Application of Hankel Transform for Solving a Fracture Problem of a Cracked Piezoelectric Strip Under Thermal Loading

Transcription

1 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng Se Ueda Oaka Inttute of Technology Japan. Introducton In th chapter, an example of the applcaton of Hankel tranform for olvng a fracture problem wll be explaned. In dcung axymmetrc problem, t advantageou to ue polar coordnate, and the Hankel tranform method powerful to olve the general equaton n polar coordnate. A bref account of the Hankel tranform wll be gven. Here f a functon of r, t tranform ndcated by a captal F, J ν the ν th order Beel functon of the frt knd, and the nature of the tranformaton ether by a uffx or by a charactertc new varable. It wll be aumed wthout comment that the ntegral n queton ext, and that, f neceary, the functon and ther dervatve tend to zero a the varable tend to nfnty. The Hankel tranform of order ν > /, Hν [ f( r)] or Fν (), of a functon f () r defned a and t nveron formula ν ν = ν H [ f ( r)] F ( ) rj ( r) f ( rdr ) Alo, ntegratng by part twce gve f () r = Jν( r) Fν() d d f df ν Hν + f F () = ν dr rdr r provded that rf ( r ) and rdf ( r)/ dr tend to zero a r and a r. The pezoelectrc materal have attracted conderable attenton recently. Owng to the couplng effect between the thermo-elatc and electrc feld n pezoelectrc materal, thermo-mechancal dturbance can be determned form meaurement of the nduced electrc potental, and the enung repone can be controlled through applcaton of an approprate electrc feld (Rao & Sunar, 994). For ucceful and effcent utlzaton of

2 8 Fourer Tranform Materal Analy pezoelectrc a enor and actuator n ntellgent ytem, everal reearche on pezothermo-elatc behavor have been reported (Tauchert, 99). Moreover a better undertandng of the mechanc of fracture n pezoelectrc materal under thermal load condton needed for the requrement of relablty and lfetme of thee ytem. Ung the Fourer tranform, the preent author tuded the thermally nduced fracture of a pezoelectrc trp wth a two-dmenonal crack (Ueda, 6a, 6b). Here the mxed-mode thermo-electro-mechancal fracture problem for a pezoelectrc materal trp wth a penny-haped crack condered. It aumed that the trp under the thermal loadng. The crack face are uppoed to be nulated thermally and electrcally. By ung the Hankel tranform (Sneddon & Lowengrub, 969), the thermal and electromechancal problem are reduced to a ngular ntegral equaton and a ytem of ngular ntegral equaton (Erdogan & Wu, 996), repectvely, whch are olved numercally (Sh, 97). Numercal calculaton are carred out, and detaled reult are preented to llutrate the nfluence of the crack ze and the crack locaton on the tre and electrc dplacement ntenty factor. The temperature, tre and electrc dplacement dtrbuton are alo preented.. Formulaton of the problem Fg.. Penny-haped crack n a pezoelectrc trp A penny-haped crack of radu c embedded n an nfnte long pezoelectrc trp of thckne h = h + h a hown n Fgure. The crack located parallel to the boundare and at an arbtrary poton n the trp, and the crack face are uppoed to be nulated thermally and electrcally. The cylndrcal coordnate ytem denoted by ( r, θ, z) wth t orgn at the center of the crack face and the plane r θ along the crack plane, where z the polng ax. It aumed that unform temperature T and T are mantaned over the tre-free boundare. In the followng, the ubcrpt r, θ, z wll be ued to refer to the drecton of coordnate. The materal properte, uch a the elatc tffne contant, the pezoelectrc contant, the delectrc contant, the tre-temperature coeffcent, the coeffcent of heat conducton, and the pyroelectrc contant, are denoted by c kl, e kl, ε kk, λ kk ( kl, =,, 3), κ r, κ z, and p z, repectvely. The conttutve equaton for the elatc feld are

