Modifications and Statistical Analysis of Acoustic ...downloads.hindawi.com/journals/amse/2018/1898937.pdf · Modifications and Statistical Analysis of Acoustic Emission Models Based

Research ArticleModifications and Statistical Analysis of Acoustic EmissionModels Based on the Damage and Fractal Characteristics

Jie Yang1 Yanna Zheng 1 and Huijing Wang2

1School of Marine and Civil Engineering Dalian Ocean University Dalian 116023 China2Department of Naval Architecture Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Yanna Zheng zhengyndloueducn

Received 15 December 2017 Accepted 31 January 2018 Published 24 April 2018

Academic Editor Alain Portavoce

Copyright copy 2018 Jie Yang et alis 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

e damage process is accompanied by the acoustic emission for quasibrittle materials And in the process of material damageevolution the length of microcracks satisfies the fractal distribution Research on their relationship in theory is helpful to revealthe law of material damage evolution and acoustic emission activities Damage variable expressions are proposed based on thedamage and fractal characteristics firstly en the statistical models for acoustic emission considering damage and fractalcharacteristics are established by deducing the relationship between acoustic emission parameters and load cycles and fractaldimensions e effects of damage and fractal effects on acoustic emission parameters are analyzed finally e results show thatthe damage accelerates the AE activity to the rougher material the opposite to the more homogeneous material It can also be seenthat the increase of the fractal dimension the homogeneity constant m will substantially increase the AE activities

1 Introduction

Material fatigue damage is a dynamic process of complexplastic-cumulative damage accompanied by acoustic emis-sion (AE) during damage degradation e study of AEmechanisms provides a physical basis for interpretation ofthe behavior of acoustic emission which helps to un-derstand the mechanism of fatigue fracture Many efforts[1ndash4] have verified and identified the AE source mecha-nisms Among them the research on acoustic emissionmechanism and acoustic emission characteristics based onmicromechanics theory has always been paid more atten-tion by researchers

Tang and Xu [5 6] put forward the viewpoint that thedamage variable is consistent with the acoustic emissionparameters by the continuous damage mechanics methodand the statistical model of acoustic emission is establishedbased on the Weibull distribution assuming damage evo-lution of each microelement defect Analytical models [7]based on the continuous damage mechanics method haveproven to be effective and have become a basis for numericalanalysis related to the fracture process of materials [8]

Fatigue and fracture of the material is the result ofdamage accumulation Internal damage to the structureduring loading can cause stress redistribution leading tonew microdefect damage and experimental studies [9 10]have confirmed that the evolution of material damage hasfractal characteristics at the same time fractal features andbehaviors exist in the distribution of microcracks thepropagation of cracks and the evolution of material damageOn the other hand AE behaviors during fracture are thereflection of the extent of damage depending on the evo-lution of internal defects and the mechanical process ofreproduction in materials It seems to suggest that thedamage variables and constitutive equations are also re-sponsible directly for the occurrence of AE in the materialHowever the current statistical models of acoustic emissiondo not account for the effects of damage and fractal featureson the constitutive equations of the materials and thestatistical models of acoustic emission are thus establishedwithout considering their effects

In this paper damage variables are redefined byAE parameters the number of cycles and the fractaldimension and the expression of damage variable is

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 1898937 6 pageshttpsdoiorg10115520181898937

corrected e Weibull distribution [5] is further used todescribe the damage evolution of microdefect and thenthe material constitutive equation and load displacementequation considering damage and fractal characteristicsare deduced to obtain the modified acoustic emissionstatistical model Finally the effects of damage andfractal characteristics on acoustic emission parametersare discussed

2 AE Parameters in Fatigue Fracture Process

21 Damage Expressions of AE Parameters Since an acousticemission represents a minor damage of the material as withstress and strain parameters some AE parameters such asthe event number of AE and the energy of AE are employedto the index of the damage degree To investigate the re-lationship between the fatigue fracture process parametersand AE parameters several definitions of damage variablesare given as follows

e material without initial damage under uniaxialloading is considered assuming that the AE rate producedby per unit area of the microelement damage is nv damageAE counts η for the microelement area of Ad is as follows

