearlyin the formation So only the new model could reflect theactual situation

5 Sensitivity Analysis

-e sensitivity analyses were carried out to study the in-fluences of cement sheath properties geostress fracturingpressure fluid temperature casing thickness and cementsheath thickness on casing stress During analyzing only oneparameter was variable and others were constants Unlessotherwise mentioned the parameters were set as mentionedin Section 4

A3

R3

R4

B3 C3

R4

R3

T3T4T4 T4 T3 T4p3 p3

R3

R4

(a)

Casing Cement sheath Formation

p2 p3

(b)

Figure 13 -ermal stress field (a) Formation stress components (b) Interface pressures pi is the interface pressure i 2 3 represented thecasing-cement sheath interface and cement sheath-formation interface


ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)

Radial displacement (mm)

New analyticalmodelndash0degExisting analyticalmodelndash0degNew FEM modelndash0degExisting FEM modelndash0deg

ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100

New analyticalmodelndash90degExisting analyticalmodelndash90degNew FEM modelndash90degExisting FEM model-90deg

(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)

Circumferential Angle (deg)

New analylical modelndashouter casingNew FEM modelndashouter casing

Existing analylical modelndashouter casingExisting FEM modelndashouter casing

(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)

Raial displacement (mm)

ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50

Existing FEM modelndash0deg

New analyticalmodelndash0degExisting analyticalmodelndash0degNew FEM modelndash0deg



(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)

Circumference angles (deg)

New analylical modelndashinner casingNew FEM modelndashinner casingExisting analylical modelndashinner casingExisting FEM modelndashinner casing

New analylical modelndashouter casingNew FEM modelndashouter casingExisting analylical modelndashouter casingExisting FEM modelndashouter casing

(d)

Figure 14 Continued

Table 1 -ermal and mechanical Parameters of fluid-casing-cement sheath-formation system

Property Casing Cement sheath Formation FluidElastic modulus Ei (GPa) 210 5 35 mdashPoissonrsquos ratio μi 03 015 025 mdashCoefficient of thermal expansion αi (10minus5middotdegCminus1) 15 10 10 mdash-ermal conductivity ki (Wmiddotmminus1middotdegCminus1) 582 10 10 173Specific heat Cpi (Jmiddotkgminus1middotdegCminus1) 460 1830 1043 3935Density ρi (kgmiddotmminus3) 7850 1800 2500 1080Note properties in parenthesis were used in the parametric study


51 Influence of Elastic Modulus Cement sheath propertiesis crucial for casing safety To evaluate the effect of elasticmodulus on casing stress the cement sheath elastic modulusof E2 was set at the range from 2GPa to 50GPa and theformation elastic modulus of E3 was set as 5 and 35GPa tosimulate a soft and hard formation -e Mises stresses atinternal casing are shown in Figure 16

From Figures 16(a) and 16(c) the maximum Mises stressappeared at the angles of 0deg and 180deg for the new model and90deg and 270deg for the existing model when the formation

modulus was small However the maximum stress allappeared at the angles of 0deg and 180deg for the new and existingmodels when the formation modulus was large FromFigure 16(b) in a soft formation (a modulus of 5GPa) withthe increase of the cement sheath modulus the maximumcasing stress increased first and then decreased for existingmodel while decreasing all the time for the new model FromFigure 16(b) in a hard formation (modulus of 35GPa) themaximum casing stress always decreased with the increase ofthe cement sheath modulus for two models In the soft

0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)


New analylicalmodelndashinner casingNew FEMmodelndashinner casingExisting analylicalmodelndashinner casingExisting FEMmodelndashinner casing

New analylicalmodelndashouter casingNew FEMmodelndashouter casingExisting analylicalmodelndashouter casingExisting FEMmodelndashouter casing

(e)

Figure 14 Comparison of numerical and analytical solutions (a) Radial stress along the radial directions of 0deg and 90deg (b) Radial stress atthe internal casing wall (c) Circumferential stress along the radial directions of 0deg and 90deg (d) Circumferential stress at the internal casingwall (e) Mises stress at inner and outer casing walls

0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300

New modelExisting model

Casing

0200

ndash02ndash04

0 50 100

Cement sheathFormation

600Radial distance from the wellbore (mm)

900 1200 1500

Figure 15 Radial displacements of the wellbore assembly along the 0deg direction


formation the stress calculated by the new model was smallerthan that by the existing model However the stress obtainedby the newmodel was larger than that by the existingmodel ina hard formation According to the fact that shale formationhad a large elastic modulus the existing model under-estimated casing stress during the fracturing operation

52 Inuence of Poissonrsquos Ratio To evaluate the eect ofPoissonrsquos ratio on casing stress cement sheath Poissonrsquosratio μ2 with a range from 005 to 045 was adopted and theformation Poissonrsquos ratio μ3 was set as 005 and 045 to

simulate a hard and soft formatione casingMises stressesare shown in Figure 17

From Figures 17(a) and 17(b) the maximum Misesstress decreased with the increase of cement sheath Pois-sonrsquos ratio for two models In a hard formation (Poissonrsquosratio of 005) the maximum stress obtained by the newmodel was larger than that by the existing model Howeverin a soft formation (Poissonrsquos ratio of 045) it was a littlesmaller than that by the existing model According to thefact that shale formation had a small Poissonrsquos ratio theexisting model underestimated casing stress during thefracturing process

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)

Circumference angle (deg)