3 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng 9 σ σ σ rr θθ zz ur ur uz φ = c + c + c3 + e3 λt, r r z z ur ur uz φ = c + c + c3 + e3 λt, r r z z ur ur uz φ = c3 + c3 + c33 + e33 λ33t, r r z z uz ur φ σ zr = c e5 r z r ( =, ) () where T ( r, z) the temperature, φ ( rz, ) the electrc potental, ur( r, z), uz( r, z) are the dplacement component, σ rr( rz, ), σ θθ( rz, ), σ zz( rz, ), σ zr( rz, )( =, ) are the tre component. The ubcrpt =, denote the thermo-electro-elatc feld n z h and h z, repectvely. For the electrc feld, the conttutve relaton are uz ur φ Dr = e5 + ε, r z r ur ur uz φ Dz = e3 + e3 + e33 ε33 + pzt r r z z ( =, ) () where D ( r, z), D ( r, z)( =, ) are the electrc dplacement component. r z The governng equaton for the thermo-electro-elatc feld of the medum may be expreed a follow: κ T T T + ( ) + = =, r r r z (3) ur ur u r ur uz φ T c + c44 ( c3 c44 ) ( e3 e5 ), r r r r = λ z rz rz r uz u z u z ur u r φ φ φ T c44 + c ( c3 + c44 ) + + e5 + + e 33 r r r z r z r z r r r = λ 33, (4) z z ( = ), uz u z u z ur u r φ φ φ T e5 e 33 ( e5 e3 ) ε ε 33 p z + r r r z + + = r z r z r r r z z where κ = κ / κ. r z The boundary condton can be wrtten a T ( r, ) = ( r < c) z (5) (, ) = (, ) ( <) T r T r c r

4 Fourer Tranform Materal Analy T( r, h) = T, T( r, ) = T( r, ), ( r <) z z T( r, h) = T (6) for thermal loadng condton and σ zz( r, ) = ( r < c) (7) u ( r, ) = u ( r, ) ( c r <) z z σ zr( r, ) = ( r < c) (8) u ( r, ) = u ( r, ) ( c r <) r r Dz( r, ) = ( r < c) (9) φ ( r, ) = φ ( r, ) ( c r <) σzz( r, ) = σzz( r, ), σzz( r, h) =, σzz( r, h) =, σzr( r, ) = σzr ( r, ), σzr( r, h) =, σzr ( r, h) =, ( r <) () D ( r, ) = D ( r, ), D ( r, h ) =, D ( r, h ) = z z z z for the electromechancal condton. 3. Temperature feld For the problem condered here, t convenent to repreent the temperature a the um of two functon. where () () T( r, z) = T ( z) + T ( r, z) ( =, ) () T () ( z ) atfe the followng equaton and boundary condton: dt dz () = () and () () = T ( h ) T, T ( h ) = T T () ( r, z)( =, ) ubjected to the relaton: κ () () () T T T + + = ( =, ) r r r z (3) (4) () d () T ( r, ) = T () ( r < c) z dz (5) T ( r, ) = T ( r, ) ( c r <) () ()

5 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng () T ( r, h) =, () () T ( r, ) = T ( r, ), ( r <) z z T ( r h ) (), = (6) It eay to fnd from Eq.() and (3) that () T ( z) = T T z+ T h + T h h + h {( ) } (7) By applyng the Hankel tranform to Eq.(4) (Sneddon & Lowengrub, 969), we have () τ (8) T ( r, z ) = D ( ) J ( r )exp( z ) d ( =, ) where D ()(, j = ), are unknown functon to be olved and τ (, =, ) are gven by τ = τ = κ, τ = τ = κ (9) Takng the econd boundary condton (5) nto conderaton, the problem may be reduced to a ngular ntegral equaton by defnng the followng new unknown functon G ( r ) (Erdogan & Wu, 996): G() r = r () () { } T ( r, ) T ( r, ) ( r < c) ( c r <) () Makng ue of the frt boundary condton (5) wth Eq.(6), we have the followng ngular ntegral equaton for the determnaton of the unknown functon () G t : { } c T T t M ( t, r) + M ( t, r) G ( t) dt = ( r < c) () () () κ h + h () In Eq.(), the kernel functon M ( tr, ) and M () ( tr, ) are gven by M () r E ( ), r < t π t t r ( t, r) = r t t E + K ( r > t) π tt ( r ) r rt r () M tr ρ ( ) ρ ( ) J rj td () (, ) = + ( ) ( ) κρ() (3) where K and E are complete ellptc ntegral of the frt and econd knd, and ρ ()( k = ),, are gven by k

