DYNAMIC PROPAGATION OF A WEAK-DISCONTINUOUS INTERFACE CRACK IN FUNCTIONALLY GRADED LAYERS UNDER ANTI-PLANE SHEAR

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1 8 TH INTERNTIONL CONFERENCE ON COMPOSITE MTERILS DYNMIC PROPGTION OF WEK-DISCONTINUOUS INTERFCE CRCK IN FUNCTIONLLY GRDED LYERS UNDER NTI-PLNE SHER J.W. Sn *, Y.S. Lee, S.C. Km, I.H. Hwng 3 Subsystem Deprtment, Kore erospce Reserc Insttute, Dejeon, Kore Deprtment of Mecncl Desgn Engneerng, Cungnm Ntonl Unversty, Dejeon, Kore 3 Rotorcrft Progrm Offce, Kore erospce Reserc Insttute, Dejeon, Kore * Correspondng utor(jeongdl@kr.re.kr Keywords: Functonlly grded mterl (FGM, Interfce movng crck, Wek-dscontnuous nterfce, Dynmc energy relese rte (DERR Introducton Functonlly grded mterls (FGMs ve been recently ttrcted extensve ttenton for use n g temperture pplctons nd wer-protectve cotngs. Te FGMs re mcroscopclly nonomogeneous becuse te mecncl propertes of te FGM vry smootly nd contnuously. Te frcture bevor of te FGM s mportnt desgn ssue. crck n te FGM my exbt complex bevor becuse of te vrton of te mecncl propertes of te mterl. Te frcture bevors of te FGMs ve been studed wdely for bot sttc nd dynmc problems. But, soluton of te dynmc crck propgton of n nterfce crck between two dssmlr functonlly grded lyers s not been presented. In ts pper, dynmc propgton of n nterfce Grfft crck between two functonlly grded lyers under nt-plne ser s nlyzed. Te propertes of te functonlly grded lyers vry contnuously long te tckness. In ddton, te propertes of te two functonlly grded lyers vry dfferently nd te two lyers re connected wekdscontnuously (L et l.[]. Te Yoffe-type model (Yoffe[] for crck propgton s dopted. Fourer trnsform s used to reduce te problem to dul ntegrl euton, wc s ten expressed n Fredolm ntegrl euton of te second knd. Numercl results of te dynmc energy relese rte re presented grpclly to sow te effect of grdent of mterl propertes, crck movng velocty, nd tckness of lyers. Problem sttement nd formultons Consder two functonlly grded lyers contnng fnte nterfce crck subjected to nt-plne ser lodng, s sown n Fg.. Mterl propertes of te two functonlly grded lyers vry dfferently. Te Crtesn coordntes ( X, Y, Z re fxed for te reference. Te FGM lyers occupy te regon, < X <, Y, nd re tck enoug n te Z -drecton. Te crck s stuted long te nterfce lne ( X, Y =. Due to te symmetry n geometry nd lodng, t s suffcent to consder te rgt-nd lf body only. Y vt y τ X x μ μ μ Fg.. Geometry of constnt movng nterfce crck between two functonlly grded lyers We ssume tt te propertes of te two functonlly grded lyers vry contnuously long te tckness nd re smplfed s follows (Delle nd Erdogn[3]: Y e β μ μ Y e β ρ ρ = ( = ( μ nd ρ re te ser modulus nd mterl densty, respectvely. μ nd ρ re te mterl

2 propertes t te nterfce nd β s te nonomogeneous mterl constnt. Subscrpt ( =, stnds for te upper nd lower lyers, respectvely. Te boundry vlue problem s smplfed consderbly f we consder only te out-of-plne dsplcement suc tt u X = u Y =, uz = w ( X, Y, t (3 u k ( k = X, Y, Z s te dsplcements. In ts cse, te consttutve relton becomes σ ( = μ (4 Zj X, Y, t w, j σ Zj ( j = X, Y s te stress component. Te dynmc nt-plne governng euton for FGM s smplfed to w w μ w + β μ = ρ (5 Y t = X + Y. By substtutng Es. ( nd ( nto te E. (5, te dynmc governng euton s trnsformed nto te followng euton: w + = Y c w w β (6 t c = μ (7 ρ nd c s te ser wve velocty. For te problem of movng crck wt constnt velocty v long te X -drecton, t s convenent to ntroduce Gllen trnsformton suc s x = X v t, y = Y, z = Z, t = t (8 ( x, y, z s te trnsltng coordnte system ttced to te center of te movng crck. In te trnsformed coordnte system, te dynmc nt-plne governng euton for FGM cn be smplfed to te followng form: w w w α + + β = (9 x y y ( v α = c ( Fourer trnsform s ppled to E. (9, nd te result s s follows: w π y y [ ( s e + ( s e ] = cos( sx ds ( β β = δ +, = δ ( β δ = α s + (3 4 nd re te unknowns to be solved. By substtutng E. ( for E. (4, we ve te followng: σ yz = μ [ ( s e π + ( s e y y ] cos( sx ds Te boundry condtons cn be wrtten s σ yz = τ ( x < (4 (5 w + = w ( < x (6 σ x, + = σ ( < x (7 yz ( yz σ x, = σ ( < x (8 yz ( yz = τ s te unform ser trcton. By pplyng te edge lodng condtons of E. (8, te unknowns n E. (4 re evluted s follows: ( s e + ( s e = (9 ( s e + ( s e ( = Te contnuty condton of E. (7 leds to te followng relton between te unknowns: s + ( s = ( s + ( ( ( s

