elastoplastic analysis of circular opening based on a new strain … · 2019. 7. 30. · epbm is...

Research ArticleElastoplastic Analysis of Circular Opening Based on a NewStrain-Softening Constitutive Model and Its EngineeringApplication in Hydraulic Fracturing

T Yang 12 and Q S Ye1

1School of Safety Engineering North China Institute of Science and Technology Yanjiao Langfang Hebei 101601 China2State Key Laboratory of Coal Resources and Safe Mining China University of Mining and Technology Beijing 100083 China

Correspondence should be addressed to T Yang yangtao585163com

Received 28 August 2018 Accepted 14 November 2018 Published 10 December 2018

Academic Editor Fan Gu

Copyright copy 2018 T Yang and Q S Ye is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Constitutive effect is extremely important for the research of the mechanical behavior of surrounding rock in hydraulic fracturingengineering In this paper based on the triaxial test results a new elastic-peak plastic-softening-fracture constitutive model(EPSFM) is proposed by considering the plastic bearing behavior of the rock mass en the closed-form solution of a circularopening is deduced with the nonassociated flow rule under the cavity expansion state Meanwhile the parameters of the load-bearing coefficient and brittles coefficient are introduced to describe the plastic bearing capacity and strain-softening degrees ofrock masses When the above two parameters take different values the new solution of EPSFM can be transformed into a series oftraditional solutions obtained based on the elastic-perfectly plastic model (EPM) elastic-brittle plastic model (EBM) elastic-strain-softening model (ESM) and elastic-peak plastic-brittle plastic model (EPBM) erefore it can be applied to a wider rangeof rock masses In addition the correctness of the solution is validated by comparing with the traditional solutions e effect ofconstitutive relation and parameters on the mechanical response of rock mass is also discussed in detail e research results showthat the fracture zone radii of circular opening presents the characteristic of EBM gt EPBM gt ESM gt EPSFM otherwise it is on thecontrast for the critical hydraulic pressure at the softening-fracture zone interface the postpeak failure radii show a linear decreasewith the increase of load-bearing coefficients or a nonlinear increase with the increasing brittleness coefficientis study indicatesthat the rock mass with a certain plastic bearing capacity is more difficult to be cracked by hydraulic fracturing the higher thestrain-softening degree of rock mass is the easier it is to be cracked From a practical point of view it provides very importanttheoretical values for determining the fracture range of the borehole and providing a design value of the minimum pumpingpressure in hydraulic fracturing engineering

1 Introduction

e stresses and plastic zone distribution of the circularopening are extremely important for evaluating the tunnelstability and hydraulic fracturing effect in undergroundengineering However the mechanical response of thesurrounding rock is closely related to rock mass lithology Infact the constitutive relation of different lithology rockmasses generally shows obviously diversity and complexityunder the effect of internal fissures joints componentsand external environment erefore it is difficult to choose

a certain simplified constitutive equation to study thisproblem [1ndash4] In the early stage the elastoplastic analysis ofthe circular opening was firstly investigated by Fenner andthen corrected by Kastner However they regarded the rockmass as the elastic-perfectly plastic material (EPM) It isobviously not reasonable for the brittle plastic and strain-softening rock masses In recent years as shown in Figure 1many studies have been carried out by using the elastic-brittle plastic model (EBM) elastic-strain-softening model(ESM) and elastic-peak plastic-brittle plastic model (EPBM)with the associated and nonassociated flow rule [5ndash9]

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 2806489 12 pageshttpsdoiorg10115520182806489

Nevertheless each constitutive model has its own applica-tion scope e EBM applies only to the poor-quality rockmass while ESM is suitable for average-quality rock massand EPM for high-quality rock mass [10ndash15] In additionEPBM is suitable for the brittle rock masses with a certainplastic bearing capacity [16ndash18] In fact many strain-softening rock masses also show a certain plastic bearingbehavior after load peak

As shown in Figure 2 the silty mudstone and marblewere respectively taken from the Yangzhuang coal mineWushan and Yarsquoan area of China A large number of rockmasses rstly showed the strain-softening characteristicsafter the peak plastic zone and then entered the fracturestage erefore according to the total stress-strain curvethe rock mass approximately experienced four stages in theprocess of the triaxial test at is elastic peak plasticsoftening and fracture stages en the elastic-peak plastic-softening-facture constitutive model (EPSFM) was proposedin this paper and then applied to the engineering practice

Most of the above investigations focus on the com-pression problems of the circular opening However theproblems of cavity expansion have also attracted much at-tention in geotechnical engineering with the application towellbore instability coal-gas exploration and hydraulicfracturing [3 4 19 20] Actually the circular opening ex-pansion is mainly applied in the hydraulic fracturing whichhas been widely used in the hard roof fracturing coalbedmethane extraction and in situ stress measurement

Since the early 1950s numerous analytical solutions tothe circular opening expansion have been studied in ma-terials and geotechnical engineering For instance Gibsonand Anderson applied the cavity expansion theory to in situmeasurement of soil properties with the pressure meter test[19] Li et al obtained the closed-form solution for thehydraulic fracturing borehole which was only applied tohard rock depending on the elastic fracture theory [20]Bishop andMott derived the quasistatic expansion equationsof cylindrical cavities in an innite medium and applied it tothe materials processing [21] Cheng discussed the errorsarising from the assumption of small displacement aroundthe cavity with no volume change in the plastic zone andmodied Kastnerrsquos formula for cylindrical cavity contractionand expansion in the MohrndashCoulomb rock masses [22] Liet al derived the stresses and plastic zone radii of the circular

borehole excavated in the strain-softening coal seam byconsidering contraction and expansion problems [23]

In this paper based on the triaxial test results a newelastic-peak plastic-softening-fracture constitutive model(EPSFM) is rstly proposed and then used to study theborehole expansion problems in underground engineeringFurthermore the validity of this solution is veried bycomparing with a series of traditional solutions based onEBM EPM ESM and EPBM Finally the inuences of theparameters and constitutive models on the mechanic re-sponses of rock mass are discussed in detail

2 Problem Description

21 Establishment of EPSFM As shown in Figure 3 aborehole with the inner radius R0 drilled in an inniteisotropic and homogeneous EPSFM rock masses is sub-jected to an inner hydraulic pressure pin at r R0 andhydrostatic pressure p0 at innite boundary Originally thesurrounding rock is in the elastic state As pin graduallyincrease the peak plastic rstly occurs around the boreholewhen pin is more than the initial yield stresses e stage isnot an innite extension whose range should be restricted bysome factors In this paper assuming the plastic shear strainincrement of the peak plastic zone reaches a certain valuethe surrounding rock of the borehole will enter the softeningstage in which the strength parameters gradually decreaseUntil a residual value is reached the surrounding rocks startto enter the fracture stage Finally it will have four zonesaround the borehole that is elastic zone peak plastic zonesoftening zone and fracture zone Meanwhile the radius ofpeak plastic softening and fracture zones are respectivelydenoted as R3 R2 and R1 e mechanical model shouldsatisfy the following assumption conditions

(i) e borehole is drilled in an innite geological body sothe problem can be regarded as a plane strain problem