6 Fourer Tranform Materal Analy { } { } ρ() = ρ() exp( κh) ρ() exp( κh), ρ() = τ τexp( κh) ( = ), (4) Once G () t obtaned from Eq.(), the temperature feld can be ealy calculated a follow: where wth () () () T ( r, z) = T ( r, z) ( =, ) (5) τ (6) T ( r, z) = R ( ) R ( ) J ( r)exp( z) d (, j =, ) c R() = tg () t J( t) dt, ρ() ρ() R() =, R() = exp( κ h), ρ() ρ() ρ() ρ() R() =, R() = exp( κ h ) ρ() ρ() (7) On the plane z =, the temperature T () ( r, )( =, ) are reduced to T () ( ) c ( ) ( r ) ( ) ( ) ( ) ( ) ( ) G t dt R, = + r j R J r d =, = (8) 4. Thermally nduced elatc and electrc feld The non-dturbed temperature fled T () ( z ) gven by Eq.(7) doe not nduce the tre and electrc dplacement component, whch affect the ngular feld. Thu, we conder the elatc and electrc feld due to the dturbed temperature dtrbuton T () ( r, z)( =, ) only. It convenent to repreent the oluton uz( r, z), ur( r, z) and φ ( rz, )( =, ) a the um of two functon, repectvely. () () uz( r, z) = uz ( r, z) + uz ( r, z), () () ur( r, z) = ur ( r, z) + ur ( r, z), ( =, ) (9) () () φ( rz, ) = φ ( rz, ) + φ ( rz, ) where u () z ( r, z), u () r ( r, z), φ () ( rz, )( =, ) are the partcular oluton of Eq.(4) replaced T by T (), and u () ( r, z), u () r ( r, z), φ () ( rz, ) ( =, ) are the general oluton of z

7 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng 3 homogeneou equaton obtaned by ettng T = ( = ), n Eq.(4). In the followng, the upercrpt () and () ndcate the partcular and general oluton of Eq.(4). Subttutng Eq.(9) nto Eq.() and (), one obtan tre σ rr( rz, ), σ θθ( rz, ), σ zz( rz, ), σ zr( rz, ) and electrc dplacement D ( r, z), D ( r, z)( =, ) expreon. r Ung the dplacement potental functon method (Ueda, 6a), the partcular oluton can be obtaned a follow: z () () σzz ( rz, ) = p R( R ) ( J ) ( r)exp( τ zd ), () () σzr ( rz, ) = p R( R ) ( J ) ( r)exp( τ zd ), () () Dz ( r, z) = p 3 R( ) R( ) J( r)exp( τ z) d, ( =, ) u r z p R () R () J ( r)exp( τ z) d, () ( ) z (, ) = 4 u r z p R R J r z d () () r (, ) = 5 ( ) ( ) ( )exp( τ ), () (), = 6 φ ( rz) p R() R() J( r)exp( τ zd ) (3) where the contant p () k (, j =,, k =,,..., 6) are gven n Appendx A. The general oluton are obtaned by ung the Hankel tranform technque (Sneddon & Lowengrub, 969): 6 () () σzz ( rz, ) = p A ( J ) ( r)exp( γzd ), 6 () () σzr ( rz, ) = p A ( J ) ( r)exp( γzd ), 6 () () Dz ( r, z) = p 3 A( ) J( r)exp( γ z) d, ( =, ) u ( r z) p A ( ) J ( r) exp( γ z) d, 6 () () z, = 4 6 () () r (, ) = 5 ( ) ( )exp( γ ), u r z p A J r z d 6 () (), = 6 φ ( rz) p A ( J ) ( r)exp( γzd ) (3) where A( ) ( =,, j =,,..., 6) are the unknown functon to be olved, and the contant γ and p () ( =,, j, k =,,..., 6) are gven n Appendx B. k