η nv middot Ad (1)

whereAd is the cross-sectional area of microdefects areaesum of ring down counts AE occurring until the totaldamage area of A failure ηt is as follows

ηt nv middot A η middot A

Ad (2)

Continuous damage variable D characterized as thedamage of material has been introduced by Robotnov [11]

D Ad

A

0 undamage state

1 damage state

⎧⎪⎨

⎪⎩(3)

where Ad is the cross-sectional area with damage and A isthe cross-sectional area without damage

Combining (2) with (3) the relationship between thedamage variable and AE count is obtained by

D ηηt

(4)

It is shown that the AE count ratio can characterize thedegree of damage evolution of the material which is themacroscopic representation of the microscopic changenamely the AE parameters that are consistent with thedamage variables

22 e Expressions of AE Parameters Based on Damage Effectand Fractal Characteristics Damage variables can track thedevelopment of the crack in real time and the fractal di-mension can describe the crack shape quantitatively Ifparameters both damage variables and fractal dimension areused to describe the damage evolution of the material theinfluence of material defects on the mechanical behaviorparameters can be further studied in a microscale

Starting from the constitutive equation of the damagevariable the damage variable expression is modified and therelationship between the damage variable defined byacoustic emission parameters and the new damage variabledefined by damage and fractal characteristics is studied

Fatigue damage is mainly caused by plastic accumulationand hardening properties for low cycle fatigue (LCF)Constitutive equations [12] derived from the damage vari-able D for the low cycle fatigue are characterized as

_D σmminus1

Bprime(Δσ)λN(1minusα)(1minusD)m_σ(LCF) (5)

where m 2s0 + 1n and Bprime nG1n(2ES0)s0 are material

constants and s0 and S0 are parameters Damage constitutiveequation for LCF is introduced by hardening state variableG G G0h assuming G with power exponent functions Δσand N G G0h G0(ΔσλN(1minus α))n Integrating (5) in oneloading cycle damage evolution equations dDdN can beobtained as

dD

dN

(Δσ)mhminus1n

BPrime(1minusD)m (LCF) (6)

where BPrime mBprime2(m+1) By integrating (6) under boundaryconditions (if N 0 D 0 else N Nf D 1) damagevariable expressions derived from LCF can be given by

D 1minus 1minusN1

Nf1113888 1113889

α

1113888 1113889

1(m+1)

(7)

With α 0 the above equations have the same patternsSo (7) can be regarded as the general expression of thefatigue damage variable which is similar to the results asdiscussed by Chaboche et al [13]

When the material is subjected to fatigue damage causedby external load the fatigue damage variable μ after N cyclescould be defined as

μ N

Nf (8)

where Nf is the fatigue failure lifetime Substituting (8) into(7) then

1minus(1minusD)m+1

μa (9)

Substituting (4) and (8) into (9) the damage variableexpression with damage can be described as

1minus 1minusηηt

1113888 1113889

m1+1

N

Nf1113888 1113889

α

(10)

e above two relationships demonstrate that AE ac-tivities are related to fatigue damage

According to the fracture surface fractal characteristicsthe damage variable with damage and fractal characteristicsin (9) is defined by Xie and Ju [14] as follows

μ Df δ( 1113857 μδDfminus2 1minus(1minus μ)δDfminus2 (11)

where Df and 1113957Df are the fractal dimensions of the damagedomain and the damage residual domain respectively

2 Advances in Materials Science and Engineering

Substituting (4) and (11) into (9) damage variable ex-pression with fractal characteristics can be described as

1minus 1minusηηt

1113888 11138891113890 1113891

m+1

N

Nf1113888 1113889δDf minus 21113890 1113891

α

(12)

e above-modified expressions ((10) and (12)) dem-onstrate that the damage can be defined by the number ofloading cycles and the fractal dimension and also can bedefined by acoustic emission parameters in AE caused byfatigue fracture e theoretical results indicated that the AEparameters are related to the damage effect and fractalcharacteristics so AE parameters can be used to evaluate theAE activity at different fatigue fracture stages