(ii) e total strain of the postpeak failure zone onlyconsists of plastic strain and the eiexclect of elastic strainis ignored

For axisymmetric plane strain problems when pin gtp0the hoop stress σθ and radial stress σr are respectively theminimum and maximum principal stresses εθ and εr are theminimum and maximum principal strains respectively

σ 1ndashσ

3

ε1

(a)σ 1

ndashσ3

ε1

(b)

σ 1ndashσ

3

ε1

(c)

σ 1ndashσ

3

ε1

(d)

Figure 1 Simplied traditional constitutive models (a) Elastic-plastic model (EPM) (b) Elastic-brittle-plastic model (EBM) (c) Elastic-strain-softening model (ESM) (d) Elastic-peak plastic-brittle plastic model (EPBM)

2 Advances in Civil Engineering

[20 21] Supposing that the rock mass satises the linearMohrndashCoulomb yield criteria the stress-strain relation atany postpeak stages can be expressed as follows [22 23]

σr Kσθ + σcc (1)

σr Kσθ + σc εr( ) (2)

σr Kσθ + σRR (3)

where σcc and σRR are respectively the initial uniaxialcompressive strength and residual compressive strengthσcc 2c3 cosφ(1minus sinφ) σRR 2c1 cosφ(1minus sinφ) c3and c1 are respectively initial and residual cohesion of rockmass and K is a constant which is related to the strengthparameter φ K (1 + sinφ)(1minus sinφ)

22 Basic Equations and Boundary Condition For theaxisymmetric plane strain problems the equilibrium

diiexclerential equation in the ldquoirdquo zone can be expressed asfollows (ignoring the body force) [7 9]

dσridr

+σri minus σθir

0 (4)

where σri and σθi are the radial and hoop stresses in the ldquoirdquozone respectively e subscript symbol ldquoirdquo representsdiiexclerent zones of surrounding rock which can be replacedby the numbers ldquo0 1 2 and 3rdquo

Based on the supposition of small deformation thegeometric equation for the axisymmetric plane strainproblem can be denoted as [12 13]

εri duridr

εθi urir

(5)

Peak plasticstage

Softening stage

Fracture stage

β2

σ1

σc

σcc

Elastic zone

Peak plasticzone

Softening zone

Fracture zone

σr + dσr

σr

In si

tu st

ress p

0

θσRR

A B

C

β1

β0

σ θ

σθ

R3R2

R1

R0

Pin

+ndash

εA32

εA12

εB32

εB12 ε1O

ε3

Figure 3 Computational mechanical model of EPSFM (note εA12 and εA32 are respectively the maximum and minimum principal strains atldquoArdquo point εB12 and εB32 are respectively the maximum and minimum principal strains at ldquoBrdquo point)

ε1 ()00 10 20 30 40

0

40

20

60

σ3 = 1882MPa

σ 1ndashσ

3 (M

Pa)

σ3 = 1246MPaσ3 = 532MPa

(a)

ε1 ()

000 02 04 06 08 10 12 14 16 18 20 22 24

10

20

30

40

50

σ 1ndashσ

3 (M

Pa)

σ3 = 0MPa σ3 = 128MPaσ3 = 235MPa

σ3 = 350MPa

(b)

ε1 ()

000 05 10 15 20 25 30 35

50

100

150

200

250

σ 1ndashσ

3 (M

Pa)

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa

(c)

Figure 2 e total stress-strain curves by diiexclerent lithology rock masses (a) Silty mudstone from Yangzhuang coal mine (b) Siltymudstone from Wushan (c) Marble from Yarsquoan area of China

Advances in Civil Engineering 3

where εri and εθi are the radial and hoop strains in the ldquoirdquo zonerespectively and uri represents the radial displacement

Supposing that the volume of rock mass is changing therelationship between hoop strain εθi and radial strain εri canbe established by adopting a nonassociated flow rule andsmall strain theory as follows [22 23]

εθi + βiεri 0 (6)

where βi (1 + sinψi)(1minus sinψi) and ψi is the dilatancyangle

Both the radial stress and radial displacement should becontinuous at the elastic-peak plastic peak plastic-softeningand softening-fracture zone interfaces erefore theboundary conditions around the borehole can be summa-rized as

r R0 σr0 pin

r R1 σr0 σr1 ur0 ur1



r⟶infin σr3 p0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

3 Closed-Form Solution of EPSFM

31 Stresses and Displacement of Elastic Zone Based on theelasticity theory the solution of a thick-walled cylinderunder hydrostatic pressure can be easily obtained estresses and displacement for the elastic zone can beexpressed as [18 23]

σr3 p0 + pexp min minusp01113872 1113873R23

r2 (8)

σθ3 p0 minus pexp min minusp01113872 1113873R23

r2 (9)

ur3 A0R23

r

εr3 minusA0R23

r2

εθ3 A0R23

r2

(10)

where pexp min is the minimum critical inner hydraulicpressure at elastic-peak plastic zone interface A0 (1 + ])

(p0 minuspexp min)E and E and ] are Youngrsquos modulus andPoissonrsquos ratio

For the borehole expansion problem both radialand circumferential stresses satisfy the MohrndashCoulombyield criteria at the elastic-peak plastic zone interfaceHence the parameters pexp min can be easily deduced bysubstituting equations (8) and (9) into equation (1) asfollows

pexp min 2Kp0 + σcc

1 + K (11)

Considering the boundary condition (ur(iminus1))rRi

(uri)rRiby equation (7) the radial displacement and strains

in the postpeak failure zones can be easily deduced based onthe small deformation supposition and volume expansionassumption by substituting equation (5) into equation (6)e calculation results are shown in Table 1

32 Stresses Distribution of Peak Plastic and Fracture ZonesWhen the inner hydraulic pressure remains at a certainvalue the surrounding rock of the borehole is in the stressequilibrium state in the peak plastic and fracture zoneserefore the principal stresses should satisfy the equations(1) and (4) in the peak plastic zone or equations (3) and (4) inthe fracture zone

In the above two zones the equilibrium differentialequation can be rewritten by substituting equation (1) orequation (3) into equation (4) as follows

dσridr

+(1minus(1K))σri + σjjK1113872 1113873

r 0 (12)

where σjj equals to σcc in the peak plastic zone or equals toσRR in the fracture zone

Solving equation (12) the stresses in the peak plasticzone can be obtained by combining with the boundarycondition (σr2)rR3

pexp min

σr2 pexp min +σcc

Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1

σθ2 Kminus1 pexp min +σcc

Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

Meanwhile the stresses in the fracture zone can be alsoeasily deduced by considering (σr0)rR0

pin

σr0 pin +σRR

Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875

σθ0 Kminus1 pin +σRR

Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

33 Stresses Distribution of Softening Zone By consideringthe condition σc(εr1) σcc at r R2 and dσc λd(εr1) thecompressive strength in the softening zone can be obtainedas

σc εr1( 1113857 σcc α(1 + v) pexp min minusp01113872 1113873βminus11 R1+βminus123 R

minusβminus12 minus12

middotR2

r1113874 1113875

1+ βminus11 minus1( )⎡⎣ ⎤⎦

(15)

where α λE which can be defined as a brittleness co-efficient and represents the strain-softening degree of rockmass and λ may be called the strain-softening modulus