8 4 Fourer Tranform Materal Analy Smlar to the temperature analy, the problem may be reduced to a ytem of ngular ntegral equaton by takng the econd boundary condton (7)-(9) nto conderaton and by defnng the followng new unknown functon Gl( r)( l =,, 3) : G() r = r () () { z z } u ( r, ) u ( r, ) ( r < c) ( c r <) () () { r r } r u ( r, ) u ( r, ) ( r < c) G() r = r r ( c r <) () () { φ φ } ( r, ) ( r, ) ( r < c) G3() r = r ( c r <) Makng ue of the frt boundary condton (7)-(9) wth Eq.(), we have the followng ytem of ngular ntegral equaton for the determnaton of the unknown functon G ()( t l = 3),, : l () { } () { } (3) (33) (34) c t[ Z M ( t, r) + M( t, r) G( t) + M( t, r) G( t) + (35) + Z M ( t, r) + M ( t, r) G ( t)] dt = σ ( r) ( r < c) zz () { } c tm [ ( trg, ) ( t) + ZM ( tr, ) + M( tr, ) G( t) + (36) + M ( t, r) G ( t)] dt = σ ( r) ( r < c) 3 3 zr () { } () { } c t[ Z 3M ( t, r) + M3( t, r) G( t) + M3( t, r) G( t) + (37) + Z M ( t, r) + M ( t, r) G ( t)] dt = D ( r) ( r < c) z () where the kernel functon M ( tr, ), Mkl ( tr, ) and the contant Zkl ( k, l=,, 3) are gven by M () 4 r r t r r K E + E K ( r t), < πrt t t πrt t r t t ( t, r) = 4 t t t K E E ( r t) + > πt r r π t r r { kl kl } { kl kl } { kl kl } { kl kl } Z () Z J ( rj ) ( td ) ( k= 3,, l= 3),, Z () Z J ( rj ) ( td ) ( k= 3,, l= ), Mkl( t, r) = Z () Z J ( rj ) ( td ) ( k=, l= 3),, Z () Z J ( rj ) ( td ) ( k=, l= ) (38) (39)

9 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng 5 6 () kl k j jl kl kl Z () = p d (), Z = lm Z () ( k, l= 3),, (4) In Eq.(4), the functon d jl()( j =,,..., 6, l= 3),, are gven n Appendx C. The functon σ zz ( r ), σ zr ( r ) and Dz ( r ), whch correpond to the tre and electrc dplacement component nduced by the dturbed temperature feld T () ( r, z)( =, ) on the r -ax n the plate wthout crack, are obtaned a follow: 6 () T () σ zz() r = R () p jdj() + p jrj() J( r) d, 6 () T () σ zr() r = R () p jdj() + p jrj() J( r) d, 6 () T () Dz() r = R () p3 jdj() p3 jrj() J ( r ) d (4) T where the functon d j() ( j =,,..., 6) are alo gven n Appendx C. Thee component are uperfcal quantte and have no phycal meanng n th analy. However, they are equvalent to the crack face tracton n olvng the crack problem by a proper uperpoton. To olve the ngular ntegral equaton () and (35)- (37) by ung the Gau-Jacob ntegraton formula (Sh, 97), we ntroduce the followng functon Φ ()( l= 3),,, : l t / c+ t Gl() t = Φl() t ( l= 3),,, c t (4) Then the tre ntenty factor K I, may be defned and evaluated a: K II and the electrc dplacement ntenty factor K D { } / / KI = lm{ π( c r)} σzz( r, ) = ( πc) ZΦ() c + Z3Φ3(), c + r c / / KII = lm { π( c r)} σzr( r, ) = ( πc) ZΦ ( c), + r c / / KD = lm{ π( c r)} Dz( r, ) = ( πc) { Z3Φ() c + Z33Φ3() c } + r c (43) 5. Numercal reult and dcuon For the numercal calculaton, the thermo-electro-elatc properte of the plate are aumed to be one of cadmum elende wth the followng properte (Ahda & Tauchert, 998). The value of the coeffcent of heat conducton for cadmum elende could not be found n the lterature. Snce the value of them for orthotropc Alumna (Al O 3 ) are κ r =.5[W/mK] and κ z = 9.8[W/mK] (Dag, 6), the value κ = κr / κz = / 5. aumed. To examne the effect of the normalzed crack ze c/ h and the normalzed crack