E2 = 5GPa existing modelE2 = 20GPa existing modelE2 = 35GPa existing modelE2 = 50GPa existing model

E2 = 5GPa new model E2 = 20GPa new model E2 = 35GPa new model E2 = 50GPa new model

(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)

Cement sheath modulus (GPa)


(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)

Figure 16 Casing Mises stress (a b) E3 5GPa and (c d) E3 35GPa


53 Inuence of In Situ Stress Nonuniformity During themultifracturing operation for shale gas wells the fracturinguid was pressed into the formation and the in situ stresseld changed abruptly to increase the nonuniformity of thestress around the wellbore To evaluate the eect of in situstress nonuniformity on casing stress the nonuniformityindex was dened as δ σHσv Dierent δ with a range of01ndash30 was adopted e casing Mises stresses calculated bytwo models are shown in Figure 18

As seen from Figure 18(a) for δ smaller than 10 themaximumMises stresses appeared at 90deg and 270deg directionsHowever for δ larger than 10 the maximum Mises stressesappeared at 0deg and 180deg directions For δ of 10 the casingMises stress around the wellbore was at a uniform stateFrom Figure 18(b) the maximum casing stress increaseddramatically with the increase of δ from 10 or decrease of δfrom 10 e solution obtained by the new model was largerthan that by the existing model for δ between 05 and 225When δ was larger than 225 or smaller than 05 the casingstress obtained by the existing model was larger than that bythe new model

54 Inuence of Fracturing Pressure A fracturing uid withhigh pressure was used to fracture a shale formation ehigh pressure depended on the formation regional tectonicstress the larger the tectonic stress the higher the pressureMoreover a high fracturing pressure posed a great potentialchallenge to casing failure Dierent fracturing pressureswith a range of 5ndash105MPa were adopted to evaluate theeect of fracturing pressure on casing stress e maximumcasing Mises stresses are shown in Figure 19

It can be seen from Figure 19 that the casing stressescalculated by the two models decreased rst and then in-creased with the increase of fracturing pressure e min-imum stress appeared at 15MPa for the new modelhowever it appeared at about 25MPa for the existing model

In addition the casing Mises stress obtained by the existingmodel was larger than that by the new model for pressurelower than 25MPa and smaller than that by the new modelfor pressure higher than 25MPa During fracturing oper-ation pressure must be large enough to fracture the for-mation so the existing model underestimated the casingstress

55 Inuence of Fluid Temperature During the cycle in-jection of fracturing uid the heat transfer coecient h wascalculated using equation (20) with a pump rate of 20m3min e corresponding casing internal Mises stress wascalculated under dierent uid temperatures at a range of10ndash100degC to evaluate the eect on casing stress Figure 20presented the maximum casing stress over temperature andthe comparison of the results obtained by the existing modeland new model

From Figure 20 the maximum Mises stress decreasedwith the increase of the injection uid temperature in-dicating that a fracturing uid with high temperature waseective to decrease casing stress Furthermore the stressobtained by the existing model was smaller than that by thenew model It revealed that the existing model under-estimated the casing Mises stress

56 Inuence of ickness e thickness of cement sheathand casing was curial for casing safety To evaluate the eectof thickness on the casing stress the cement thickness wasset at a range of 2ndash50mm and the casing thickness was set ata range of 5ndash15mm e comparisons of maximum casingMises stress obtained by the two models are shown inFigure 21

As shown in Figure 21 the maximum casing Mises stressincreased with the increase of cement sheath thickness andhowever decreased with the increase of casing thickness Soa thicker casing wall and thinner cement sheath were

0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)

Cement sheath Poissonrsquos ratio


(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)

Figure 17 Casing Mises stresses for dierent Poissonrsquos ratios (a) μ3 005 (b) μ3 045


eective to ensure the casing integrity Meanwhile casingstress obtained by the existing model was smaller than thatby the new model indicating that the existing modelunderestimated casing stress

6 Conclusions

A new analytical model considering drilling constructionwas established to assess the casing stress under dierentconditions considering thermal-pressure coupling e so-lutions were obtained by dividing the model into three partssuch as initial stress eld wellbore disturbance eld andthermal stress eld Sensitivity analyses of dierent factorswere conducted to evaluate the inuences on casing stress

Some conclusions were drawn from the comparisons be-tween new model and existing model

(1) e results of radius stress tangential stress andcasing Mises stress calculated by the analyticalmethod were in good agreement with the solutionsby a nite element solution e minor deviationsdid not exceed 139 Moreover the analytical so-lutions were in-line with the actual boundary con-ditions of shale gas wells

(2) e casing stress calculated by the existing modelwas smaller than that by the new model for hard

0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)


δ = 01 existing modelδ = 05 existing modelδ = 10 existing modelδ = 20 existing modelδ = 30 existing model

δ = 01 new model δ = 05 new model δ = 10 new model δ = 20 new model δ = 30 new model P110 yield stress

(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)

Stress nonuniformity index

New modelExisting modelP110 yield stress

(b)

Figure 18 Casing Mises stress for dierent stress nonuniformity indexes (a) casing internal stress and (b) maximum casing stress

0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)

Fracturing pressure (MPa)


Figure 19 Maximum casing Mises stress for dierent pressures

400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)

Fluid temperature (degC)


Figure 20 Casing Mises stress for dierent temperatures


formation with larger modulus or low Poissonrsquosratio geostress heterogeneity index at a range of05ndash225 and fracturing pressure larger than25MPa