Introducing equations (2) and (15) into equation (4) theequilibrium differential equation in the softening zone canbe deduced as

dσr1dr

+1minusKminus1( 1113857σr1

r+

Kminus1 σcc minus α(1 + ]) pexp min minusp01113872 1113873βminus11 middot R1+βminus123 R

minusβminus12 minus12 R2r( 1113857

1+βminus11 minus 11113876 11138771113882 1113883

r 0

(16)

e radial stress at the peak plastic-softening interfacemust be coincided thus it can be obtained by solvingequation (16) and considering the boundary condition σr1

σr2 at r R2

σr1 pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+σcc1minusK

+αKminus1βminus11 (1 + ]) pexp min minusp01113872 1113873

βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +

αβminus11 (1 + ]) pexp min minusp01113872 1113873

1minusK

+R3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891

(17)

en by introducing equations (15) and (17) intoequation (2) the hoop stress is

σθ1 1K

⎧⎨

⎩ pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+Kσcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

middotR3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891 + α(1 + ])

middot pexp min minusp01113872 1113873βminus11 R1+βminus123 R


R2

r1113874 1113875

1+βminus11minus 1⎡⎣ ⎤⎦

⎫⎬

⎭

(18)

34 Radius (R3 R2 R1) of Postpeak Failure Zones As theinner hydraulic pressure pin gradually increasing the sur-rounding rock of the borehole will experience four stagesat is elastic stage elastic-peak plastic stage elastic-peakplastic-softening stage and elastic-peak plastic-softening-fracture stage

341 Elastic-Peak Plastic Stage In this stage the sur-rounding rock of the borehole only consists of elastic andpeak plastic zones e range of the peak plastic zonegradually increases with the increase of the inner hydraulicpressure As shown in Figure 3 when the plastic shearstrain increment of the peak plastic zone increases to aparticular value the rock mass will reach the maximumpeak plastic state in which the softening zone is just notarisen Hence we can define a load-bearing coefficient Δcwhich can be calculated by the difference of the plasticshear strain in section ldquoABrdquo of Figure 3 to describe theplastic bearing capacity of rock masse parameter Δc canbe expressed as follows

Δc cB minus cA εB12 minus εB321113872 1113873minus εA12 minus ε

A321113872 1113873

εr2 minus εθ2( 1113857rR0minus εr2 minus εθ2( 1113857rR3

(19)

where cB and cA represent the plastic shear strain at pointsldquoBrdquo and ldquoArdquo respectively ey can easily be determined bythe experiment Hence the radius of the peak plastic zonecan be obtained as

R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R0 TR0 (20)

Presently the middle critical inner hydraulicpressure pexpmid at the peak plastic-softening zone in-terface can be solved by introducing equation (20) intoequation (13)

Table 1 Radial displacement and strain of the postpeak failure zones

State Calculated variable Peak plastic zone Softening zone Fracture zone

Expansion

Displacement uri A0R1+βminus123 rminusβ

minus12 A0R

1+βminus123 R

βminus11 minusβminus12

2 r+βminus11 A0R1+βminus123 R


2 Rβminus10 minusβ

minus11

1 rminusβminus10

Radial strain εri minusA0βminus12 R

1+βminus123 rminusβ

minus12 minus1 minusA0β

minus11 R

1+βminus123 R


2 rminusβminus11 minus1 minusA0β

minus10 R

1+βminus123 R



minus11

1 rminusβminus10 minus1

Hoop strain εθi A0R1+βminus123 rminusβ

minus12 minus1 A0R

1+βminus123 R


2 rminusβminus11 minus1 A0R

1+βminus123 R



minus11



pexpmid pexp min +σcc

Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦

minus Kminus1minus1( ) 1+βminus12( )( )

minusσcc

Kminus 1

(21)

342 Elastic-Peak Plastic-Softening Stage When R3R0 gtTthe softening zone appears If assuming that the surroundingrock is in the critical state where the fracture zone is not yetarisen equation (20) can be rewritten as

R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)

By integrating equation (15) according to σc(εr1) σRRat r R0 the relationship between R3 and R2 can be ob-tained as follows

R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889

1+βminus11minus 1⎡⎢⎢⎣ ⎤⎥⎥⎦

σcc minus σRRα(1 + ]) pexp min minusp01113872 1113873βminus11

(23)

en by substituting equation (22) into equation (23)the softening zone radii can be expressed as

R2 σcc minus σRR

α(1 + ]) pexp min minusp01113872 1113873βminus11 T1+βminus12+ 1⎛⎝ ⎞⎠

1 1+βminus11( )

middot R0 tR0

(24)

At this state introducing equations (22) and (24) intoequation (17) the maximum critical inner hydraulic pres-sure pexp max can be calculated as follows

pexp max pexp min +σcc

Kminus 11113874 1113875T

1minusKminus1t1minusKminus1

+σcc1minusK


βminus11 + Kminus1T1+βminus12

middot t1minusKminus1 minus t

1+βminus111113876 1113877 +αβminus11 (1 + ]) pexp min minusp01113872 1113873

1minusKT1+βminus12

middot t1minusKminus1 minus 11113876 1113877

(25)

343 Elastic-Peak Plastic-Softening-Fracture Stage WhenR2R0 gt t it means that the rock mass has entered into thefracture stage According to equations (22) and (24) therelationship of R3 TR2 and R2 tR1 is easily deduced Inaddition the radial stress should be consistent at thesoftening-fracture zone interface erefore we can obtain

pexp min +σcc

Kminus 11113874 1113875T


+σcc minus σRR1minusK


βminus11 + Kminus1T1+βminus12 middot t

1minusKminus1 minus t1+βminus111113874 1113875

+αβminus11 (1 + ]) pexp min minusp01113872 1113873

1minusKT1+βminus12 t

1minusKminus1 minus 11113874 1113875

minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)

Integrating equation (26) the fracture zone radius can beobtained as follows

R1 R0pexp min + σcc(Kminus 1)( 11138571113872 1113873T1minusKminus1t1minusK

minus1+ σcc minus σRR( 1113857(1minusK)( 1113857 + αKminus1βminus11 (1 + ]) pexp min minusp01113872 11138731113872 1113873 βminus11 + Kminus11113872 11138731113872 1113873T1+βminus12 middot t1minusK

minus1 minus t1+βminus111113872 1113873 + αβminus11 (1 + ]) pexp min minusp01113872 11138731113872 1113873(1minusK)1113872 1113873T1+βminus12 t1minusKminus1 minus 11113872 1113873

pin + σRR(Kminus 1)( 1113857

⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)

en the radius of peak plastic and softening zones canalso be calculated by introducing equation (27) intoR3 TR2 and R2 tR1