3 Modifications of AE Statistical ModelsBased on Damage and Fractal Characteristics

e main contributor to damage is the rate of strain energydensity release and is always associated with irreversiblestrain at the microscopic or macroscopic level More gen-erally the damage rate is described by deformation variablessuch as the strain ε of the following form [7]

dD f(ε) dε (ε ζ and dεgt 0)

0 (εlt ζ and dεlt 0)1113896 (13)

where f(ε) is a continuous positive definite function of εnamely the damage evolution function and ζ is thevariable damage strain threshold value Assuming that thedamage process of material is continuous the Weibulldistribution function is used to describe the damageevolution of these defects in this paper In what follows wewill express the AE parameters by statistical methodaccording to the AE parameters are consistent with and thedamage variables

It is assumed that the microunit strength of the materialsobeys the Weibull distribution

f(ε) m

ε0

εε0

1113888 1113889

mminus1

exp minusεε0

1113888 1113889

m

1113890 1113891

or fu

L1113874 1113875

mL

u0

u

u01113888 1113889

mminus1

exp minusu

u01113888 1113889

m

1113890 1113891

(14)

F(ε) 1minus exp minusεε0

1113888 1113889

m

1113890 1113891

or Fu

L1113874 1113875 1minus exp minus

u

u01113888 1113889

m

1113890 1113891

(15)

where u0 and ε0 are the reference mean displacement andstrain when the specimen reaches the peak load respectively(u0 Lε0) the displacement-strain relationship in one di-mension is u Lε where L is the length of the specimen andm is the material heterogeneity (the larger the value themore uniform the material)

For monotonic loading the first expression in (13) isalways establishede initial damage condition is considered

namely D ε ζ 0 and in terms of (4) (13) and (15) thecumulative number of AE can be expressed as

ηηt

D F(ε) 1113946ε

0f(x) dx 1minus exp minus

εε0

1113888 1113889

m

1113890 1113891 (16)

Here we intend to show that the modified constitutiveequations and load expressions can be proposed by asuitably chosen damage variable with damage and fractalcharacteristics

31 Modifications of Constitutive Equations and Load Expres-sions with Damage Effect and Fractal Characteristics Combin-ing the effective stress concept [11] with (10) (14) and (15)the improved constitutive equation with damage of thematerial is expressed by the AE parameter as follows

σ Eε(1minus μ) Eε 1minus 1minus 1minusηηt

1113888 1113889

m1+1⎡⎣ ⎤⎦

1α⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

Eε11138981minus 1minus exp minus m1 + 1( 1113857εε0

1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(17)

where σ is the stress E is the elastic modulus and m1 and αare the material constants

e relation between the load R and deformation u withdamage is obtained by

R g(u) EA

L11138841minus 1minus[1minusF(ε)]m1+1

1113966 11139671α

1113885u

k0u11138981minus 1minus 1minusFu

L1113874 11138751113876 1113877

m1+11113896 1113897

1α

1113899

k0u11138981minus 1minus exp minus m1 + 1( 1113857u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(18)

where k0 is the specimen of initial stiffness k0 EAL forthe weakened properties of materials namely σprime(ε0)lt 0then mgt 1

In terms of (12) (14) and (15) the modified constitutiveequation with fractal characteristics of the material isexpressed by the AE parameter as follows

σ σ(1minus 1113957μ) Eε 1minus μδDfminus21113872 1113873 Eε 1minusN

NfδDfminus21113888 1113889

Eε 1minus δDfminus2 1minus 1minus ηηt

1113874 1113875m1+1

1113890 1113891

1α⎧⎨

⎩

⎫⎬

⎭

Eε11138981minus δDfminus2 1minus exp minus m1 + 1( 1113857εε0

1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(19)

e relation between the load R and deformation u withfractal characteristics is obtained by

Advances in Materials Science and Engineering 3

R g(u) k0u11138981minus δDfminus2 1minus 1minusFu

L1113874 11138751113874 1113875

m1+11113896 1113897

1α

1113899

k0u11138981minus δDfminus2 1minus exp minus m1 + 1( 1113857u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(20)

32 Modifications of AE Statistical Model Based on DamageEffect e following expressions of AE rates are modified

according to the constitutive equations and load ex-pressions with damage and fractal characteristics in twocases

In one case loaded at displacement rate c1 as a constantthe deformation rate of the specimen is given by