(3) e casing stress increased with the increase of the insitu stress nonuniformity index With the increase offracturing pressure casing stress decreased rst andthen increased

(4) Cement sheath with appropriate modulus and largerPoissonrsquos ratio thinner cement sheath thicker cas-ing and higher uid temperature were eective todecrease the casing stress

In conclusion the new analytical model can accuratelypredict casing stress and become an alternative method ofcasing integrity evaluation for shale gas wells It is a usefuland ecient method for a preliminary design being capableof simulating the actual situations in order to assess thecasing stresses and integrity

Data Availability

e data of each gure used to support the ndings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare that they have no conicts of interest

Acknowledgments

is research was nancially supported by the NationalNatural Science Funds of China (51674272) the Key Programof National Natural Science Foundation of China (U1762211)and China Petrochemical Corporation (HX20180001) e

assistance of Dr Wei Lian in contribution to modify thelanguage of the manuscript and the pictures and editablegure les is gratefully acknowledged

References

[1] Z Lv L Wang S Deng et al ldquoChinarsquos marine qiongzhusishale play rst deep Asia pacic region horizontal multiplestage frac case history operation amp executionrdquo in Proceedingsof the International Petroleum Technology Conference 2013

[2] W Yan L Zou H Li J Deng H Ge and H Wang ldquoIn-vestigation of casing deformation during hydraulic fracturingin high geo-stress shale gas playrdquo Journal of Petroleum Scienceand Engineering vol 150 pp 22ndash29 2017

[3] G E King and R L Valencia ldquoWell integrity for fracturingand re-fracturing what is needed and whyrdquo in Proceedings ofthe SPE Hydraulic Fracturing Technology Conference Feb-ruary 2016

[4] R J Davies S Almond R S Ward et al ldquoOil and gas wellsand their integrity implications for shale and unconventionalresource exploitationrdquo Marine and Petroleum Geologyvol 56 pp 239ndash254 2014

[5] C Atkinson and D A Eftaxiopoulos ldquoA plane model for thestress eld around an inclined cased and cemented wellborerdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 20 no 8 pp 549ndash569 1996

[6] F Yin and D Gao ldquoMechanical analysis and design of casingin directional well under in-situ stressesrdquo Journal of NaturalGas Science and Engineering vol 20 pp 285ndash291 2014

[7] J Fang Y Wang and D Gao ldquoOn the collapse resistance ofmultilayer cemented casing in directional well under aniso-tropic formationrdquo Journal of Natural Gas Science and Engi-neering vol 26 pp 409ndash418 2015

[8] Z Chen W Zhu and Q Di ldquoElasticity solution for the casingunder linear crustal stressrdquo Engineering Failure Analysisvol 84 pp 185ndash195 2018

[9] X Huang M Mihsein K Kibble and R Hall ldquoCollapsestrength analysis of casing design using nite element

300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)

Cement sheath thickness (mm)


(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)

Casing thickness (mm)


(b)

Figure 21 Casing Mises stress for dierent thicknesses of casing and cement sheath (a) Dierent cement thickness (b) Dierent casingthickness


methodrdquo International Journal of Pressure Vessels and Pipingvol 77 no 7 pp 359ndash367 2000

[10] A Berger W W Fleckenstein A W Eustes andG -onhauser ldquoEffect of eccentricity voids cement chan-nels and pore pressure decline on collapse resistance ofcasingrdquo in Proceedings of the SPE Annual Technical Confer-ence and Exhibition March 2004

[11] W W Fleckenstein A W Eustes W J Rodriguez et alldquoCemented casing the true stress picturerdquo in Proceedings ofthe AADE 2005 National Technical Conference and Exhibitionthe American Association of Drilling Engineers the WyndamGreenspoint in Houston Society of Petroleum EngineersHouston TX USA April 2005

[12] A R McSpadden O D Coker and G C Ruan ldquoAdvancedcasing design with finite-element model of effective doglegseverity radial displacements and bending loadsrdquo in Pro-ceedings of the SPE Production and Operations SymposiumMarch 2011

[13] G Wang Z Chen J Xiong and K Yang ldquoStudy on the effectof non-uniformity load and casing eccentricity on the casingstrengthrdquo Energy Procedia vol 14 pp 285ndash291 2012

[14] X Zhu S Liu and H Tong ldquoPlastic limit analysis of defectivecasing for thermal recovery wellsrdquo Engineering FailureAnalysis vol 27 pp 340ndash349 2013

[15] P D Pattillo and T G Kristiansen ldquoAnalysis of horizontalcasing integrity in the valhall fieldrdquo in Proceedings of the SPEISRM Rock Mechanics Conference October 2002

[16] K E Gray E Podnos and E Becker ldquoFinite-element studiesof near-wellbore region during cementing operations part IrdquoSPE Drilling amp Completion vol 24 no 1 pp 127ndash136 2009

[17] F Mackay and S A B Fontoura ldquo-e description of a processfor numerical simulations in the casing cementing of pe-troleum salt wellsmdashpart I from drilling to cementingrdquo inProceedings of the 48th US Rock MechanicsGeomechanicsSymposium American Rock Mechanics Association Min-neapolis MN USA June 2014

[18] W Zhang A Eckert and X Liu ldquoNumerical simulation ofmicro-annuli generation by thermal cyclingrdquo in Proceedingsof the 51st US Rock MechanicsGeomechanics SymposiumAmerican Rock Mechanics Association San Francisco CAUSA June 2017