35 Discussion and Transformation with Traditional Modele new closed-form solution based on the EPSFM can bedegenerated for different traditional solutions based on theEPM EBM ESM and EPBM in a particular situation Forinstance only when Δc 0 the results of EPSFM can betranslated into the results of ESM [23] when Δc 0α⟶infin the EPSFM converts to the EBM if assuming thatΔc 0 and α 0 the EPSFM solution degenerates for EPMsolution [22] only when α⟶infin the EPSFM solutionchanges to the EPBM solution It includes not only the

traditional results but also a series of new results comparedwith the traditional ones Hence it can be regarded as aunified analytical solution In other words the new closed-form solution can generate a broad range of theoretical andpractical values in circular opening expansion engineeringespecially in the hydraulic fracturing

When load-bearing coefficient Δc and brittleness co-efficient α take special values the new analytical solution willdegenerate for a series of traditional solutions It mainlyincludes four different cases

Case 1 When Δc 0 and T limΔc⟶0

T 1 the peak plasticzone will disappear and then the EPSFM degenerates intothe elastic-strain-softening model


In this state the softening and fracture zones radius canbe obtained by solving equation (27)

R2 σcc minus σRR

α(1 + ]) pexp min minusp01113872 1113873βminus11+ 1⎛⎝ ⎞⎠

1 1+βminus11( )

R1 tR1 (28)

R1 R0pexp min + σcc(Kminus 1)( 11138571113872 1113873t1minusK

minus1+ σcc minus σRR( 1113857(1minusK)( 1113857 + αKminus1βminus11 (1 + ]) pexp min minusp01113872 11138731113872 1113873 βminus11 + Kminus11113872 1113873 middot t1minusK

minus1 minus t1+βminus111113872 1113873 + αβminus11 (1 + ]) pexp min minusp01113872 11138731113872 1113873(1minusK) t1minusKminus1 minus 11113872 1113873


⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)

When β1 1 equations (28) and (29) are the solutionsobtained by Li et al [23] for the circular opening expansion

en integrating equation (25) the maximum criticalinner hydraulic pressure pexp max at r R1 can be rewrittenas follows


Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK


βminus11 + Kminus1middot t



1minusKt1minusKminus1 minus 11113876 1113877

(30)

Case 2 When Δc 0 α⟶infin T limΔc⟶ 0

T 1 andt lim

α⟶infint 1 the EPSFM converts to the elastic-brittle

plastic model e stress at the elastic-fracture zone interfacepresents instantaneous dropping characteristics Howeverthe radius of the fracture zone cannot be given directly efracture zone radius can be deduced by considering theboundary condition pexp min (σr0)rR1

as follows

R1 R02Kp0 + σcc( 1113857(1 + K)( 1113857 + σRR( 1113857(Kminus 1)( 1113857

pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)

Case 3 When α⟶infin and t limα⟶infin

t 1 the softeningzone will disappear us the EPSFM degenerates into theelastic-peak plastic-brittle plastic model Meanwhile themaximum principal stress between peak plastic and fracturezones shows obvious drop characteristics In this state theradius of peak plastic and fracture zones can be deduced byintegrating equations (22) and (27)

R1 R0pexp min + σcc(Kminus 1)( 11138571113872 1113873T1minusKminus1 + σcc minus σRR( 1113857(1minusK)( 1113857

pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

R3 TR0pexp min + σcc(Kminus 1)( 11138571113872 1113873T1minusKminus1 + σcc minus σRR( 1113857(1minusK)( 1113857

pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)

Case 4 When Δc 0 α 0 T limΔc⟶0

T 1 andσcc σRR the surrounding rock is only composed of theelastic and peak plastic zones erefore the EPSFM be-comes the elastic-perfectly plastic model e radius of thepeak plastic zone can also be deduced by considering theboundary condition (σr2)rR0

pin

R3 R0pin + σcc(Kminus 1)( 1113857

2Kp0 + σcc( 1113857(1 + K)( 1113857 + σcc(Kminus 1)( 11138571113888 1113889

1 1minusKminus1( )

(33)

e analytical solution of equation (33) is the same withreference results (Cheng [22])

4 Case Studies

41 Case I Comparative Analysis Constitutive effect is ex-tremely important for researching the mechanics and de-formation behavior of rock mass To validate the developedmodel in this paper and study the influence of constitutiverelation on the mechanics response of the rock mass thegeometrical and physical parameters of a circular openingare shown in Table 2 Moreover the load-bearing coefficientis assumed as 0004

e circular opening expansion theory is mainly appliedto hydraulic fracturing in underground engineering estresses distribution law under different constitutive modelsis shown in Figure 4 In addition Table 3 presents themaximum inner hydraulic pressure pexp max at the softening-fracture zone interface It can be seen from Figure 4 andTable 3 that the maximum critical pressure shows thecharacteristics of EBM lt EPBM lt ESM lt EPSFM Bycomparing with the EBM EPBM and ESM rock masses themaximum critical pressure of EPSFM increases by9895MPa 7752MPa and 1286MPa respectively It meansthat the EPSFM rock mass is the hardest to be crackedwhereas the EBM rock mass is the easiest in the process ofhydraulic fracturing

e influence of constitutive relation on the postpeakfailure radii is shown in Figure 5 When the inner hydraulicpressure is equal to 40MPa the radii of R1 R2 and R3 showthe characteristics of EBM gt EPBM gt ESM gt EPSFMerefore the above results indicate that the rock mass witha certain plastic bearing capacity is more difficult to becracked in hydraulic fracturing engineering In other words


the design of hydraulic fracturing pressure should take fullaccount of the inuence of lithology to achieve the best crackeiexclect

42 Case II Parameter Analysis A case of hydraulic frac-turing in coal seam is used to study the mechanical responseof rock masses with the change of hydraulic pressure einuence of parameters on the surrounding rock state is alsodiscussede hydraulic fracturing case was implemented inNo 7601 coal seam with high gas in Wuyang Coal Mine ofChina for improved gas extraction e coalbed was buriedat about 480m underground e average value of hydro-static pressure p0 is 716MPa the radius of the borehole R0 is01m Youngrsquos modulus E and Poissonrsquos ratio ] are 30GPaand 028 respectively the initial cohesion c3 and the internalfriction angle φ are 15MPa and 30deg and σcc and σRR arerespectively about 52MPa and 12MPa Moreover theload-bearing coecopycient Δc and brittleness coecopycient α are00006 and 12 respectively It should be noted that theinuence of the dilatancy coecopycient is ignored (βi 1) inorder to avoid the errors arising from the volume change ofpostpeak rock mass

421 Stresses and Postpeak Failure Radii Evolution LawFigure 6 shows the stress evolution law with the change of thecritical hydraulic pressure In the present example it can beseen that there is only elastic zone around the borehole when716MPalepin le 12039MPa (Figure 6(a)) ere are elasticand peak plastic zones when 12039MPalepin le 12711MPa(Figure 6(b)) en the surrounding rock of the borehole iscomposed of elastic peak plastic and softening zones if12711MPalepin le 14917MPa (Figure 6(c)) Finally thesurrounding rock consists of four zones if pin ge 14917MPa(Figure 6(d)) In addition σr gt σθ is commonly found inFigure 6 for the borehole expansion

e radius of the postpeak failure zone is also signi-cantly important for evaluating the hydraulic fracturingeiexclect and optimizing the layout of the boreholes e radiusof the peak plastic softening and fracture zones evolu-tion law under diiexclerent hydraulic pressures are shown inFigure 7 It is clear that there is no postpeak failure zonewhen p0 lepin lepexp min e radius gradually increases withthe increasing of the hydraulic pressure in the rangepin gepexp min for the circular opening expansion Figure 7 isof great practical signicance because the threshold of thecritical hydraulic pressure pexp max has an important theo-retical value for providing a design value of the minimumpumping pressure compared with the traditional empiricism[23] In this case the threshold of calculation is 14917MPaand is in good accordance with the eld test results(1454MPa)