_u c1

1 + gprime(u)Km (21)

where Km is the load stiffness of the test machine eexpression of the AE rate with damage can be described as

nt c1

L

f(uL)

1 + gprime(u)Km

mc1

u0

middotuu0( 1113857

mminus1 exp minus uu0( 1113857m

1113858 1113859

1 + k0Km( 1113857lfloorlang1minus 1minus exp minus m1 + 1( 1113857 uu0( 1113857m

1113858 11138591113864 11138651αrang minus m1 + 1( 1113857m( 1113857α( 1113857 uu0( 1113857

m 1minus exp minus m1 + 1( 1113857 uu0( 1113857m

1113858 11138591113864 11138651(αminus1) exp minus m1 + 1( 1113857 uu0( 1113857

m1113858 1113859rfloor

(22)

e relationship between the deformation and the timeof the specimen is obtained by

t 1c1

u +g(u)

Km1113888 1113889

u

c1 ordm1 +k0

Km11138981minus 1minus exp minus m1 + 1( 1113857

u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

Oslash

(23)

In the other case loaded at stress increase rate c2as aconstant the deformation rate of the specimen isgiven by

_u c2

gprime(u) (24)

e expression of the AE rate with damage can be de-scribed as

nt c2f(uL)

Lgprime(u)

mc2

k0u0

middotuu0( 1113857


1113858 1113859

lfloorlang1minus 1minus exp minus m1 + 1( 1113857 uu0( 1113857m




m1113858 1113859rfloor

(25)


t g(u)

c2

k0

c2u11138981minus 1minus exp minus m1 + 1( 1113857

u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(26)

33 Modification of AE Statistical Model Based on FractalCharacteristics In what follows the statistical analysis of AEmodels are investigated considering damage and fractalcharacteristics

In one case loaded at displacement rate c1 as a constantthe expression of the AE rate with fractal characteristics canbe described as

nt c1L

f(uL)

1 + gprime(u)Km

mc1u0

middotuu0( 1113857


1113858 1113859

1 + k0Km( 1113857lfloorlang1minus δDfminus2 1minus exp minus m1 + 1( 1113857 uu0( 1113857m

1113858 11138591113864 11138651αrang minus δDfminus2 m1 + 1( 1113857m( 1113857α( 1113857 uu0( 1113857



m1113858 1113859rfloor

(27)



t1c1

u+g(u)Km

( )u

c1 ordm1+ k0Kmlang1minusδDfminus2 1minusexp minus m1 +1( )

u

u0( )

m

[ ] 1α

rang

Oslash

(28)

In the other case loaded at stress increase rate c2 asa constant the expression of the AE rate with fractalcharacteristics can be described as

nt c2f(uL)Lgprime(u)

mc2k0u0

middotuu0( )mminus1 exp minus uu0( )m[ ]

lfloorlang1minus δDfminus2 1minus exp minus m1 + 1( ) uu0( )m[ ] 1αrang minus δDfminus2 m1 + 1( )m( )α( ) uu0( )m 1minus exp minus m1 + 1( ) uu0( )m[ ] 1(αminus1) exp minus m1 + 1( ) uu0( )m[ ]rfloor

(29)


t g(u)c2

k0c2ulang1minus δDfminus2 1minus exp minus m1 + 1( )

u

u0( )

m

[ ] 1α

rang

(30)

4 Damage and Fractal Characteristics Analysisand Effect on AE Parameters

e eects of damage and fractal characteristics on AEparameters are discussed belowe results in Figure 1 showthe curve of the AE rate (u0c1)nt versus time (c1u0)t withdierent homogeneity m of quasibrittle materials whenconsidering the initial stiness ratio Kmrarrinfin of the system

ere are three cases the rst one does not consider thedamage and the result of the AE rate is taken from Zhang et al[7] the second one considers the damage and the calculationformula of the AE rate can be seen from (22) and (23) and thethird considers the fractal damage and the calculation formulaof the AE rate can be seen from (27) and (28)