10 6 Fourer Tranform Materal Analy locaton h / h on the tre and electrc dplacement ntenty factor, the oluton of the ytem of the ngular ntegral equaton have been computed numercally. 9 9 c = 74. [ N/ m ], c = 45. [ N/ m ], 9 9 c3 = [ N/ m ], c33 = [ N/ m ], 9 c44 = 3. [ N/ m ], e3 =. 6[ C/ m ], e33 =. 347[ C/ m ], e5 =. 38[ C/ m ], ε =. / ε33 =. / 6 6 λ =. 6 [ N/ Km λ33 =. N/ Km 6 p z 94 [ CK m ] 8 6 [C Vm], 9 3 [C Vm], ], 55 [ ], =.. (44) In the frt et of calculaton, we conder the temperature feld and the electro-elatc feld wthout crack. Fgure how the normalzed temperature ( T ( x) T )/ T ( =, ) on the ± crack face ( r < c, z ) and the crack extended lne ( c r c, z= ) for h / h = 5. and c/ h =. 5, where T = T T. The maxmum local temperature dfference acro the crack occur at the center of the crack. Fgure 3 exhbt the normalzed tre component ( σzz( r), σzr( r)) / λ33t and the electrc dplacement component Dz( r) / pzt on the r -ax n the trp wthout crack due to the temperature hown n Fgure. The maxmum abolute value of σ zz ( r ) and Dz ( r ) occur at the center of the crack ( r / c =. ), wherea the maxmum value of σ zr ( r ) occur at the crack tp ( r/ c = )...8 (T -T )/T.6.4 h /h=.5 c/h=.5 : = (z + ) : = (z - ). r/c Fg.. The temperature on the crack face and the crack extended lne for c/ h =. 5 and h / h = 5. In the econd et of calculaton, we tudy the nfluence of the crack ze on the tre and electrc dplacement ntenty factor. Fgure 4(a)-(c) how the plot of the normalzed / tre and electrc dplacement ntenty factor ( KI, KII) / λ33t( πc), KD pzt ( πc ) / / veru c / h for h / h = 5.,.5 and.75. Becaue of ymmetry, the value of K I and K D

11 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng 7. (σ zz, σ zr )/λ 33 T.6.4. h /h=.5 c/h=.5 D z σ zr D z /p z T σ zz -.4 r/c Fg. 3. The tre component σ zz, σ zr and the electrc dplacement component Dz r -ax wthout crack due to the temperature hown n Fg. on the / K I/λ 33T (πc).. -. (a) h /h=.5 h /h=.5 h /h=.75 / K II/λ 33T (πc) (b) h /h=.5 h /h=.5, c/h c/h.. (c) h /h=.75 K D /p T (πc) Z / -. h /h=.5 h /h=.5 -. c/h Fg. 4. (a) The effect of the crack ze on the tre ntenty factor K I. (b) The effect of the crack ze on the tre ntenty factor K II. (c) The effect of the crack ze on the electrc dplacement ntenty factor K D

12 8 Fourer Tranform Materal Analy / =. are zero, and [ K ] h for h h 5 [ K ] h D / h.5 = [ ] D h / h=.75 I / h.5 = K. The abolute value of = = [ KI ], [ K ] h / h=.75 h.75 monotoncally ncreae wth ncreang c/ h, but the value of / II / h.5 = K = [ ] h II / h=.75 K I λ 33 T ( π c) / / for h / h = 5. and K II λ 33 T ( π c) / / and the abolute value of KD / pzt ( πc ) ncreae at frt, reach maxmum value and then decreae wth ncreang c / h. The value of K I for h / h = 75. become negatve o that the contact of the crack face would occur. The reult preented here wthout conderng th effect may not be exact but would be more conervatve. Snce the contact of the crack face wll ncreae the frcton between the face and make thermo-electrcal tranfer acro the crack face eaer, the tre and electrc dplacement ntenty factor would be lowered by thee two factor. In the fnal et of calculaton, we nvetgate the nfluence of the crack locaton on the ntenty factor. Fgure 5 ndcate the effect of the crack locaton on K I, K II and K D for c/ h = 5.. A h / h ncreae, the value of K I and K D tend to decreae or ncreae monotoncally. The value of K II λ 33 T ( π c) / / decreae f the crack approache the free boundare ( h / h.or.), and the peak value of K II λ 33 T ( π c) / / =.77 occur at h / h = K II (K I, K II )/λ 33 T (πc) /.. K D K I. -. K D /p z T (πc) / -. c/h= h /h Fg. 5 The effect of the crack locaton on the tre ntenty factor dplacement ntenty factor K 6. Concluon D K I, K II and the electrc An example of the applcaton of Hankel tranform for olvng a mxed-mode thermoelectro-elatc fracture problem of a pezoelectrc materal trp wth a parallel penny-haped crack explaned. The effect of the crack ze ( c / h) and the crack locaton ( h / h) on the fracture behavor are analyzed. The followng fact can be found from the numercal reult.. The large hear tre occur n the trp wthout crack due to the dturbed temperature feld.. The normalzed ntenty factor are under the great nfluence of the geometrc parameter h / h and c / h.