[19] M D Simone F L G Pereira and D M Roehl ldquoAnalyticalmethodology for wellbore integrity assessment consideringcasing-cement-formation interactionrdquo International Journalof Rock Mechanics and Mining Sciences vol 94 pp 112ndash1222017

[20] W Liu B Yu and J Deng ldquoAnalytical method for evaluatingstress field in casing-cement-formation system of oilgaswellsrdquo Applied Mathematics and Mechanics vol 38 no 9pp 1273ndash1294 2017

[21] T L Chow Mathematical Methods for Physicists-A ConciseIntroduction Cambridge University Press New York NYUSA 2010

[22] X Wang Z Qu Y Dou and W Ma ldquoLoads of casing andcement sheath in the compressive viscoelastic salt rockrdquoJournal of Petroleum Science and Engineering vol 135pp 146ndash151 2015

[23] Z Wu and S Li ldquo-e generalized plane strain problem and itsapplication in three-dimensional stress measurementrdquo In-ternational Journal of Rock Mechanics and Mining Sciences ampGeomechanics Abstracts vol 27 no 1 pp 43ndash49 1990

[24] H Jo and K E Gray ldquoMechanical behavior of concentriccasing cement and formation using analytical and numericalmethodsrdquo in Proceedings of the 44th US Rock Mechanics

Symposium and 5th US-Canada Rock Mechanics SymposiumAmerican Rock Mechanics Association Salt Lake City UTUSA June 2010

[25] C Yan J Deng Y Cheng M Li Y Feng and X Li ldquoMe-chanical properties of gas shale during drilling operationsrdquoRock Mechanics and Rock Engineering vol 50 no 7pp 1753ndash1765 2017

[26] L Helfen F Dehn P Mikulık and T Baumbach ldquo-ree-dimensional imaging of cement microstructure evolutionduring hydrationrdquo Advances in Cement Research vol 17no 3 pp 103ndash111 2005

[27] L Zhang X Yan X Yang and X Zhao ldquoEvaluation ofwellbore integrity for HTHP gas wells under solid-temperature coupling using a new analytical modelrdquo Jour-nal of Natural Gas Science and Engineering vol 25pp 347ndash358 2015

[28] A Lavrov and M Torsaeligter Physics and Mechanics of PrimaryWell Cementing Springer Berlin Germany 1st edition 2016

[29] S P Timoshenko and J N Goodier 13eory of ElasticityMcGraw-Hill Book Company Inc New York NY USA 3rdedition 1970

[30] D W Marshall and R G Bentsen ldquoA computer model todetermine the temperature distributions in a wellborerdquoJournal of Canadian Petroleum Technology vol 21 no 1pp 63ndash75 1982

[31] J B J Fourier 13eorie Analytique de la Chaleur CambridgeUniversity Press New York NY USA 1822


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Submit your manuscripts atwwwhindawicom

distribution in a casing-cement-formation system under an-isotropic in situ stresses Fang et al [7] developed a multilayercemented casing system in the directional well under aniso-tropic formation to investigate the collapse resistance of casingunder nonuniform in situ stress and anisotropic formationBased on the stress function method a three-dimensionalmodel of the casing-cement sheath-formation (CCF) systemwas proposed subjected to linear crustal stress and then ananalytical solution of the model was obtained [8]

A finite element method (FEM) had been proposed toachieve a better understanding of the ultimate collapsestrength of casing [9] Using this method casing stress wasbetter analyzed subjected to external and nonuniformloading caused by void and pressure [10] -e results of aseries of finite element studies of the cemented casing undera variety of stress conditions for both burst and collapse werepresented demonstrating the inadequacy of accepted designequations under many cemented conditions [11] Apurpose-built finite element model was applied to simulatethe radial displacement of a casing string constrained withinan outer wellbore [12] Casing stress on the inner wall underthe condition of elliptic casing was calculated and analyzedto improve the designing and correction of casing strength[13] -e main controlling factor to the plastic limit load ofdefective casing was analyzed by the combination of hightemperature and high steam injection pressure [14]

However the above analytical or FEM solutions areobtained by setting the casing and cement together instantlyat the beginning of the analysis -e strains induced by theinitial stress are included in the model which is not in-linewith the actual situation In fact initial stress has alreadyexisted in the formation before wellbore excavation -ewellbore stress redistributed after removing the rocks thatoriginally occupied the borehole volume and just affected thestress and displacement near wellbore zones In view of thismore sophisticated solutions have been developed withadditional parameters and appropriate assumptions re-garding the drilling completion and fracturing phases

Pattillo and Kristiansen [15] developed a finite modelstarting with the virgin reservoir which considered pre-wellbore depletion drilling the wellbore installation of casingand cement and subsequent draw down Behavior of variousconfigurations during subsequent draw down permitted themto be ranked according to the life expectancy of the resultingcompletion Gray et al [16] presented a staged-finite elementprocedure during the well construction which consideredsequentially the stress states and displacements at and near thewellbore But temperature flow and poroelasticity effectswere not included Mackay and Fontoura [17] carried anumerical analysis to determine the effect of salt creep andcontacts amongst the materials before during and afterdrilling the well focusing on the drilling of the wellbore andon the hardening of the cement Zhang et al [18] used thestaged-finite element modeling approach to simulate the wellconstruction processes and injection cycle using a ldquorealisticrdquobottom-hole state of stress to simulate the microannuligeneration by the tensile debonding of the cement-formationinterface Simone et al [19] developed an analytical solutionof single and double casing configurations to assess the

stresses during the drilling construction and productionphases But the nonuniform boundary stress was not con-sidered in this model Liu et al [20] presented an analyticalmethod for evaluating the stress field within a casing-cement-formation system of oilgas wells under anisotropic in situstresses in the rock formation and uniform pressure withinthe casing However the temperature was not considered inthis model