10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone

EPSFMCritical stress state

Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone

EBMCritical stress state

(b)

Figure 4 Stress distribution law for borehole expansion (notebecause the maximum critical inner hydraulic pressure pexp max istaken as the calculated inner pressure the fracture zone does notappear)

Table 2 Geometrical and physical parameters of circular opening(data from Li et al [23])

Parameter ValueRadius of opening R0 (m) 01In situ stress p0 (MPa) 15Inner pressures pin (MPa) 0 or 40Youngrsquos modulus E (MPa) 1500Dilatancy coecopycient βi 10Poissonrsquos ratio ] 03Brittleness coecopycient α 05Internal friction angle φ (deg) 30Initial compressive strength σcc (MPa) 8Residual compressive strength σRR (MPa) 1

Table 3 e maximum critical inner hydraulic pressure pexp max(MPa)

Model EBM EPBM ESM EPSFMValue 24500 26643 33109 34395


Maximum hydraulic pressure (40MPa)

10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)

Figure 5 e radii distribution laws for circular opening expansion

σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)

Figure 6 Stresses evolution law for the borehole expansion


422 Inuence of Load-Bearing Coecient e load-bearing coecopycient Δc reects the plastic bearing capacityof rock mass and is extremely important for determining thefracture range and the critical hydraulic pressure in theprocess of hydraulic fracturing e radii of the postpeakfailure zone evolution law are shown in Figure 8 It can beseen that the postpeak failure radii obviously decrease withthe increase of the load-bearing coecopycient However thedecreasing rate of softening zone radii is the maximum Forinstance when Δc transforms from 2 times 10minus3 to 5 times 10minus4 theradii R1 R2 and R3 respectively decrease by 158mm259mm and 18mm It means that the greater the Δc is thestronger the plastic bearing capacity of the rockmass and thesmaller the fracture range of the drill hole are Here theinner hydraulic pressure is set at 20MPa (gt15986MPa)(Table 4) in order to make the rock mass enter the residualstate

In addition the load-bearing coecopycient also has a veryimportant eiexclect on the critical hydraulic pressure As shownin Table 4 pexpmid and pexp max respectively decrease by1987MPa and 1525MPa with the load-bearing coecopycientΔc decreasing from 2 times 10minus3 to 5 times 10minus5 e conclusion canprovide exceedingly important reference for determining thethreshold of maximum critical hydraulic pressure in hy-draulic fracturing engineering

423 Inuence of Brittleness Coecient Figure 9 shows theinuence of brittleness coecopycients (α) on the postpeakfailure radii With the parameter (α) increasing the postpeakfailure radii show a nonlinear increase characteristicHowever the increase rate is gradually decreasing For in-stance when α changes from 06 to 2 the radii R1 R2 andR3 respectively increase by 356mm 67mm and 72mmIn addition as shown in Table 5 the maximum criticalhydraulic pressure pexp max is negatively correlated with the

brittleness coecopycient (α) e above result shows that thehigher the strain-softening degree of rock mass is the easierit is to be cracked by hydraulic fracturing

5 Conclusions

Based on the triaxial test results a new elastic-peak plastic-softening-fracture constitutive model (EPSFM) is proposedby considering the plastic bearing behavior of the siltymudstone en the closed-form solution of a circularopening based on the new proposed constitutive model isdeduced with the nonassociated ow rule under the cavityexpansion state e correctness of the solution is alsoveried by comparing with the traditional solutions eeiexclect of the constitutive relation and parameters on themechanical response of rock mass is also discussed in detaile primary conclusions can be summarized as follows

(1) e new closed-form solution based on EPSFMconsidering the eiexclect of plastic bearing capacity ofrock masses can be regarded as a uniform solutioncompared with the traditional research results Onlywhen the load-bearing coecopycient is equal to zero thecalculated results of the EPSFM can be converted tothe ESMrsquos solution only when the brittleness co-ecopycient is large enough or zero the EPSFMrsquos so-lution turned to the result by EPBM or EPMMeanwhile when the load-bearing coecopycient is zeroand the brittleness coecopycient is large enough thecalculated results of the EPSFM was found to be inaccordance with the closed-form solution of theEBM

(2) In hydraulic fracturing engineering when thehydraulic pressure remains at a certain values

10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2

Figure 7 Radius of postpeak failure zones evolution law withhydraulic pressure

0 05 10 15 20013

014

015

016

017

018

019

020

Load-bearing coefficient (10minus3)

R 1 R

2 R 3

(m)

R1

R2

R3

Figure 8 e radii of the postpeak failure zone under diiexclerentload-bearing coecopycients


the fracture zone radii of circular opening presentthe characteristic of EBM gt EPBM gt ESM gt EPSFMotherwise it is on the contrast for the critical hy-draulic pressure at the softening-fracture zone in-terfaceerefore the EPSFM rockmass is hardest tobe cracked whereas the EBM rock mass is easiest inthe process of hydraulic fracturing

(3) e postpeak failure radii show obviously a lineardecrease with the increase of load-bearing co-ecopycients or a nonlinear increase with the increasingbrittleness coecopycient It means that for the bestfracturing eiexclects the design of hydraulic fracturingpressure should take full account of the inuence ofrock mass lithology load-bearing coecopycient andbrittleness coecopycient

Data Availability

e article data used to support the ndings of this study areincluded within the article

Conflicts of Interest

e authors declare that there are no conicts of interestregarding the publication of this paper

Acknowledgments

e authors would like to thank the nancial support fromthe National Natural Science Foundation for Young Sci-entists of China (51604116) State Key Laboratory of CoalResources and Safe Mining (China University of Mining andTechnology) (SKLCRSM16KFB10) Fundamental ResearchFunds for the Central Universities (3142018028) NaturalScience Foundation of Hebei Province (E2016508036) andState Key Laboratory Cultivation Base for Gas Geology andGas Control (Henan Polytechnic University) (WS2017B07)

References

[1] L Placidi and E Barchiesi ldquoEnergy approach to brittlefracture in strain-gradient modellingrdquo Proceedings of theRoyal Society A Mathematical Physical and Engineering Sci-ences vol 474 no 2212 article 20170878 2018

[2] A H Wilson ldquoA method of estimating the closure andstrength of lining required in drivages surrounded by a yieldzonerdquo International Journal of Rock Mechanics and MiningSciences and Geomechanics Abstracts vol 17 no 6pp 349ndash355 1980

[3] Y J Ning J Yang and P W Chen ldquoNumerical simulation ofrock blasting in jointed rock mass by DDA methodrdquo Rock ampSoil Mechanics vol 31 no 7 pp 2259ndash2263 2010