From Figure 1 it can be seen that the AE rate curvesoverlap when m takes dierent values that is the values ofthe AE rate are identical for the three cases e results showthat the eects of damagefractal eects on the AE pa-rameters can be neglected when the initial stiness is greaterthan Km to innity

e change of AE rates related with dierent homoge-neity m for constant stress increase rate c2 is presented inFigure 2e dotted line represents the case without damageand the calculation formula of acoustic emissivity is takenfrom [7] e solid line represents the case with damage theresult from (25) and (26) It can be seen that the AE activitydoes not appear aftershock during the deformation process

m = Inf

m = 100

m = 50

m = 30m = 20

m = 10m = 13

0

1

2

3

4

(u0c 1

)nt

05 1 15 2 250(c1u0)t

Figure 1 Relationship between AE rates (u0c1)nt versus (c1u0)tof the material with homogeneity m loaded at displacement rate c1as a constant with Kmrarrinfin of the testing machine

m gt 1increasing

02 04 06 08 10[c2(k0u0)]t

0

1

2

3

4

5

(k0u

0c 2

)nt

CurrentReference [1]

Figure 2 AE hit rates (k0u0c2)nt versus [c2(k0u0)]t of thematerial with m for constant stress increase rate c2


and when the material becomes unstable and the mainrupture occurs in the state of ultimate strength the AE ratesapproach innity e rougher the material the more ob-vious the precursor of main shock of AE activity edamage accelerates the AE activity to the rougher materialthe opposite to the more homogeneous material

As shown in Figure 3 with m1 4 α 02 m 5δ 05 loaded at stress increase rate c2 as a constant the AErate nt relates to time twith dierent fractal dimensionDf Itis also shown that the AE activity will increase with theincrease in the fractal dimension Df

5 Conclusions

Modications of the AE statistical models based on the damageand fractal characteristics are presented and the eects ofdamage and fractal eects on acoustic emission parameters areanalyzed e following conclusions can be drawn

(1) e expressions of damage variables taking into accountdamage and fractal characteristics are modiedeAE parameter in addition to its dependence on theload cycles is shown to be a function of the fractaldimension

(2) e statistical analysis of the AEmodels are modiedconsidering damage and fractal characteristics eresults show that for the constant stress increase ratethe damage accelerates the AE activity to the roughermaterial the opposite to the more homogeneousmaterial It can also be seen that the increase of thefractal dimension the homogeneity constant m willsubstantially increase the AE activities

Conflicts of Interest

e authors declare that they have no conicts of interest

Acknowledgments

is research was supported by the grant (no 20170540105)from the Natural Science Foundation of Liaoning Provinceof China

References

[1] M N Bassim and M Houssny-Eman ldquoAcoustic emissionduring the low cycle fatigue of AISI4340 steelrdquo MaterialsScience and Engineering vol 68 no 1 pp 79ndash83 1984

[2] D Fang and A Berkovits ldquoFatigue design model basedon damage mechanisms revealed by acoustic emission mea-surementsrdquo Journal of Engineering Materials and Technologyvol 117 no 4 pp 200ndash208 1995

[3] C Barile C Casavola G Pappalettera and C PappalettereldquoAnalysis of crack propagation in stainless steel by comparingacoustic emissions and infrared thermography datardquo Engi-neering Failure Analysis vol 69 pp 35ndash42 2016

[4] A Monti A El Mahi Z Jendli and L Guillaumat ldquoMechanicalbehaviour and damage mechanisms analysis of a ax-brereinforced composite by acoustic emissionrdquo Composites PartA Applied Science andManufacturing vol 90 pp100ndash110 2016

[5] C A Tang and X H Xu ldquoEvolution and propagation ofmaterial defects and Kaiser eect functionrdquo Journal of Seis-mological Research vol 13 no 2 pp 203ndash213 1990

[6] Z H Chen and C A Tang ldquoeoretical and experimentalstudies for Kaiser eect in rockrdquo e Chinese Journal ofNonferrous Metals vol 7 no 1 pp 9ndash12 1997

[7] M Zhang Z Li Q Yang and X Feng ldquoA damage model andstatistical analysis of acoustic emission for quasi-brittle ma-terialsrdquo Chinese Journal of Rock Mechanics and Engineeringvol 25 pp 2493ndash2501 2006

