FAR FIELD SOLUTION OF SH-WAVE BY CIRCULAR INCLUSION AND LINEAR CRACK

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The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia FAR FIELD SOLUTION OF SH-WAVE BY CIRCULAR INCLUSION AND LINEAR CRACK HogLiag Li,GuoHui Wu, Associate Pofesso, Depatmet of Egieeig Mechaics, Habi Egieeig Uivesity, Habi. Chia Email: leehl@sia.com ABSTRACT : Cicula iclusio is used widely i stuctue egieeig. I this pape, the method of Gee s fuctio is used to ivestigate the poblem of fa field solutio of cicula iclusio ad liea cack impacted by icidet SH-wave. Fistly, a Gee s fuctio is costucted fo the poblem, which is a fudametal solutio of displacemet field fo a elastic space possessig a cicula iclusio while beaig out-of-plae hamoic lie souce foce at ay poit; Secodly, i tems of the solutio of SH-wave s scatteig by a elastic space with a cicula iclusio, ati-plae stesses which ae the same i quatity but opposite i diectio to those metioed befoe, ae loaded at the egio whee the liea cack is i existet actually, we called this pocess cack-divisio ; Fially, the expessios of the displacemet ad stesses ae give whe the cicula iclusio ad liea cack exist at the same time. The, whe the special Gee s fuctio has bee costucted ad close field solutio has bee illustated, the fa field of scatteed wave is studied. The displacemet mode of scatteed wave at fa field ad scatteig coss-sectio ae give. Numeical esults ae illustated ad the ifluece of wave umbe, icidet agles of SH-wave, ad the combiatio of diffeet media paametes ae discussed. The esults ca be applied i the study of factue, ad udamaged fame cack detectio. KEYWORDS: cack, cicula iclusio, Gee s Fuctio, SH-wave scatteig, displacemet mode of scatteed wave at fa field, scatteed coss-sectio. INTRODUCTION Cicula iclusio exists widely i atual media, egieeig mateials ad stuctues, ad defects ae usually foud aoud the iclusio. Whe a composite mateial with cicula iclusio ad cacks is impacted by the dyamic load, o the oe had, the scatteig field poduced by the cicula iclusio ad cacks detemies the dyamic stess cocetatio facto aoud the cicula iclusio, ad theefoe detemies whethe the mateial is damaged o ot; o the othe had, the scatteig field also pesets may chaacteistic paametes of the iclusio ad cacks such as defect compositio, locatio ad shape, so the eseach o the scatteig fa-field is impotat to the geological pospects, seismological ivestigatio, o-destuctio evaluatio ad the othe fields. I the ocea acoustics, the scatteig fa-field of the acoustic wave is also used i the ude-wate suvey, object distiguishig ad so o. I theoy, the scatteig solutio of elastic waves is oe of the basic topics of evese poblems o elastic wave. O the basis of liteatue, few pape cocetates o the scatteig fa-field solutio of SH-wave by a cicula iclusio ad a liea cack aoud the iclusio. I the pape a ew model ad a ew method ae peseted i ode to ivestigate deeply o this kid poblem. At peset, to obtai the theoetical solutio of the poblem coceed i this thesis is of geat iteest ad cetaily it has some difficulties. The developmet of computatioal mechaics has povided may methods to solve the poblem, but a theoetical aswe is still expected i ode to ivestigate the chaacteistics of the cicula iclusio ad cack. The pape uses the Gee s fuctio to study the scatteig fa-field of elastic wave by a cicula iclusio ad a liea cack. The Gee s fuctio should be a fudametal solutio of displacemet field fo a elastic space possessig a cicula iclusio while beaig out-of-plae hamoic lie souce foce at ay poit. I tems of the solutio of SH-wave s scatteig by a elastic space with a cicula iclusio, ati-plae stesses which ae the same i quatity but opposite i diectio to those metioed befoe,

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia ae loaded at the egio whee the liea cack is i existet actually, we called this pocess cack-divisio. The, the expessios of the displacemet ad stesses ae give whe the cicula iclusio ad liea cack exist at the same time. The, whe the special Gee s fuctio has bee costucted ad close field solutio has bee illustated, the fa field of scatteed wave is studied. The displacemet mode of scatteed wave at fa field ad scatteig coss-sectio ae give. At last, a example is give ad its umeical esults ae discussed.. MODEL AND GOVERNING EQUATION The model is show as Fig., a elastic space cotaiig a cicula iclusio ad a liea cack aoud the iclusio. I this pape, the ati-plae shea SH wave model is studied. The displacemet i the elastic space is expessed as W( x, y, t),the displacemet i the iclusio is expessed as W( x, y, t). The goveig equatio of Wi ca be witte i the pola coodiate system as: W W W + + + kw = θ W W W + + + kw = θ (.) (.) ω μ whee ki =, C i Si =,ω C ρ is the cicula fequecy of the displacemet W( i x, y, t), Csi stads fo the shea Si i wave velocity, ρi ad μ i ae the mass desity ad the shea modulus of elasticity espectively. Figue Model of Poblem 3. GREEN S FUNCTION The Gee s fuctio used i this pape is egaded as the displacemet espose to the elastic space cotaiig a cicula iclusio impacted by ati-plae hamoic liea souce foce at ay poit. The depedece of the i t displacemet fuctiog i o time t is e ω.i the pola coodiate system, the goveig equatio of Gi ca be witte as: G G G ( ), G k G G + + + = δ + + G + k G = (3.) θ θ stads fo the positio of the liea souce foce i pola coodiates. The bouday coditios ca be expessed as below:

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia G = G τ = τ (3.), = R z z = R = R = R The basic solutio which satisfies the cotol equatio (3.) ad the bouday coditios (3.) should iclude two pats of motio: the distubace of ati-plae liea souce foce ad the scatteig wave icited by the cicula iclusio. The wave displacemet of the complete elastic space due to the lie souce load δ ( ) o the abitay positio ca be give: () i i () G = H ( k ) 4μ (3.3) () Whee H () is the fist kid of Hakel fuctio ad zeo-ode. The scatteig wave i the elastic space ad i the cicula iclusio ca be witte as: ( s) () ( i) = m m θ θ = m m θ θ m= m= (3.4) G A H ( k ) cos[ m( )], G B J ( k ) cos[ m( )] whee A, B ae ukow coefficiets. m m Theefoe, G G G () i ( s) = +, wave field G of this poblem ca be obtaied. G ( i) = G.Accodig to the bouday coditios, we ca obtai m A, B. So, the m 4. EXPRESSION OF DISPLACEMENT AND STRESS FOR THE MODEL The stess o the cack aoud the iclusio poduced by icidet SH-wave ad the scatteig wave icited by the cicula iclusio ca be obtaied. A pai of opposite foces is applied to the cack; theefoe the esultat foce o the cack is zeo, which ca be thought as cack.. The above costuctig pocess is called cack-divisio techique which ca be used to obtai the expessio of displacemet ad stess fo the model. The detail ca be discussed as follows. Fistly, we coside the icidece of SH-wave o the ifiite liea-elastic space cotaiig a cicula iclusio. () i The icidet displacemet field W hamoic to time ca be witte as follows: () i W = W εi cos[ ( θ α)] J( k) (4.) = whee α is the icidet agle. =, ε = ;, ε =. The scatteig wave i the elastic space ad i the cicula iclusio ca be witte as: ( s) () ( i) = = (4.) W = A H ( k)cos[ ( θ α)], W = B J ( k )cos[ ( θ α)] By usig the bouday coditios, we ca obtai A, B. The displacemet field W ca be give as: () i ( s) W = W + W (4.3)

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia The, we coside the scatteig poblem of icidet SH-wave whe the cicula iclusio ad cack exist at the same time. Accodig to icidet field ad scatteig field i the elastic space cotaiig oly a iclusio, the cack-divisio techique is used to costuct the model of SH-wave scatteig by a elastic space cotaiig a cicula iclusio ad a liea cack. The costuctig pocess is that: the space is sepaated alog the cack ad a pai of ati-plae opposite foces with the multitude τ θ z ae applied to up ad dow sectio of the egio whee cack will appea, theefoe the esultat foce o up (o dow) sectio of the egio is zeo, which ca be thought as cack. The above costucted Gee s fuctio idicates that the basic displacemet solutio ca be obtaied wheeve the ati-plae liea souce foce s positio it is. Cosequetly, we ca obtai the total displacemet field ad stess field ude the iteactio of the cicula iclusio ad the cack fo icidet SH-wave. The foce is applied o the cack ad the tectoic additioal displacemet field ca be obtaied: τ θ z = τ = θ z G (,, θθ, ) (4.4) Itegatig alog the lie of cack, we ca obtai: G (,, θ, θ ) d τ θ z = (4.5) Hece, the total displacemet field ca be witte as follows: W = W G (,, θ, θ ) d τ θ z = (4.6) 5. THE SCATTERING DISPLACEMENT MODE AT FAR FIELD ( s) The total scatteig wave field icludes the scatteig wave W poduced by the cicula iclusio, ad τ G (,, θ, θ ) d poduced by the liea cack, that is θ z = W = W G (,, θ, θ ) d ( zs) ( s) ( t ) τ θ z = (5.) The scatteig wave ca be expessed as a seies of Hakel fuctio, ad thei commo item H () ( K) ca be abstacted. Make use of the asymptotic expessio of Hakel fuctio as the idepedet vaiable is lage eough: () ( / 4) ( ) ( ) i z π H z = i e (5.) π z The scatteig fa-field displacemet ca be expessed as: π ( ) ( ) 8 ik zs π 4 W (, θ ) = e F( θ ) (5.3) k whee