In this work to evaluate the thermal and mechanicalstresses of casing an analytical model of casing-cementsheath-formation system was established considering well-bore construction -e boundary stresses in the wellborecoordinate system were obtained through three-dimensionalrotations from principal in situ stresses-e casing stress wasobtained by dividing the model into three parts such asinitial stress field disturbance stress field and thermal stressfield -e continuous homogeneous isotropy and linearelasticity were taken into consideration Solutions werevalidated by a finite element method Sensitivity analyseswere conducted to estimate the influence of different factorson casing stress such as property of cement sheath andcasing fracturing pressure fluid temperature and initialgeostress Useful countermeasures were put forward todecrease casing stress during the fracturing operation

2 Method and Basic Conditions Analysis

21 Method Comparison

211 Existing Method For existing method the casingcement sheath and formation were set together at the be-ginning of analysis -en temperature boundary and loadswere added to the system to calculate wellbore stress dis-tribution as shown in Figure 1-is method did not considerthe process of wellbore construction and hydraulic frac-turing So the wellbore stress distribution could not accu-rately be calculated under the condition of formation straininduced by the initial in situ stress

212 New Method To exactly predict the wellbore stressdistribution a casing-cement sheath-formation model usinga new analysis method was provided -e analysis processwas divided into four steps (Figure 2) Before drilling theinitial stresses such as normal stress and shear stress wereloaded in the model and initial strains were produced -econstrained boundary conditions were assigned to the entiremodel which reached initial mechanical equilibrium Duringdrilling the rock was excavated causing stress concentrationaround the borehole -e drilling mud pressure Pm wasapplied to the internal face of the wellbore During casingand cementing casing and cement sheath elements wereadded simultaneously to the model and the cement hard-ening procedure was not considered Initial stress and strainin casing and cement sheath were ignored-e cement slurrypressure Pc was applied to the internal casing wall Duringfracturing a low temperature Tn and the fracturing pressurePf were assigned in the internal casing wall -e wellborestress field was obtained under the condition of thermal-mechanical coupling




Cxαx

1 0 0

0 cos αx sin αx


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cyαy


0 1 0

sin αy 0 cos αy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Czαz

cos αz sin αz 0


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

σ0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)


zjCTzβC

TyαC

Tz90 (3)



ForamtionCasing


ForamtionCasing


Y

σy

σxX

τyx

τxy

Tf

TnPf

Y

σy

σxX

τyx

τxy

Tf

Tn

Pm

Y

σy

σxX

τyx

τxy

Tf

Tn

Pc

Y

σy

σxX

τyx

τxy

Tf

TnPf


Before loading

Y Y

σy

σxX

τyx

τyx

Tf

TnPfX

After loading

ForamtionCasing

Cement sheathFluid











minusσx 0

0 minusσy


0 minusp01113890 1113891 +

minuss 0

0 s1113890 1113891 (4)





(5)


Wellbore trajectory

σv

σh

σH

YPrime

Yprimeh (East)

Xprimeh (North)

YPrime Y XXPrime

XPrime

ZPrime Z ZvZ Z

X

ZPrime

Y

Oβ

α

φ

Xh

XH





σprime30r


τprime30rθ

11138681113868111386811138681113868rinfin s sin(2θ)

⎧⎪⎨

⎪⎩(6)






uprimei

r 12Gi


1113876 1113877qi

+12Gi



(7)



1113874 1113875qi+1



1113874 1113875qi+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(8)



i0r are


Y

Y

R4

R3 R2 R1

σy

σx

TnPf σx

σy

X

τyx

τyx

τxy

Tf

θ

Z

X

ForamtionCasing

Cement sheathFluid



Y

R4

R3 R2 R1

Pf

p0

p0

X R3 R2 R1R3 R3

R2 R2R1 R1

R4 R4 R4

Tn

Tfndashs

s

Y Y Y

X X

τyx

τxy

X


ForamtionCasing


ForamtionCasing


ForamtionCasing


ForamtionCasing




i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2





3 wereobtained

uprime3

r (r) 1

2G3

R23






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)






uprime1r 1


1r

1113876 1113877pi

+1


1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)




⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1



where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Cxαx

1 0 0

0 cos αx sin αx


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cyαy


0 1 0

sin αy 0 cos αy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Czαz

cos αz sin αz 0


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

σ0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)


zjCTzβC

TyαC

Tz90 (3)



ForamtionCasing


ForamtionCasing


Y

σy

σxX

τyx

τxy

Tf

TnPf

Y

σy

σxX

τyx

τxy

Tf

Tn

Pm

Y

σy

σxX

τyx

τxy

Tf

Tn

Pc

Y

σy

σxX

τyx

τxy

Tf

TnPf


Before loading

Y Y

σy

σxX

τyx

τyx

Tf

TnPfX

After loading

ForamtionCasing

Cement sheathFluid











minusσx 0

0 minusσy


0 minusp01113890 1113891 +

minuss 0

0 s1113890 1113891 (4)