[8] C A Tang and X H Xu ldquoA cusp catastrophic model of rockunstable failurerdquo Chinese Journal of Rock Mechanics andEngineering vol 9 no 2 pp 100ndash107 1990

[9] B B Mandelbrot e Fractal Geometry of Nature W HFreeman and Company New York NY USA 1982

[10] K Falconer Fractal Geometry Mathematical Foundations andApplications Wiley Chichester UK 1990

[11] Y N Robotnov ldquoCreep rupturesrdquo in Proceedings of the 12thInternational Congress of Applied Mechanics M Hetenyi andW G Vincenti Eds pp 342ndash349 Standford-Springer-VerlagBerlin Germany 1968

[12] J Lemaitre ldquoFormulation and identication of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanicseory and Applications D Krajcinovic and J Lemaitre Edspp 37ndash89 Springer Verlag New York NY USA 1987

[13] J L Chaboche H Kaczmarek and P Raine ldquoOn the in-teraction of hardening and fatigue damage in the 316 stainlesssteel Advances in fracture researchrdquo in Proceedings of the 5thInternational Conference on Fracture pp 1381ndash1393 CannesFrance March 1981

[14] P H Xie and Y Ju ldquoA study of damage mechanics theory infractional dimensional spacerdquo Chinese Journal of eoreticaland Applied Mechanics vol 31 no 3 pp 300ndash310 1999

Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt

Figure 3 AE hit rates (k0u0c2)nt versus time [c2(k0u0)]t of thematerial with various Df for constant stress increase rate c2


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Submit your manuscripts atwwwhindawicom

corrected e Weibull distribution [5] is further used todescribe the damage evolution of microdefect and thenthe material constitutive equation and load displacementequation considering damage and fractal characteristicsare deduced to obtain the modified acoustic emissionstatistical model Finally the effects of damage andfractal characteristics on acoustic emission parametersare discussed

2 AE Parameters in Fatigue Fracture Process

21 Damage Expressions of AE Parameters Since an acousticemission represents a minor damage of the material as withstress and strain parameters some AE parameters such asthe event number of AE and the energy of AE are employedto the index of the damage degree To investigate the re-lationship between the fatigue fracture process parametersand AE parameters several definitions of damage variablesare given as follows

e material without initial damage under uniaxialloading is considered assuming that the AE rate producedby per unit area of the microelement damage is nv damageAE counts η for the microelement area of Ad is as follows

η nv middot Ad (1)

whereAd is the cross-sectional area of microdefects areaesum of ring down counts AE occurring until the totaldamage area of A failure ηt is as follows

ηt nv middot A η middot A

Ad (2)

Continuous damage variable D characterized as thedamage of material has been introduced by Robotnov [11]

D Ad

A

0 undamage state

1 damage state

⎧⎪⎨

⎪⎩(3)

where Ad is the cross-sectional area with damage and A isthe cross-sectional area without damage

Combining (2) with (3) the relationship between thedamage variable and AE count is obtained by

D ηηt

(4)

It is shown that the AE count ratio can characterize thedegree of damage evolution of the material which is themacroscopic representation of the microscopic changenamely the AE parameters that are consistent with thedamage variables

22 e Expressions of AE Parameters Based on Damage Effectand Fractal Characteristics Damage variables can track thedevelopment of the crack in real time and the fractal di-mension can describe the crack shape quantitatively Ifparameters both damage variables and fractal dimension areused to describe the damage evolution of the material theinfluence of material defects on the mechanical behaviorparameters can be further studied in a microscale

Starting from the constitutive equation of the damagevariable the damage variable expression is modified and therelationship between the damage variable defined byacoustic emission parameters and the new damage variabledefined by damage and fractal characteristics is studied

Fatigue damage is mainly caused by plastic accumulationand hardening properties for low cycle fatigue (LCF)Constitutive equations [12] derived from the damage vari-able D for the low cycle fatigue are characterized as