[4] J F Zou W Q Tong and J Zhao ldquoEnergy dissipation ofcavity expansion based on generalized non-linear failurecriterion under high stressesrdquo Journal of Central SouthUniversity vol 19 no 5 pp 1419ndash1424 2012

[5] H Zhang Z Wan D Ma Y Zhang J Cheng and Q ZhangldquoExperimental investigation on the strength and failure be-havior of coal and synthetic materials under plane-strainbiaxial compressionrdquo Energies vol 10 no 4 p 500 2017

[6] E Hoek and E T Brown ldquoPractical estimates of rock massstrengthrdquo International Journal of Rock Mechanics andMining Science amp Geomechanics Abstracts vol 34 no 8pp 1165ndash1186 1997

[7] K H Park B Tontavanich and J G Lee ldquoA simple procedurefor ground response curve of circular tunnel in elastic-strainsoftening rock massesrdquo Tunnelling and Underground SpaceTechnology vol 23 no 2 pp 151ndash159 2008

[8] Y K Lee and S Pietruszczak ldquoA new numerical procedure forelasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and UndergroundSpace Technology vol 23 no 5 pp 588ndash599 2008

[9] Q Zhang B S Jiang S L Wang X R Ge andH Q Zhang ldquoElasto-plastic analysis of a circular openingin strain-softening rock massrdquo International Journal of

Table 4 e critical hydraulic pressure under diiexclerent load-bearing coecopycients

State Δc pconmax pexp min (MPa) pconmid pexp max (MPa) pconmin pexp max (MPa)

Expansion

2 times 10minus3 12039 14084 159861 times 10minus3 12039 13128 152355 times 10minus4 12039 12603 148365 times 10minus5 12039 12097 14461

05 1 15 201

012

014

016

018

02

022

Brittleness coefficient

R 1 R

2 R 3

(m)

R1

R2

R3

Figure 9 e radii of the postpeak failure zone under diiexclerentbrittleness coecopycients

Table 5 Maximum critical hydraulic pressure under diiexclerentbrittleness coecopycients (MPa)

α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679


Rock Mechanics and Mining Sciences vol 50 no 1pp 38ndash46 2012b

[10] S L Wang H Zheng C G Li and X R Ge ldquoA finite elementimplementation of strain-softening rock massrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 48 no 1pp 67ndash76 2011

[11] S K Sharan ldquoExact and approximate solutions for dis-placements around circular openings in elastic-brittle-plasticHoek-Brown rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 42 no 4 pp 542ndash549 2005

[12] K H Park and Y J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 43 no 4pp 616ndash622 2006

[13] Q Zhang B S Jiang X S Wu H Q Zhang and L J HanldquoElasto-plastic coupling analysis of circular openings inelasto-brittle-plastic rock massrdquo Georetical and AppliedFracture Mechanics vol 60 no 1 pp 60ndash67 2012a

[14] S L Wang X T Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[15] Q Zhang B S Jiang and H J Lv ldquoAnalytical solution for acircular opening in a rock mass obeying a three-stage stress-strain curverdquo International Journal of Rock Mechanics andMining Sciences vol 86 pp 16ndash22 2016

[16] B S Jiang Q Zhang Y N He et al ldquoElastioplastic analysis ofcracked surrounding rocks in deep circular openingsrdquo Chi-nese Journal of Rock Mechanics and Engineering vol 26 no 5pp 982ndash986 2007 in Chinese

[17] M H Yu S Y Yang S C Fan and G W Ma ldquoUnifiedelastoplastic associated and non-associated constitutivemodeland its engineering applicationsrdquo Computers and Structuresvol 71 no 6 pp 627ndash636 1999

[18] C G Zhang J F Wang and J H Zhao ldquoUnified solutions forstresses and displacements around circular tunnels using theunified strength theoryrdquo Science China Technological Sciencesvol 53 no 6 pp 1694ndash1699 2010

[19] R E Gibson and W F Anderson ldquoIn-situ measurement ofsoil properties with the pressuremeterrdquo Civil Engineering andPublic Works Review vol 56 pp 615ndash618 1961

[20] Y Li N Fantuzzi and N Tornabene ldquoOn mixed mode crackinitiation and direction in shafts strain energy density factorand maximum tangential stress criteriardquo Engineering FractureMechanics vol 109 no 1 pp 273ndash289 2013

[21] R F Bishop and N F Mott ldquoe theory of indentation andhardnessrdquo Proceedings of the Physical Society vol 57 no 3pp 147ndash159 1945

[22] Y M Cheng ldquoModified Kastner formula for cylindrical cavitycontraction in Mohr-Coulomb medium for circular tunnel inisotropic mediumrdquo Journal of Mechanics vol 28 no 1pp 163ndash169 2012

[23] Y Li S G Cao F Nicholas and Y Liu ldquoElastoplastic analysisof a circular borehole in elastic-strain softening coal seamsrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 80 pp 316ndash324 2015


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Submit your manuscripts atwwwhindawicom

Nevertheless each constitutive model has its own applica-tion scope e EBM applies only to the poor-quality rockmass while ESM is suitable for average-quality rock massand EPM for high-quality rock mass [10ndash15] In additionEPBM is suitable for the brittle rock masses with a certainplastic bearing capacity [16ndash18] In fact many strain-softening rock masses also show a certain plastic bearingbehavior after load peak

As shown in Figure 2 the silty mudstone and marblewere respectively taken from the Yangzhuang coal mineWushan and Yarsquoan area of China A large number of rockmasses rstly showed the strain-softening characteristicsafter the peak plastic zone and then entered the fracturestage erefore according to the total stress-strain curvethe rock mass approximately experienced four stages in theprocess of the triaxial test at is elastic peak plasticsoftening and fracture stages en the elastic-peak plastic-softening-facture constitutive model (EPSFM) was proposedin this paper and then applied to the engineering practice

Most of the above investigations focus on the com-pression problems of the circular opening However theproblems of cavity expansion have also attracted much at-tention in geotechnical engineering with the application towellbore instability coal-gas exploration and hydraulicfracturing [3 4 19 20] Actually the circular opening ex-pansion is mainly applied in the hydraulic fracturing whichhas been widely used in the hard roof fracturing coalbedmethane extraction and in situ stress measurement

Since the early 1950s numerous analytical solutions tothe circular opening expansion have been studied in ma-terials and geotechnical engineering For instance Gibsonand Anderson applied the cavity expansion theory to in situmeasurement of soil properties with the pressure meter test[19] Li et al obtained the closed-form solution for thehydraulic fracturing borehole which was only applied tohard rock depending on the elastic fracture theory [20]Bishop andMott derived the quasistatic expansion equationsof cylindrical cavities in an innite medium and applied it tothe materials processing [21] Cheng discussed the errorsarising from the assumption of small displacement aroundthe cavity with no volume change in the plastic zone andmodied Kastnerrsquos formula for cylindrical cavity contractionand expansion in the MohrndashCoulomb rock masses [22] Liet al derived the stresses and plastic zone radii of the circular

borehole excavated in the strain-softening coal seam byconsidering contraction and expansion problems [23]