13 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng 9 3. For the cae of h / h >.5, mode I tre ntenty factor become negatve o that the contact of the crack face would occur. 4. The ntenty factor of crack near the free urface due to the thermal load are not o large. Appendx A The contant where p () (, j =,, k =,,..., 6) are k ( ) () ( ) ( τ ) () p = c3 c33kτ C e33τ N λ33, p = c44 + k τ C + e5n, () p3 = e3 e33k C + ε33τ N + pz, () () () p4 = kτ C, p5 = C, p6 = N (, =, ) (A.) wth b b τ C =, () () m + km () () n + kn N =, ( ) () (), j =, m + km () () τ n + n k = () () τ n + n (A.) () m = a 4 a 4τ, () m = ( a 44τ a 43) τ, () () n = {( Hr + H3r) ( b b τ) + H4rm } τ, () () n = {( Hz + H3z) ( b b τ) + H4zm } τ, () () n = {( Hr Hrτ )( b b τ ) + H4rm } τ, () () n = {( Hz Hzτ ) ( b ) } b τ + H 4 zm τ (, j =,) (A.3) c44ε33 + e5e33 c44ε + e5 c33ε33 + e33 c33ε + e5e 33 Hr =, Hz =, Hr =, Hz =, e33ε e5ε33 e5ε33 e33ε e33ε e5ε33 e5ε33 e33ε c3ε33 + e3e33 c3ε + e5e3 pe z 33 λ33ε33 pe z 5 λ33ε H3r =, H3z =, H4r =, H4z = e33ε e5ε33 e5ε33 e33ε e33ε e5ε33 e5ε33 e33ε (A.4)

14 Fourer Tranform Materal Analy a = c, a = c + ( e + e )( H + H ), r 3r 43 = c3 + c44 + ( e5 + e3) Hr, a 44 = ( e5 + e3) Hr, = λ, b = ( e5 + e3) H 4r a b (A.5) Appendx B The contant γ ( =,, j =,,..., 6) are the root of the followng charactertc equaton: 6 4 ( f4g g4f ) γ ( f4g fg g4f gf ) γ ( fg fg gf gf ) γ ( fg gf ) = j ( =,, j =,,..., 6) (B.) where R [ γ j] <R [ γ j + ], R [ γ j] >R [ γ j+ ]( j =,,..., 5) and f4 = c44e33, f = ( c3 + c44)( e5 + e3) ce33 c44e5, f = ce5, f = c33( e5 + e3) e33( c3 + c44), f = c44( e5 + e3) + e5( c3 + c44) g4 = c44ε 33, g = ( e5 + e3) cε33 c44ε, g = cε, g = e33( e5 + e3) + ε 33( c3 + c44), g = e5( e5 + e3) ε( c3 + c44) (B.) (B.3) The functon p () ()( =,, j, k =,,..., 6) are k () () 3 () γ ( ), ( γ ), γ ( ε ) p = c a + c e b p = c a + e b 44 5 p e a e b = p p () 4 () 5 p () 6 =, = a = b where a and b ( =,, j =,,..., 6) are gven by b,, g γ + g a =, 4 g4γ + gγ + g ( c44γ c) a c3 c44 = e5 + e3 ( =,, j =,,..., 6) ( =,, j =,,..., 6) (B.4) (B.5)