3 DYNMIC PROPGTION OF WEK-DISCONTINUOUS INTERFCE CRCK IN FUNCTIONLLY GRDED LYDERS UNDER NTI-PLNE SHER It s convenent to use te followng defntons: s ( s + ( s ( s = ( ( ( s Usng te Es. (9 to (, we cn obtn te followng reltons: π τ Ω( ξ + K( ξ, η Ω( η dη = (3 α μ [ F( s ] J ( sη J ( sξ ds K ξ, η = η s (3 ( δ ( e ( s = ( (3 δ δ δ δ ( e ( + e + ( e ( + e s ( s δ δ e ( e ( s δ δ δ δ ( e ( + e + ( e ( + e ( s δ δ e ( e ( s δ δ δ δ ( e ( + e + ( e ( + e ( s δ ( e ( s δ δ δ δ ( e ( + e + ( e ( + e = (4 = (5 = (6 Te mxed boundry condtons of Es. (5 nd (6 led to dul ntegrl eutons n te followng form: π τ s F( s ( s cos( sx ds = ( x < α μ ( s cos( sx ds = ( < x (7 We ntroduce te followng dmensonless vrbles nd functon for numercl nlyss s follows: S Β s =, β =, Β Δ β =, Δ δ =, δ = (3 η = Η, ξ = Ξ (33 π τ Ψ( Ξ Ω( ξ = α μ Ξ (34 F( s = α s ( e δ ( + e ( e δ δ + ( e ( e δ δ ( + δ e (8 Te dul ntegrl E. (7 my be solved by usng new functon Ω (ξ defned by ( s = ξ Ω( ξ J ( sξ dξ (9 J s te zero-order Bessel functon of te frst knd. By nsertng E. (9 nto E. (7, we cn fnd tt te uxlry functon Ω (ξ s gven by Fredolm ntegrl euton of te second knd n te followng form: By substtutng Es. (3 to (34 for Es. (3 nd (3, we cn obtn Fredolm ntegrl euton of te second knd n te followng form: Ψ( Ξ + L ( Ξ, Η Ψ( Η dη = Ξ (35 3

4 L ( Ξ, Η = S ΞΗ S F J ( S Η J ( S Ξ ds (36 Δ Δ ( ( S QQQQ e e F( = α S Δ Δ Δ Δ QQ( e ( Q + Q e + QQ ( e ( Q + Q e (37 Β Β Β Β Q = Δ +, Q = Δ +, Q = Δ, Q = Δ (38 Te mode III dynmc stress ntensty fctor K III (v nd dynmc energy relese rte (v re defned nd determned n te followng forms: ( v = τ π Ψ( (39 K III π ( v = τ Ψ ( (4 μ E. (4 cn be mde dmensonless s follow: ( v μ = Ψ ( τ π (4 s benefcl to ncrese of te resstnce of te nterfce crck propgton of FGM. π GIII(v μ/τ v/c =.4 / = / =. B =. B =. B =. B = -. B = -. n wc te functon Ψ ( cn be clculted from E. (35. 3 Dscussons To nvestgte te effect of te grdent of mterl propertes, crck movng velocty nd tckness of lyers on te dynmc energy relese rte (DERR, numercl nlyses re crred out. Fg. dsplys te vrton of te normlzed DERR ( v μ τ π gnst te normlzed nonomogeneous mterl constnt of te upper lyer Β wt vrous normlzed non-omogeneous mterl constnts of te lower lyer Β t v / c =.4 nd / = / =.. Te normlzed DERR decreses wen te grdents of mterl propertes of te upper nd lower lyers ncrese. For te upper lyer, te grdent of mterl propertes ncreses s te non-omogeneous mterl constnt ncreses. But for te lower lyer, te grdent of mterl propertes ncreses s te non-omogeneous mterl constnt decreses becuse vlue of te y-xs s negtve. Increse of te grdent of mterl propertes from te nterfce B Fg.. Vrton of te normlzed DERR ( vμ τ π wt Β Fg. 3 sows te vrton of te normlzed DERR gnst te normlzed crck movng velocty v / c wt te vrous normlzed non-omogeneous mterl constnts. ccordng to te vlues of te grdent of mterl propertes of te upper nd lower lyer, we cn clssfy nto tree ctegores s follows: Cse I : mterl propertes ncrese wen te tckness of upper nd lower lyer ncreses from te nterfce, Cse II : mterl propertes decrese wen te tckness of upper nd lower lyer ncreses from te nterfce, Cse III : mterl propertes ncrese from te lower surfce ( y = to upper surfce