_D σmminus1

Bprime(Δσ)λN(1minusα)(1minusD)m_σ(LCF) (5)

where m 2s0 + 1n and Bprime nG1n(2ES0)s0 are material

constants and s0 and S0 are parameters Damage constitutiveequation for LCF is introduced by hardening state variableG G G0h assuming G with power exponent functions Δσand N G G0h G0(ΔσλN(1minus α))n Integrating (5) in oneloading cycle damage evolution equations dDdN can beobtained as

dD

dN

(Δσ)mhminus1n

BPrime(1minusD)m (LCF) (6)

where BPrime mBprime2(m+1) By integrating (6) under boundaryconditions (if N 0 D 0 else N Nf D 1) damagevariable expressions derived from LCF can be given by

D 1minus 1minusN1

Nf1113888 1113889

α

1113888 1113889

1(m+1)

(7)

With α 0 the above equations have the same patternsSo (7) can be regarded as the general expression of thefatigue damage variable which is similar to the results asdiscussed by Chaboche et al [13]

When the material is subjected to fatigue damage causedby external load the fatigue damage variable μ after N cyclescould be defined as

μ N

Nf (8)

where Nf is the fatigue failure lifetime Substituting (8) into(7) then

1minus(1minusD)m+1

μa (9)

Substituting (4) and (8) into (9) the damage variableexpression with damage can be described as

1minus 1minusηηt

1113888 1113889

m1+1

N

Nf1113888 1113889

α

(10)

e above two relationships demonstrate that AE ac-tivities are related to fatigue damage

According to the fracture surface fractal characteristicsthe damage variable with damage and fractal characteristicsin (9) is defined by Xie and Ju [14] as follows

μ Df δ( 1113857 μδDfminus2 1minus(1minus μ)δDfminus2 (11)

where Df and 1113957Df are the fractal dimensions of the damagedomain and the damage residual domain respectively



1minus 1minusηηt

1113888 11138891113890 1113891

m+1

N

Nf1113888 1113889δDf minus 21113890 1113891

α

(12)





0 (εlt ζ and dεlt 0)1113896 (13)



f(ε) m

ε0

εε0

1113888 1113889

mminus1

exp minusεε0

1113888 1113889

m

1113890 1113891

or fu

L1113874 1113875

mL

u0

u

u01113888 1113889

mminus1

exp minusu

u01113888 1113889

m

1113890 1113891

(14)


1113888 1113889

m

1113890 1113891

or Fu


u

u01113888 1113889

m

1113890 1113891

(15)




ηηt

D F(ε) 1113946ε


εε0

1113888 1113889

m

1113890 1113891 (16)




1113888 1113889

m1+1⎡⎣ ⎤⎦

1α⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(17)



R g(u) EA


1113966 11139671α

1113885u


L1113874 11138751113876 1113877

m1+11113896 1113897

1α

1113899


u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(18)




NfδDfminus21113888 1113889


1113874 1113875m1+1

1113890 1113891

1α⎧⎨

⎩

⎫⎬

⎭


1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(19)




L1113874 11138751113874 1113875

m1+11113896 1113897

1α

1113899


u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(20)




_u c1



nt c1

L

f(uL)

1 + gprime(u)Km

mc1

u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(22)


t 1c1

u +g(u)

Km1113888 1113889

u

c1 ordm1 +k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

Oslash

(23)


_u c2

gprime(u) (24)


nt c2f(uL)

Lgprime(u)

mc2

k0u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(25)


t g(u)

c2

k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(26)



nt c1L

f(uL)

1 + gprime(u)Km

mc1u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(27)



t1c1

u+g(u)Km

( )u


u

u0( )

m

[ ] 1α

rang

Oslash

(28)



mc2k0u0



(29)


t g(u)c2


u

u0( )

m

[ ] 1α

rang

(30)






m = Inf

m = 100

m = 50

m = 30m = 20

m = 10m = 13

0

1

2

3

4

(u0c 1

)nt

05 1 15 2 250(c1u0)t


m gt 1increasing

02 04 06 08 10[c2(k0u0)]t

0

1

2

3

4

5

(k0u

0c 2

)nt






5 Conclusions






Acknowledgments


References















Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt





Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





1minus 1minusηηt

1113888 11138891113890 1113891

m+1

N

Nf1113888 1113889δDf minus 21113890 1113891

α

(12)