15 Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng Appendx C The functon d ( ) ( =,, j =,,..., 6, k =,, 3) are gven by k djk() = qj, k+ 9(), ( j =,,..., 6, k =,, 3) djk() = qj+ 6, k+ 9() (C.) where the functon qjk, ()( j, k =,,..., ) are the element of a quare matrx Q = Δ of order. The element δ, ( ) ( j, k =,,..., ) of the quare matrx Δ are gven by jk () δjk, ( ) = pjkexp( γkh) ( j =,, 3), () δj+ 3, k+ 6() = pjkexp( γkh) ( j = 3),,, ( 6) () k =,,..., δ j+ 6, k() = pjk ( j =,,..., 6), () δ j+ 6, k+ 6() = pjk ( j =,,..., 6) (C.) T The functon d ( ) ( =,, j =,,..., 6) are T d j() = qj, k() uk(), k= ( 6) j =,,..., T dj() = qj+ 6, k() uk() k= (C.3) where u () = p R ()exp( h ) ( k = 3),,, R() () k k j j τ j R() () k+ 3 k j j τ j u () = p R ()exp( h ) ( k = 3),,, R() () () uk+ 6() = { pkjrj() pkjrj() } ( k 6) =,,..., (C.4) 7. Reference Ahda, F. & Tauchert, T.R. (998). Tranent Repone of a Pezothermoelatc Crcular Dk under Axymmetrc Heatng. Acta Mechanca, Vol. 8, pp. -4, -597 Dag, S., Ilhan, K.A. & Erdogan, F. (6), Mxed-Mode Stre Intenty Factor for an Embedded Crack n an Orthotropc FGM Coatng, Proceedng of the Internatonal Conference FGM IX, , Oahu Iland, Hawa, October 6 Erdogan, F. & Wu, B.H. (996). Crack Problem n FGM Layer under Thermal Stree. Journal of Thermal Stree, Vol. 9, pp ,

16 Fourer Tranform Materal Analy Rao, S.S. & Sunar, M. (994). Pezoelectrcty and It Ue n Dturbance Senng and Control of Flexble Structure: a Survey. Appled Mechanc Revew, Vol. 47, pp. 3-3, 3-69 Sh, G.C. (Ed.). (97). Method of Analy and Soluton of Crack Problem, Noordhoff, Internatonal Publhng, , Leyden Sneddon, I.N. & Lowengrub, M. (969). Crack Problem n the Clacal Theory of Elatcty, John Wley & Son, Inc., , New York Tauchert, T.R. (99). Pezothermoelatc Behavor of a Lamnated Plate. Journal of Thermal Stree, Vol. 5, pp. 5-37, Ueda, S. (6a). The Crack Problem n Pezoelectrc Strp under Thermoelectrc Loadng. Journal of Thermal Stree, Vol. 9, pp , Ueda, S. (6b). Thermal Stre Intenty Factor for a Normal Crack n a Pezoelectrc Strp. Journal of Thermal Stree, Vol. 9, pp. 7-6,

17 Fourer Tranform - Materal Analy Edted by Dr Salh Salh ISBN Hard cover, 6 page Publher InTech Publhed onlne 3, May, Publhed n prnt edton May, The feld of materal analy ha een explove growth durng the pat decade. Almot all the textbook on materal analy have a ecton devoted to the Fourer tranform theory. For th reaon, the book focue on the materal analy baed on Fourer tranform theory. The book chapter are related to FTIR and the other method ued for analyzng dfferent type of materal. It hoped that th book wll provde the background, reference and ncentve to encourage further reearch and reult n th area a well a provde tool for practcal applcaton. It provde an applcaton-orented approach to materal analy wrtten prmarly for phyct, Chemt, Agrculturalt, Electrcal Engneer, Mechancal Engneer, Sgnal Proceng Engneer, and the Academc Reearcher and for the Graduate Student who wll alo fnd t ueful a a reference for ther reearch actvte. How to reference In order to correctly reference th cholarly work, feel free to copy and pate the followng: Se Ueda (). Applcaton of Hankel Tranform for Solvng a Fracture Problem of a Cracked Pezoelectrc Strp Under Thermal Loadng, Fourer Tranform - Materal Analy, Dr Salh Salh (Ed.), ISBN: , InTech, Avalable from: InTech Europe Unverty Campu STeP R Slavka Krautzeka 83/A 5 Reka, Croata Phone: +385 (5) Fax: +385 (5) InTech Chna Unt 45, Offce Block, Hotel Equatoral Shangha No.65, Yan An Road (Wet), Shangha, 4, Chna Phone: Fax: