TWO INTERFACIAL COLLINEAR GRIFFITH CRACKS IN THERMO-ELASTIC COMPOSITE MEDIA

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1 THERMAL SCIENCE: Yer 8, Vol., No. B, pp TWO INTERFACIAL COLLINEAR GRIFFITH CRACKS IN THERMO-ELASTIC COMPOSITE MEDIA y Prshnt Kumr MISHRA nd Sur DAS * Deprtment of Mthemtcl Scences, Indn Insttute of Technology, Vrns, Ind Introducton Orgnl scentfc pper The ojectve of the rtcle s to fnd the stress ntensty fctors nd crc energy for pr of collner Grffth crcs stuted t the nterfce of the two orthotropc mterls under stedy-stte temperture feld. The prolem s reduced to pr of sngulr ntegrl equtons, whch re solved usng Jco s polynomls. Numercl computtons re crred out for two dfferent prs of orthotropc mterls for dfferent prtculr cses, whch re depcted through fgures. The effect of mterl constnts nd temperture coeffcents on the ehvor of physcl qunttes vz., stress ntensty fctors nd crc energy of the nterfcl crcs s the ey feture of the present rtcle. Key words: therml stresses, collner Grffth crcs, strn energy relese rte, het flux In mny engneerng dscplnes vz., electroncs, erospce, nd nucler energy, lot of reserch hs lredy een done durng the study of ehvor of the stress nd dsplcement felds t the vcnty of the crc tp stuted t the nterfce of the composte mterls suject to therml lodng. Orthotropc composte mterls re wdely used n structurl mterls due to ther lght weght nd strong n nture. When crced orthotropc composte mterl s used n hgh or low temperture regon, then het flows through mterl. In ths cse, t s mportnt to determne the therml stress ntensty round the crc, whch occurs due to the dsturnce n the het flux. The nvestgton of thermo-elstc feld nd therml stress concentrton round the crc help to understnd the stlty nd lfe of the crced engneerng mterls nd structures. Accordng to lner elstc frcture mechncs, stress t the vcnty of the crc tp s sngulr. It s drectly proportonl to the nverse of squre root of dstnce from the crc tp. Mny oservtons of thermo-elstc crced surfces show tht the therml stress sngulrty t the vcnty of the crc tps re sme s those wth mechncl stresses. However, the nture of sngulrty ecomes dfferent for n nterfcl crc. The occurrence of the nterfcl crcs t the surfce of structurl components, due to therml nd mechncl lodng, ecme n mportnt reserch topc n frcture mechncs. For nlyzng nterfcl crcs, mny studes were conducted under therml stedy-stte condtons for orthotropc composte mterls. Sh [] determned the stress ntensty fctor (SIF) of crc n n nfnte plte when the het flows perpendculr to the crc surfce. Lter, Sene [] determned the SIF of crc due to het flux. The therml stresses n n nfnte plte due to the het flux, for two * Correspondng uthor, e-ml: sds.pm@thu.c.n

2 44 THERMAL SCIENCE: Yer 8, Vol., No. B, pp crcs hve een determned y the sme uthor [3]. The SIF round the two collner crcs were evluted y Chen nd Zhng [4] n n orthotropc plte under the het flux. Therml stress for sngle crc n n nfnte elstc lyer nd therml stress round two prllel crcs hd een determned y Itou nd Rengen [5]. Chen nd Zhng [6] hve determned the therml stress n n orthotropc strp contnng two collner crcs. Itou [7] evluted SIF for two prllel crcs n n nfnte orthotropc plte due to the het flux. Bs et l. [8] hve solved the prolem of determnng the therml stresses nd dsplcement felds n n orthotropc plne contnng pr of equl collner Grffth crcs usng the ntegrl trnsform technque sed upon dsplcement potentl under stedy-stte temperture feld. Zhong et l. [9] exmned the ehvor of two collner crcs emedded n n orthotropc sold, usng the Fourer ntegrl trnsform technque, under unform het flux nd mechncl lodng on the crced surfces. Prolems relted to therml stress nd strn cn lso e found n the reserch rtcles [-5]. In the present rtcle the uthors hve mde n endevor to determne the SIF t the tps of pr of collner Grffth crcs stuted t the nterfce of two orthotropc thermo-elstc hlf plnes sujected to unform het flux nd lso to determne the energy requred for cretng two new surfces nd plstc deformton of the crcs under the stedy-stte temperture feld. The prolem hs een reduced to pr of second nd Fredholm ntegrl equtons, whch re solved numerclly usng Jco polynomls. Numercl vlues of the SIF t the tps of the crcs for dfferent prescred crc lengths re presented through grphs for dfferent prtculr cses. Numercl vlues of other physcl quntty crc energy, otned through dfferent forms of the dsplcement potentl functons, re lso presented grphclly. Prolem formulton Let us consder mthemtcl model of two onded homogeneous orthotropc elstc hlf plnes, y < nd < y, contnng pr of collner Grffth crcs stuted symmetrclly t the nterfce y, when Crtesn co-ordnte xes concde wth the xes of symmetry of the elstc mterl. When therml condtons re ppled to the surfce of n rtrry -D orthotropc hlf plnes, then the temperture feld only depends on n-plne co-ordntes under the stedy-stte condton. The temperture dstruton functons T () (x,y) re ssumed to stsfy the followng het conducton equton n the orthotropc med. T x T y K where (K () ) K () y / K () x nd K () y nd K () x, (, ) re the therml conductve coeffcents long y- nd x-drectons, respectvely, for ech hlf plne. The generl soluton of T (j) (x, y) s (c. f., Clements nd Tuchert [6]): j j j ( j) [ p( xy / K )] ( j) [ p( xy / K )] T ( xy, ) { A ( p)e A ( p)e }d p, π () where ( ) /, j, nd A (j) (p) nd Ā (j) (p) re the rtrry functons of p. Here we hve ssumed tht () T () (x, ) h () (x) (3) nd hence the Fourer ntegrl form of temperture dstruton my e wrtten: ( ) [ p ] [ px ] ( ) [ p ] [ p( x)] T ( x,) h e ξ ξ ξ e h ( ξ)e e dξ dp π (4)

3 THERMAL SCIENCE: Yer 8, Vol., No. B, pp From eqs. () nd (4), we get: pξ A ( p) h ( ξ)e dξ, p A ( p) h ( ξ) e ξ dξ (5) From eqs. () nd (5), the temperture dstruton T () (x, y) s otned: T y K h ξ ξ (6) ( y/ K ) ( ξ x) / d ( xy, ) π If we consder h () (x) δ(x) (7) where h () (x) s the prescred temperture dstruton ecome lne source long y-xs nd δ(x) s the Drc delt functon, the resultnt temperture dstruton s otned: ( yk / ) T ( xy, ) π ( yk / ) x (8) The reltons etween the plne stress, nduced y the dstruton of temperture, nd dsplcement components u () (x, y) nd ν () (x, y) long x- nd y-drectons re gven y: u x v y xx ( xy, ) C C βx T u x v y yy ( xy, ) C C βy T u v xy ( xy, ) C66 x y where C () j re the elstc constnts, β () x nd β () ν re the stress temperture coeffcents. The dsplcement equtons of equlrum re gven y: u u 66 v T 66 β x (9) () () C C C C () x y xy x v v 66 u T 66 β y C C C C y x (3) xy y The qunttes wth superscrpts, refer to those for the hlf plne-() nd hlf plne-(), respectvely. It s ssumed tht t the nterfce y, the crcs defned y < x < re opened y nternl norml nd sherng trctons p (x) nd p (x), respectvely, fg.. For the descred prolem the oundry condtons on y re gven y: () yy ( x,) p ( x), x () xy ( x,) p ( x), x () () u x u x (4) (5) (,) (,), x <, x > (6) () () v x v x (,) (,), x <, x > (7)