In this paper based on the triaxial test results a newelastic-peak plastic-softening-fracture constitutive model(EPSFM) is rstly proposed and then used to study theborehole expansion problems in underground engineeringFurthermore the validity of this solution is veried bycomparing with a series of traditional solutions based onEBM EPM ESM and EPBM Finally the inuences of theparameters and constitutive models on the mechanic re-sponses of rock mass are discussed in detail

2 Problem Description

21 Establishment of EPSFM As shown in Figure 3 aborehole with the inner radius R0 drilled in an inniteisotropic and homogeneous EPSFM rock masses is sub-jected to an inner hydraulic pressure pin at r R0 andhydrostatic pressure p0 at innite boundary Originally thesurrounding rock is in the elastic state As pin graduallyincrease the peak plastic rstly occurs around the boreholewhen pin is more than the initial yield stresses e stage isnot an innite extension whose range should be restricted bysome factors In this paper assuming the plastic shear strainincrement of the peak plastic zone reaches a certain valuethe surrounding rock of the borehole will enter the softeningstage in which the strength parameters gradually decreaseUntil a residual value is reached the surrounding rocks startto enter the fracture stage Finally it will have four zonesaround the borehole that is elastic zone peak plastic zonesoftening zone and fracture zone Meanwhile the radius ofpeak plastic softening and fracture zones are respectivelydenoted as R3 R2 and R1 e mechanical model shouldsatisfy the following assumption conditions

(i) e borehole is drilled in an innite geological body sothe problem can be regarded as a plane strain problem

(ii) e total strain of the postpeak failure zone onlyconsists of plastic strain and the eiexclect of elastic strainis ignored

For axisymmetric plane strain problems when pin gtp0the hoop stress σθ and radial stress σr are respectively theminimum and maximum principal stresses εθ and εr are theminimum and maximum principal strains respectively

σ 1ndashσ

3

ε1

(a)σ 1

ndashσ3

ε1

(b)

σ 1ndashσ

3

ε1

(c)

σ 1ndashσ

3

ε1

(d)

Figure 1 Simplied traditional constitutive models (a) Elastic-plastic model (EPM) (b) Elastic-brittle-plastic model (EBM) (c) Elastic-strain-softening model (ESM) (d) Elastic-peak plastic-brittle plastic model (EPBM)









dσridr

+σri minus σθir

0 (4)



εri duridr

εθi urir

(5)

Peak plasticstage

Softening stage

Fracture stage

β2

σ1

σc

σcc

Elastic zone

Peak plasticzone

Softening zone

Fracture zone

σr + dσr

σr

In si

tu st

ress p

0

θσRR

A B

C

β1

β0

σ θ

σθ

R3R2

R1

R0

Pin

+ndash

εA32

εA12

εB32

εB12 ε1O

ε3


ε1 ()00 10 20 30 40

0

40

20

60

σ3 = 1882MPa

σ 1ndashσ

3 (M

Pa)

σ3 = 1246MPaσ3 = 532MPa

(a)

ε1 ()

000 02 04 06 08 10 12 14 16 18 20 22 24

10

20

30

40

50

σ 1ndashσ

3 (M

Pa)

σ3 = 0MPa σ3 = 128MPaσ3 = 235MPa

σ3 = 350MPa

(b)

ε1 ()

000 05 10 15 20 25 30 35

50

100

150

200

250

σ 1ndashσ

3 (M

Pa)

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa

(c)








r R0 σr0 pin




r⟶infin σr3 p0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)




r2 (8)


r2 (9)

ur3 A0R23

r

εr3 minusA0R23

r2

εθ3 A0R23

r2

(10)





1 + K (11)






dσridr

+(1minus(1K))σri + σjjK1113872 1113873

r 0 (12)



pexp min

σr2 pexp min +σcc

Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1


Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


pin

σr0 pin +σRR

Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875


Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)




middotR2

r1113874 1113875


(15)




dσr1dr


r+



1+βminus11 minus 11113876 11138771113882 1113883

r 0

(16)


σr2 at r R2

σr1 pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+σcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

+R3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891

(17)


σθ1 1K

⎧⎨

⎩ pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+Kσcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

middotR3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891 + α(1 + ])



R2

r1113874 1113875


⎫⎬

⎭

(18)




A321113872 1113873


(19)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R0 TR0 (20)




Expansion


minus12 A0R

1+βminus123 R





minus11

1 rminusβminus10




minus11 R

1+βminus123 R



minus10 R

1+βminus123 R



minus11



minus12 minus1 A0R

1+βminus123 R



1+βminus123 R



minus11




Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦


minusσcc

Kminus 1

(21)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)


R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889



(23)


R2 σcc minus σRR


1 1+βminus11( )

middot R0 tR0

(24)



Kminus 11113874 1113875T


+σcc1minusK





1minusKT1+βminus12


(25)


pexp min +σcc

Kminus 11113874 1113875T









minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)






⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)









R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









dσridr

+σri minus σθir

0 (4)



εri duridr

εθi urir

(5)

Peak plasticstage

Softening stage

Fracture stage

β2

σ1

σc

σcc

Elastic zone

Peak plasticzone

Softening zone

Fracture zone

σr + dσr

σr

In si

tu st

ress p

0

θσRR

A B

C

β1

β0

σ θ

σθ

R3R2

R1

R0

Pin

+ndash

εA32

εA12

εB32

εB12 ε1O

ε3


ε1 ()00 10 20 30 40

0

40

20

60

σ3 = 1882MPa

σ 1ndashσ

3 (M

Pa)

σ3 = 1246MPaσ3 = 532MPa

(a)

ε1 ()

000 02 04 06 08 10 12 14 16 18 20 22 24

10

20

30

40

50

σ 1ndashσ

3 (M

Pa)

σ3 = 0MPa σ3 = 128MPaσ3 = 235MPa

σ3 = 350MPa

(b)

ε1 ()

000 05 10 15 20 25 30 35

50

100

150

200

250

σ 1ndashσ

3 (M

Pa)

σ3 = 10MPa

σ3 = 20MPa

σ3 = 0MPa

(c)








r R0 σr0 pin




r⟶infin σr3 p0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)




r2 (8)


r2 (9)

ur3 A0R23

r

εr3 minusA0R23

r2

εθ3 A0R23

r2

(10)





1 + K (11)






dσridr

+(1minus(1K))σri + σjjK1113872 1113873

r 0 (12)



pexp min

σr2 pexp min +σcc

Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1


Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


pin

σr0 pin +σRR

Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875


Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)




middotR2

r1113874 1113875


(15)




dσr1dr


r+



1+βminus11 minus 11113876 11138771113882 1113883

r 0

(16)


σr2 at r R2

σr1 pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+σcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

+R3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891

(17)


σθ1 1K

⎧⎨

⎩ pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+Kσcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

middotR3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891 + α(1 + ])



R2

r1113874 1113875


⎫⎬

⎭

(18)




A321113872 1113873


(19)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R0 TR0 (20)