0 (εlt ζ and dεlt 0)1113896 (13)



f(ε) m

ε0

εε0

1113888 1113889

mminus1

exp minusεε0

1113888 1113889

m

1113890 1113891

or fu

L1113874 1113875

mL

u0

u

u01113888 1113889

mminus1

exp minusu

u01113888 1113889

m

1113890 1113891

(14)


1113888 1113889

m

1113890 1113891

or Fu


u

u01113888 1113889

m

1113890 1113891

(15)




ηηt

D F(ε) 1113946ε


εε0

1113888 1113889

m

1113890 1113891 (16)




1113888 1113889

m1+1⎡⎣ ⎤⎦

1α⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭


1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(17)



R g(u) EA


1113966 11139671α

1113885u


L1113874 11138751113876 1113877

m1+11113896 1113897

1α

1113899


u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(18)




NfδDfminus21113888 1113889


1113874 1113875m1+1

1113890 1113891

1α⎧⎨

⎩

⎫⎬

⎭


1113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(19)




L1113874 11138751113874 1113875

m1+11113896 1113897

1α

1113899


u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(20)




_u c1



nt c1

L

f(uL)

1 + gprime(u)Km

mc1

u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(22)


t 1c1

u +g(u)

Km1113888 1113889

u

c1 ordm1 +k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

Oslash

(23)


_u c2

gprime(u) (24)


nt c2f(uL)

Lgprime(u)

mc2

k0u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(25)


t g(u)

c2

k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(26)



nt c1L

f(uL)

1 + gprime(u)Km

mc1u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(27)



t1c1

u+g(u)Km

( )u


u

u0( )

m

[ ] 1α

rang

Oslash

(28)



mc2k0u0



(29)


t g(u)c2


u

u0( )

m

[ ] 1α

rang

(30)






m = Inf

m = 100

m = 50

m = 30m = 20

m = 10m = 13

0

1

2

3

4

(u0c 1

)nt

05 1 15 2 250(c1u0)t


m gt 1increasing

02 04 06 08 10[c2(k0u0)]t

0

1

2

3

4

5

(k0u

0c 2

)nt






5 Conclusions






Acknowledgments


References















Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt





Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





L1113874 11138751113874 1113875

m1+11113896 1113897

1α

1113899


u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(20)




_u c1



nt c1

L

f(uL)

1 + gprime(u)Km

mc1

u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(22)


t 1c1

u +g(u)

Km1113888 1113889

u

c1 ordm1 +k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

Oslash

(23)


_u c2

gprime(u) (24)


nt c2f(uL)

Lgprime(u)

mc2

k0u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(25)


t g(u)

c2

k0


u

u01113888 1113889

m

1113890 11138911113896 1113897

1α

1113899

(26)



nt c1L

f(uL)

1 + gprime(u)Km

mc1u0

middotuu0( 1113857


1113858 1113859





m1113858 1113859rfloor

(27)



t1c1

u+g(u)Km

( )u


u

u0( )

m

[ ] 1α

rang

Oslash

(28)



mc2k0u0



(29)


t g(u)c2


u

u0( )

m

[ ] 1α

rang

(30)






m = Inf

m = 100

m = 50

m = 30m = 20

m = 10m = 13

0

1

2

3

4

(u0c 1

)nt

05 1 15 2 250(c1u0)t


m gt 1increasing

02 04 06 08 10[c2(k0u0)]t

0

1

2

3

4

5

(k0u

0c 2

)nt






5 Conclusions






Acknowledgments


References















Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt





Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





t1c1

u+g(u)Km

( )u


u

u0( )

m

[ ] 1α

rang

Oslash

(28)



mc2k0u0



(29)


t g(u)c2


u

u0( )

m

[ ] 1α

rang

(30)






m = Inf

m = 100

m = 50

m = 30m = 20

m = 10m = 13

0

1

2

3

4

(u0c 1

)nt

05 1 15 2 250(c1u0)t


m gt 1increasing

02 04 06 08 10[c2(k0u0)]t

0

1

2

3

4

5

(k0u

0c 2

)nt






5 Conclusions






Acknowledgments


References















Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt





Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






5 Conclusions






Acknowledgments


References















Df

m = 5

02 04 06 08 10[c2(k0u0)]t

0

05

1

15

2

25

(k0u

0c 2

)nt





Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




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