(5)


Wellbore trajectory

σv

σh

σH

YPrime

Yprimeh (East)

Xprimeh (North)

YPrime Y XXPrime

XPrime

ZPrime Z ZvZ Z

X

ZPrime

Y

Oβ

α

φ

Xh

XH





σprime30r


τprime30rθ

11138681113868111386811138681113868rinfin s sin(2θ)

⎧⎪⎨

⎪⎩(6)






uprimei

r 12Gi


1113876 1113877qi

+12Gi



(7)



1113874 1113875qi+1



1113874 1113875qi+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(8)



i0r are


Y

Y

R4

R3 R2 R1

σy

σx

TnPf σx

σy

X

τyx

τyx

τxy

Tf

θ

Z

X

ForamtionCasing

Cement sheathFluid



Y

R4

R3 R2 R1

Pf

p0

p0

X R3 R2 R1R3 R3

R2 R2R1 R1

R4 R4 R4

Tn

Tfndashs

s

Y Y Y

X X

τyx

τxy

X


ForamtionCasing


ForamtionCasing


ForamtionCasing


ForamtionCasing




i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2





3 wereobtained

uprime3

r (r) 1

2G3

R23






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)






uprime1r 1


1r

1113876 1113877pi

+1


1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)




⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1



where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










minusσx 0

0 minusσy


0 minusp01113890 1113891 +

minuss 0

0 s1113890 1113891 (4)





(5)


Wellbore trajectory

σv

σh

σH

YPrime

Yprimeh (East)

Xprimeh (North)

YPrime Y XXPrime

XPrime

ZPrime Z ZvZ Z

X

ZPrime

Y

Oβ

α

φ

Xh

XH





σprime30r


τprime30rθ

11138681113868111386811138681113868rinfin s sin(2θ)

⎧⎪⎨

⎪⎩(6)






uprimei

r 12Gi


1113876 1113877qi

+12Gi



(7)



1113874 1113875qi+1



1113874 1113875qi+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(8)



i0r are


Y

Y

R4

R3 R2 R1

σy

σx

TnPf σx

σy

X

τyx

τyx

τxy

Tf

θ

Z

X

ForamtionCasing

Cement sheathFluid



Y

R4

R3 R2 R1

Pf

p0

p0

X R3 R2 R1R3 R3

R2 R2R1 R1

R4 R4 R4

Tn

Tfndashs

s

Y Y Y

X X

τyx

τxy

X


ForamtionCasing


ForamtionCasing


ForamtionCasing


ForamtionCasing




i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2





3 wereobtained

uprime3

r (r) 1

2G3

R23






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)






uprime1r 1


1r

1113876 1113877pi

+1


1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)




⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1



where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



σprime30r


τprime30rθ

11138681113868111386811138681113868rinfin s sin(2θ)

⎧⎪⎨

⎪⎩(6)






uprimei

r 12Gi


1113876 1113877qi

+12Gi



(7)



1113874 1113875qi+1



1113874 1113875qi+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(8)



i0r are


Y

Y

R4

R3 R2 R1

σy

σx

TnPf σx

σy

X

τyx

τyx

τxy

Tf

θ

Z

X

ForamtionCasing

Cement sheathFluid



Y

R4

R3 R2 R1

Pf

p0

p0

X R3 R2 R1R3 R3

R2 R2R1 R1

R4 R4 R4

Tn

Tfndashs

s

Y Y Y

X X

τyx

τxy

X


ForamtionCasing


ForamtionCasing


ForamtionCasing


ForamtionCasing




i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2





3 wereobtained

uprime3

r (r) 1

2G3

R23






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)






uprime1r 1


1r

1113876 1113877pi

+1


1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)




⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1



where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



i (R2i+1 minusR2

i )Biprime R2

i+1(R2i+1 minusR2





3 wereobtained

uprime3

r (r) 1

2G3

R23






⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(9)






uprime1r 1


1r

1113876 1113877pi

+1


1r

1113876 1113877 minusp2prime( 1113857

uprime2r 1


1r

1113876 1113877p2prime

+1


1r

1113876 1113877 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp2prime( 1113857



1113874 1113875 minusp3prime( 1113857



1113874 1113875 minusp3prime( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)


uprime1r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR2

uprime2r1113868111386811138681113868rR3

uprime3r1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩(12)




⎧⎨

⎩ (13)

Ri

qi

Ri + 1

qi + 1



where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


where

AA 1


1R2

1113890 1113891

+1


1R2

1113890 1113891

BB 1


1R2

1113890 1113891

CC 1


1R2

1113890 1113891

DD 1


1R3

1113890 1113891

EE 1


1R3

1113890 1113891 +1

2G3R31113896 1113897

FF minusR3

2G3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



ϕ APrimei r4

+ BPrimei r2


1113888 1113889cos(2θ) (15)