Expansion


minus12 A0R

1+βminus123 R





minus11

1 rminusβminus10




minus11 R

1+βminus123 R



minus10 R

1+βminus123 R



minus11



minus12 minus1 A0R

1+βminus123 R



1+βminus123 R



minus11




Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦


minusσcc

Kminus 1

(21)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)


R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889



(23)


R2 σcc minus σRR


1 1+βminus11( )

middot R0 tR0

(24)



Kminus 11113874 1113875T


+σcc1minusK





1minusKT1+βminus12


(25)


pexp min +σcc

Kminus 11113874 1113875T









minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)






⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)









R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







r R0 σr0 pin




r⟶infin σr3 p0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)




r2 (8)


r2 (9)

ur3 A0R23

r

εr3 minusA0R23

r2

εθ3 A0R23

r2

(10)





1 + K (11)






dσridr

+(1minus(1K))σri + σjjK1113872 1113873

r 0 (12)



pexp min

σr2 pexp min +σcc

Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1


Kminus 11113874 1113875

r

R31113888 1113889

Kminus1minus1

minusσcc

Kminus 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)


pin

σr0 pin +σRR

Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875


Kminus 11113874 11138751113874 1113875

r

R01113888 1113889

Kminus1minus1

minusσRR

Kminus 11113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)




middotR2

r1113874 1113875


(15)




dσr1dr


r+



1+βminus11 minus 11113876 11138771113882 1113883

r 0

(16)


σr2 at r R2

σr1 pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+σcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

+R3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891

(17)


σθ1 1K

⎧⎨

⎩ pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+Kσcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

middotR3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891 + α(1 + ])



R2

r1113874 1113875


⎫⎬

⎭

(18)




A321113872 1113873


(19)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R0 TR0 (20)




Expansion


minus12 A0R

1+βminus123 R





minus11

1 rminusβminus10




minus11 R

1+βminus123 R



minus10 R

1+βminus123 R



minus11



minus12 minus1 A0R

1+βminus123 R



1+βminus123 R



minus11




Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦


minusσcc

Kminus 1

(21)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)


R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889



(23)


R2 σcc minus σRR


1 1+βminus11( )

middot R0 tR0

(24)



Kminus 11113874 1113875T


+σcc1minusK





1minusKT1+βminus12


(25)


pexp min +σcc

Kminus 11113874 1113875T









minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)






⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)









R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



dσr1dr


r+



1+βminus11 minus 11113876 11138771113882 1113883

r 0

(16)


σr2 at r R2

σr1 pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+σcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

+R3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891

(17)


σθ1 1K

⎧⎨

⎩ pexp min +σcc

Kminus 11113874 1113875

R3

r1113874 1113875

1minusKminus1

+Kσcc1minusK


βminus11 + Kminus1

R3

R21113888 1113889

1+βminus12

middotR2

r1113874 1113875

1minusKminus1

minusR2

r1113874 1113875

1+βminus11⎡⎣ ⎤⎦ +


1minusK

middotR3

R21113888 1113889

1+βminus12 R2

r1113874 1113875

1minusKminus1

minus 11113890 1113891 + α(1 + ])



R2

r1113874 1113875


⎫⎬

⎭

(18)




A321113872 1113873


(19)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R0 TR0 (20)




Expansion


minus12 A0R

1+βminus123 R





minus11

1 rminusβminus10




minus11 R

1+βminus123 R



minus10 R

1+βminus123 R



minus11



minus12 minus1 A0R

1+βminus123 R



1+βminus123 R



minus11




Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦


minusσcc

Kminus 1

(21)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)


R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889



(23)


R2 σcc minus σRR


1 1+βminus11( )

middot R0 tR0

(24)



Kminus 11113874 1113875T


+σcc1minusK





1minusKT1+βminus12


(25)


pexp min +σcc

Kminus 11113874 1113875T









minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)






⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)









R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Kminus 11113874 1113875 1minus

ΔcA0 1 + βminus121113872 1113873

⎡⎢⎣ ⎤⎥⎦


minusσcc

Kminus 1

(21)


R3 1minusΔc

A0 1 + βminus121113872 1113873⎡⎢⎣ ⎤⎥⎦

1 1+βminus12( )

R2 TR2 (22)


R3

R21113888 1113889

1+βminus12 R2

R01113888 1113889



(23)


R2 σcc minus σRR


1 1+βminus11( )

middot R0 tR0

(24)



Kminus 11113874 1113875T


+σcc1minusK





1minusKT1+βminus12


(25)


pexp min +σcc

Kminus 11113874 1113875T









minus pin +σRR

Kminus 11113874 1113875

R1

R01113888 1113889

Kminus1minus1

0

(26)






⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(27)









R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



R2 σcc minus σRR


1 1+βminus11( )

R1 tR1 (28)





⎧⎨

⎩

⎫⎬

⎭

1 Kminus1minus1( )

(29)




Kminus 11113874 1113875t

1minusKminus1+

σcc1minusK






(30)


T 1 andt lim



as follows


pin + σRR(Kminus 1)( 11138571113888 1113889

1 Kminus1minus1( )

(31)




pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )


pin + σRR(Kminus 1)( 1113857⎡⎣ ⎤⎦

1 Kminus1minus1( )

(32)



pin



1 1minusKminus1( )

(33)


4 Case Studies









10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






10 12 14 16 180

01

02

03

04

05

06

07

08

09

10

rR0

σ θp

0

Peak plastic zone


Softening zone

ESMEPBMEBM

Elastic zone

(a)

rR0

σ rp

0

10 12 14 16 1810

12

14

16

18

20

22

2425

Softening zone

EPSFM

EPBMESM

Elastic zone

Peak plastic zone


(b)








10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



10 12 14 16 18 2016

18

20

22

24

26

28

R3R0R2R0R1R0

p inp 0 R2 of EPSFM

R3 of EPSFM

EBMR1 of EPSFM

R2 of ESM

R1 of ESM

(a)

R3R0R2R0R1R0

p inp 0

10 12 14 16 18 2016

18

20

22

24

26

28

R1 of EPSFM

R2 of EPSFM

R1 of EPBM

R2 of EPBM

R3 of EPSFM

EBM


(b)


σrσθ

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

σ (M

Pa)

Elastic zone

(a)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

r (m)

Peak plastic zone

Elastic zone

(b)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

r (m)

Peak plastic zone

Softening zone

Elastic zone

(c)

σrσθ

σ (M

Pa)

0 01 02 03 04 05 06 072

4

6

8

10

12

14

16

18

20

r (m)

Peak plastic zone

Softening zone

Elastic zone

Fracture zone

(d)







5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






5 Conclusions




10 11 12 13 14 15 16 17 1801

011

012

013

014

015

016

017

018

019

pin (MPa)

R 3 R

2 R 1

(m)

R3

R1

R2


0 05 10 15 20013

014

015

016

017

018

019

020


R 1 R

2 R 3

(m)

R1

R2

R3





Data Availability




Acknowledgments


References












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















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












Expansion


05 1 15 201

012

014

016

018

02

022


R 1 R

2 R 3

(m)

R1

R2

R3



α 2 16 12 08 06pexp max 14103 14418 14917 15840 16679




















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


elastoplastic analysis of circular opening based on a new strain … · 2019. 7. 30. · epbm is...

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