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia i A [ ] θ z = = μ = A μw m A () θ α ε m α m m = = m= A F( θ) = { W ( i) A cos ( θ α ) τ ε (cos θ)[ J ( k ) + ] d } π 4 = { W ( i) A cos [ ( )] i (si m ) m[ J ( k ) H ( k )] π i 4 A A ( i) ε(cos θ)[ J( k) ] d } μ + (5.4) 6.THE SCATTERING CROSS SECTION (SCS) The time aveage eegy flow of the wave ove oe peiod T ca be defied as: ( ) Ave( E ) = iω i ij j ij j 4 σ u σ u da A ω = Im iσiju j da A (6.) Im( ) is the imagiay pat of a complex fuctio. The above-metioed fomula ca be used to calculate the time aveage eegy flow of the elastic wave. The time aveage eegy flow passig though the suface = R( which is axis Z diectio is a uit log) is: (s) z π W (s) z μ Ave( E ω ) = Im( W ) Rdθ ωμ = Im π ( zs) W ( zs) W Rd θ (6.) Substitute (5.) ito fomula(6.), ad make use of ()' () iπ ( m ) / H ( k) Hm ( k) e, (6.3) π we ca obtai the esult at fa distace R: Ave( E ωμ ) = W Im( E) (6.4) π The scatteig coss sectio is the atio of the total eegy of fa field scatteig wave to the time aveage eegy flow pe uit aea of the icidet wave. Fo the plae icidet SH wave, the time aveage of eegy flow pe uit aea is: Ave( E ) Ave( e ) = = μkωw = σωw (6.5) A ad letγ expesses the atio of these two eegy, we ca obtai the followig esult: γ = Im( E),whee, γ is the π k scatteig coss sectio.

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia 7.EXAMPLE I this pape, we pay attetio to a epesetative kid of models, which is show as Figue. The adius of the cicula iclusio is, ad the legth of the cack is. I Fig. ad Fig.3, the distace betwee the ie tip of the cack ad the cete of the iclusio is. Fig. ad Fig.3 show that sice thee is a liea cack compaed with the displacemet mode of fa field poduced by the cicula iclusio scatteig wave, the displacemet mode of fa field poduced by the cicula iclusio ad cack scatteig wave is chaged a lot. Whe the icidet wave is vetical to the cack thee is the most chage. I Fig.4, it shows the ifluece of μ μ to the Displacemet Mode whe k = α = 9. It ca be foud that the moe diffeet the mateial of elastic space with the mateial of iclusio is, the bigge ifulece the cack have. I Fig.5, the chage of the scatteig coss sectio goig togethe with the chage of the icidet wave umbe is give. It ca be foud that whe thee is a liea cack low fequecy sympathetic vibatio come ito beig. (a) α = 3 (b) α = 6 (c) α = 9 Figue Ifluece of Cack to the Displacemet Mode whe k = μ μ =

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia (a) α = 3 (b) α = 6 (c) α = 9 Figue 3 Ifluece of Cack to the Displacemet Mode whe k = μ μ = (a) μ μ = 8 (b) μ μ =.5 Figue 4 Ifluece of μ μ to the Displacemet Mode whe k = α = 9

The 4 th Wold Cofeece o Eathquake Egieeig Octobe -7, 8, Beijig, Chia Figue 5 Vaiatio of SCS vs. kr whe μ μ = Fom the istaces above-said, it ca be show that the ifluece of the cack should ot be eglected, ad by usig the coclusio we should be capable to estimate the positio of the cack though aalysig test data of the displacemet mode. 8.SUMMARY I this pape, by usig the techique of cack-divisio, fa field solutio of cicula iclusio ad liea cack impacted by icidet SH-wave is give.. By usig the method a example is solved, ad some ew coclusio is give. The method i the pape could be used to study some othe coelative poblem. REFERENCES Pao Y. H. (983).Elastic Waves i Solids. ASME Joual of Applied Mechaics 5:4, 5-64. Zheg Zhemi, Zhou Heg, Zhag Haxi. (995) Teds of Developmet i Mechaics i the Ealy st Cetuy. Advaces i Mechaics 5:4, 433-449 Wag Duo, Wag Yuesheg. (993). Recet Pogess i Dyamics of Iteface. Shaghai Joual of Mechaics 4:4, -5 Liu Diakui, Liu Hogwei. (999).Scatteig of SH-wave by Cacks Oigiatig at A Cicula Hole Edge ad Dyamic Stess Itesity Facto. Acta Mechaica Siica 3:3, 9 99 Liu Diakui, Liu Hogwei. (998).Scatteig ad Dyamic Stess Cocetatio of SH-wave by Iteface Cicula Hole. Acta Mechaica Siica 3:5, 597-64 Liu hogwei, Liu Diakui. (999).Fa Field Solutio of SH-wave Scatteed by Iteface Cicula Hole. Acta Mechaica Solida Siica :4, 349-355 Li HogLiag, Liu DiaKui. (4).Iteactio of SH-Waves by Cacks with Cicula Iclusio.Joual of Habi Egieeig Uivesity 5:5, 68-6. Li HogLiagi. (4).The Iteactio of Cicula Cavity, Iclusio with Beelie Cacks by SH-wave, Habi Egieeig Uivesity.

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