+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


Ei

12υiAPrimei r


4CPrimeir2

+6DPrimeir4

1113890 1113891


εPrimeiθ 1 + μi

Ei


+ 2BPrimei + μi

4CPrimeir2

+6DPrimeir4

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

A1 B1 C1 F1

p0

p0

p0

R4 R4 R4 R4

R3 R3 R3

pprime3 pprime3 p0

p0

R3

p0


Casing




RiRi+1





εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




εPrimeir zuPrime

ir

zr

εPrimeiθ 1r

zuPrimei

θzθ

+uPrime

ir

r

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(17)



i

θ were obtained as

uPrimei

r minus1 + μi

Ei

4μiAPrimei r



1113890 1113891


uPrimei

θ 1 + μi

2Ei



r+4DPrimeir3

1113890 1113891


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εPrimei0r 1 + μi

Ei


εPrimei0θ 1 + μi

Ei


r1113876 1113877

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)





uPrime3r minus1


2CPrime3rminus

DPrime3r3

1113890 1113891cos 2θ

uPrime3θ 1


CPrime3r

+DPrime3r3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(19)


+6DPrime3r4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(20)


uPrimei

r minus1Gi

2μiAPrimei r


2CPrimeirminus

DPrimeir3

1113890 1113891cos 2θ

uPrimei

θ 1Gi



+DPrimeir3

1113890 1113891sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(21)


+6DPrimeir4

1113888 1113889cos 2θ


1113888 1113889cos 2θ


1113888 1113889sin 2θ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)


σPrime1r

1113868111386811138681113868rR1 0

τPrime1rθ

11138681113868111386811138681113868rR1 0

⎧⎪⎪⎨

⎪⎪⎩

σPrime1r

1113868111386811138681113868rR2σPrime2r

1113868111386811138681113868rR2

τPrime1rθ

11138681113868111386811138681113868rR2 τPrime2rθ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

σPrime2r

1113868111386811138681113868rR3σPrime3r

1113868111386811138681113868rR3

τPrime2rθ

11138681113868111386811138681113868rR3 τPrime3rθ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

uPrime1r

1113868111386811138681113868rR2 uPrime2r

1113868111386811138681113868rR2

uPrime1θ

11138681113868111386811138681113868rR2 uPrime

2θ

11138681113868111386811138681113868rR2

⎧⎪⎪⎨

⎪⎪⎩

uPrime2r

1113868111386811138681113868rR3 uPrime3r

1113868111386811138681113868rR3

uPrime2θ

11138681113868111386811138681113868rR3 uPrime

3θ

11138681113868111386811138681113868rR3

⎧⎪⎪⎨

⎪⎪⎩

(23)




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




R21minus6DPrime1

R41

1113888 1113889cos 2θ 0


2R21CPrime1 minus

6R41DPrime11113888 1113889sin 2θ 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(24)


R22

+6DPrime1

R42

1113888 1113889 + 2BPrime2 +4CPrime2

R22

+6DPrime2

R42

1113888 1113889 0


2CPrime1

R22minus6DPrime1

R42

1113888 1113889


2CPrime2

R22minus6DPrime2

R42

1113888 1113889 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


R23

+3DPrime2

R43

1113888 1113889 +2CPrime3

R23

+3DPrime3

R43

minuss


CPrime2

R23minus3DPrime2

R43

1113888 1113889 +CPrime3

R23

+3DPrime3

R43

s

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(26)

minus1

G12μ1R

32APrime1 + R2B


2R2

CPrime1 minus1

R32DPrime11113890 1113891

+1

G22μ2R

32APrime2 + R2B


2R2

CPrime2 minus1

R32DPrime21113890 1113891 0

1G1



1R2

CPrime1 +1


1G2



1R2

CPrime2 +1

R32DPrime21113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

minus1

G22μ2R

33APrime2 + R3B


2R3

CPrime2 minus1

R33DPrime21113890 1113891

+1


2R3

CPrime3 minus1

R33DPrime31113890 1113891 0

1G2



1R3

CPrime2 +1

R33DPrime21113890 1113891

minus1


1R3

CPrime3 +1

R33DPrime31113890 1113891 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)




ir uPrimeprime

i


Ω 12arctan minus

2τxy

σx minus σy

1113888 1113889 π4

(29)

σπ4 τxy




A2 B2 F2C2

ndashscos2θ

ndashscos2θ

RiRi+1

scos2θ

RiRi + 1

pcos2θRi

Ri + 1

ndashscos2θ

pcos2θ

RiRi + 1




h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



h Stkm

D 00107

km

D


K((3n + 1)4n)n 32QπD3( )nminus11113896 1113897

067

middotK((3n + 1)4n)n 32QπD3( 1113857

nminus1Cm

km1113890 1113891

033

(31)



d2T

dr2+1r

dT

dr 0 (32)




Ti(r) A

Ti ln r + B

Ti (33)

ATi

Ti+1 minusTn


BTi



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(34)


(m) and ATi andBT



ki

dTi(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

ki+1dTi+1(r)

dr

11138681113868111386811138681113868111386811138681113868rRi+1

(35)


1 + β1( 1113857T2 minus β1T3 T1

minusT2 + 1 + β2( 1113857T3 β2T41113896 (36)

where

β1 k2

k1

ln R2R1( 1113857 + k1R1h( 1113857

ln R3R2( 1113857

β2 k3

k2

ln R3R2( 1113857

ln R4R3( 11138571113890 1113891

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(37)


εTr

1 + μi

Ei


Tθ1113960 1113961 + 1 + μi( 1113857αiT

εTθ

1 + μi

Ei


Tr1113960 1113961 + 1 + μi( 1113857αiT

εTz 0

cTrθ

2 1 + μi( 1113857

Ei

τTrθ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(38)