5 DYNMIC PROPGTION OF WEK-DISCONTINUOUS INTERFCE CRCK IN FUNCTIONLLY GRDED LYDERS UNDER NTI-PLNE SHER ( y =, nd vce verse. For te Cse II nd III, te normlzed DERR ncreses s te crck movng velocty ncreses. But for te Cse I, te trend s opposte. Te normlzed DERR decreses wen te crck movng velocty ncreses. Tt s, ncrese of te stffness from te nterfce to te upper nd lower surfce s elpful to ncrese of te resstnce of te nterfce crck propgton of FGM. s te bsolute vlue of Β Β ncreses, tt s, te dfference between te grdents of te mterl propertes of upper nd lower lyer s gettng bgger, te normlzed DERR ncreses or decreses more rpdly. π GIII(v μ/τ / = / = B = -., B =. B =., B =. B =., B =. B =., B = -. B =., B = v/c Fg. 3. Vrton of te normlzed DERR ( v μ τ π wt v / c Te effect of te crck movng velocty on te vrton of te normlzed DERR s sown n Fg. 4 wt vrous tcknesses of te lyers. Te normlzed DERR ncreses wt te ncrese of te crck movng velocty. But te normlzed DERR decreses wen te tckness of lyer ncreses. Increse of te tckness of FGM lyer s lso benefcl to ncrese of te resstnce of te nterfce crck propgton of FGM. Fg. 5 presents te vrton of te normlzed DERR gnst te normlzed tckness of te lower FGM lyer wt te vrous non-omogeneous mterl constnts. Smlr to Fg. 4, te normlzed DERR decreses s te tckness of te lower lyer ncreses. But, over certn vlue of te tckness of te lower lyer (bout 3., te effect of decrese of te normlzed DERR s neglgble. s seen n Cse I of Fg. 3, Fg. 5 lso sows tt ncrese of te stffness from te nterfce to te upper nd lower surfce s elpful to ncrese of te resstnce of te nterfce crck propgton of FGM. π GIII(v μ/τ π GIII(v μ/τ...8 Β =., B =. / =., / =. / =., / =. / =., / = v/c Fg. 4. Vrton of te normlzed DERR ( v μ τ π wt v / c v/c =., / =. B =., B =. B =., B =. B =., B =. B =., B = -. B =., B = / Fg. 5. Vrton of te normlzed DERR ( vμ τ π wt / 4 Conclusons Te problem of dynmc propgton of wek- 5

6 dscontnuous nterfce crck between two functonlly grded lyers under nt-plne ser lodng ws nlyzed by te ntegrl trnsform pproc. Te ser modulus nd mss densty of te FGM vry contnuously long te tckness. Fredolm ntegrl euton s solved numerclly. Te computed results sow tt te followngs re elpful to ncrese of te resstnce of te nterfce crck propgton of FGM: Increse of te grdent of mterl propertes, b Increse of te mterl propertes from te nterfce to te upper nd lower free surfce, c Increse of te tckness of FGM lyer. Te normlzed DERR ncreses or decreses wt ncrese of crck movng velocty. References [] Y.D. L, B. J, N. Zng, L.Q. Tng nd Y. D, Dynmc stress ntensty fctor of te wek/mcrodscontnuous nterfce crck of FGM cotng. Int. J. Solds Strut., Vol. 43, pp , 6. [] E.H. Yoffe, Te movng Grfft crck. Plos. Mgzne 7, Vol. 4, pp , 95. [3] F. Delle nd F. Erdogn, Te crck problem for nonomgeneous plne. SME J. ppl. Mec., Vol. 5, pp 69-64, 983.

4. Eccentric axial loading, cross-section core

4. Eccentric axial loading, cross-section core . Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we