Shear stress field

R3 R3

R4

R2 R2R1 R1

R4


σπ4 = τxy

XX

τyx

τxy




R4 R3 R2 R1

T4 T3 T2 T1








uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







uTir

1 + μi( 1113857

1minus μi( 1113857

αi

r1113946

r

Ri

rΔTidr + C

Ti1 r +

CTi2rminus rεTi0

r (39)

σTir minus

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

minusCTi2

r21113890 1113891

σTiθ

αiEi

1minus μi

1r2

1113946r

Ri

rΔTidr +

Ei

1 + μi

CTi1

1minus 2μi

+CTi2

r21113890 1113891

minusαiEi

1minus μi

ΔTi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(40)

where CTi1 andCTi





uT1r

1113868111386811138681113868rR2 uT2

r

1113868111386811138681113868rR2

uT2r

1113868111386811138681113868rR3 uT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR2σT2

r

1113868111386811138681113868rR2

σT2r

1113868111386811138681113868rR3σT3

r

1113868111386811138681113868rR3

⎧⎪⎨

⎪⎩

σT1r

1113868111386811138681113868rR1 0

σT3r

1113868111386811138681113868rR4 0

⎧⎪⎨

⎪⎩

(41)


CT11 R2 +

CT12

R2minusC

T21 R2 minus

CT22

R2 minus

1 + μ1( 1113857

1minus μ1( 1113857

α1R2

1113946R2

R1

rT1dr

CT21 R3 +

CT22

R3minusC

T31 R3 minus

CT32

R3

1 + μ2( 1113857

1minus μ2( 1113857

α2R3

1113946R3

R2

rT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(42)

E1

1 + μ1

CT11

1minus 2μ1minus

CT12

R22

1113890 1113891minusE2

1 + μ2

CT21

1minus 2μ2minus

CT22

R22

1113890 1113891

α1E1

1minus μ11

R22

1113946R2

R1

rΔT1dr

E2

1 + μ2

CT21

1minus 2μ2minus

CT22

R23

1113890 1113891minusE3

1 + μ3

CT31

1minus 2μ3minus

CT32

R23

1113890 1113891

α2E2

1minus μ21

R23

1113946R3

R2

rΔT2dr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

C11R1 +

C12

R1 0

C31R4 +

C32

R4 minus

1 + μ2( 1113857

1minus μ2( 1113857

α2R4

1113946R4

R3

rT dr

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(44)


2 CT21 CT2

2 CT31 andCT3




r


θ

σiz μi σi

r + σiθ( 1113857


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(45)



iz is



f J2 k( 1113857 J2

1113968minus k 0

J2 16

σ11 minus σ22( 11138572

+ σ22 minus σ33( 11138572

+ σ33 minus σ11( 11138572

1113960 1113961

+ σ212 + σ223 + σ231(46)



1113968 σ

3




σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


σMises

12


+ σθ minus σz( 11138572

+ σz minus σr( 11138572

1113960 1113961 + 3τ2rθ + 3τ2θz + 3τ2zr

1113970

(47)

4 Model Validation










A3

R3

R4

B3 C3

R4

R3


R3

R4

(a)


p2 p3

(b)



ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ndash90

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

0 200 400 600 800 1000

Radi

al st

ress

(MPa

)



ndash80

ndash60

ndash40

ndash20

0 20 40 60 80 100


(a)

ndash70

ndash65

ndash60

ndash55

ndash50

ndash45

ndash40

ndash35

ndash30

ndash25

ndash20

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(b)

ndash200

ndash100

0

100

200

300

400

0 200 400 600 800 1000

Tang

entia

l stre

ss (M

Pa)


ndash200

0

200

400

0 2 4 6 8 10ndash40

ndash30

ndash20

ndash10

0

10 20 30 40 50





(c)

ndash300

ndash200

ndash100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(d)

Figure 14 Continued







0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0

50

100

150

200

250

300

350

400

450

0 30 60 90 120 150 180 210 240 270 300 330 360

Mise

s stre

ss (M

Pa)




(e)


0

05

00

ndash05

ndash10

ndash15

Radi

al d

ispla

cem

ent (

mm

)

ndash20

ndash25300


Casing

0200

ndash02ndash04

0 50 100



900 1200 1500







0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(a)

0

200

400

600

800

0 10 20 30 40 50

Mise

s str

ess (

MPa

)



(b)

0

100

200

300

400

500

600

700

800

0 100 200 300 400

Mise

s str

ess (

MPa

)




(c)

200

300

100

400

500

600M

ises s

tres

s (M

Pa)

00 10 20 30 40 50



(d)












0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











0

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(a)

100

200

300

400

500

600

0 01 02 03 04 05

Mise

s str

ess (

MPa

)



(b)




6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



6 Conclusions





0

200

400

600

800

1000

1200

0 60 120 180 240 300 360

Mise

s str

ess (

MPa

)




(a)

400

600

800

1000

0 05 1 15 2 25 3

Mise

s str

ess (

MPa

)



(b)


0

200

400

600

800

0 15 30 45 60 75 90 105

Mise

s str

ess (

MPa

)




400

500

600

700

0 20 40 60 80 100

Mise

s str

ess (

MPa

)









Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Data Availability




Acknowledgments



References










300

500

400

600

700

0 10 20 30 40 50

Max

imum

mise

s str

ess (

MPa

)



(a)

400

800

600

1000

4 6 8 10 12 1614

Max

imum

mise

s str

ess (

MPa

)



(b)






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


new analytical method to evaluate casing integrity during

Documents