4 46 THERMAL SCIENCE: Yer 8, Vol., No. B, pp p( x) p( x) y () p( x) () p( x) Fgure. Geometry of the prolem x () () yy ( x,) yy ( x,), < x < (8) () () xy ( x,) xy ( x,), < x < (9) Soluton of the prolem Durng the soluton of the prolem, we frst ntroduce dsplcement potentls ψ () (x, y) nd ϕ () j (x, y) s, Shrm [7]: p{ x( y/ K )} p{ x( y/ K )} ψ ( xy, ) { A ( pb ) ( p)e A ( pb ) ( p)e }dp π () Potentl functons for the hlf plnes re gven y: φ sy () φ ( x, y) s A ( s)e cos( sx)ds π () sy () ( x, y) s A ( s)e cos( sx) d s,, π () The dsplcement components u () nd ν () re wrtten: u ψ φ φ x x x nd The correspondng therml stresses re: ψ φ φ v y y y xx ( xy, ) φ φ ψ ( () ) ( ) η C66 y y y ( xy, ) φ φ ψ yy ( () ) ( ) η C66 y y x ( xy, ) φ φ ψ xy ( () ) ( ) ( η ) C66 xy xy xy The dsplcement eqs. () nd (3) re stsfed y eq. () for non-trvl ϕ () j f: (3) (4) (5) (6) η β C K C β C C K βx C K C66 β y C C 66 y 66 x 66 β C K C β C C ( ) x 66 y 66 pb pb K C K C 66 C66 K C K C C 66 Here, the potentl functons ϕ () j stsfes the followng dfferentl equtons: x φ j ( xy, ),,, j, y (7) (8)

5 THERMAL SCIENCE: Yer 8, Vol., No. B, pp where () nd () re the rel roots of the equton: wth C C [( C ) C C C C ] C C (9) C C j 66 j C66 C A () (s) nd A () (s) (,) re the undetermned functons. Applyng the oundry condtons (8) nd (9), we get: ( K ) () () () () () () () () () () () () ( η ) K C66 ( η ) () () () () () () () () () 66 ( ) K C K ( ) A A () A ( ) () () () () ( ) () () () () () () () () ( ) ( ) () A () () () () ( ) ( ) ( K ) () () () () () () () () () () () () ( η ) K C66 ( η ) () () () () () () () () () 66 ( ) K C K ( ) A () ( ) () () () () ( ) () () () () () () () () ( ) ( ) () A () () () () ( ) ( ) Boundry condtons (6) nd (7), wth the help of the prevous equtons, gve rse to: () n( sx) { α αa α3a } dx, < x<, < x< s () () cos( sx) { β βa β3a } dx,, < x<, < x< s Now settng: () () 3 α α A ( s) α A ( s) f ( t)cos( st)dt () () 3 β β A β A f ()sn( t st)dt where () () () () () () () () () () () () () ( ) K C66 K η η α () () () () () () () () () 66 ( )( ) K C K ( ) ( ) () () () () () () () () () () () ( ) K C66 K η η () () () () () () () () () 66 ( )( ) K C K ( ) ( ) (3)

6 48 THERMAL SCIENCE: Yer 8, Vol., No. B, pp ( ) ( ) α ( ) ( ) () () () () () () () () () () () () () () () () ( ) ( ) α3 ( ) ( ) () () () () () () () () () () () () () () () () η η β K K () () () () () () ( ) ( ) β () () () () () () () () () () () () () () () () () () () () () () ( ) ( ) ( ) ( ) β3 () () () () () () () () () () () () () () () () () () () () () () ( ) ( ) nd fter lengthy process of mthemtcl mnpultons, oundry condtons (4) nd (5) fnlly led to the followng sngulr ntegrl equtons: where f () t f( x) d t p( x) x (3) π ( tx) π f () t c cf( x) d t p( x) π x (3) tx π πx d () () β 3 () β C66 ( ) ( ) π βα 3 αβ 3 βα 3 αβ3 () () α 3 () α C66 ( ) ( ) π βα 3 αβ 3 βα 3 αβ3 ( ) α ( ) α () () () 3 C 66 π () () βα 3 αβ 3 βα 3 αβ3 c () () () ( ) β 3 ( ) β C 66 d () () π βα 3 αβ 3 βα 3 αβ3 ( c ) αβ αβ ( ) αβ αβ ( η ) () () () () 3 3 () () () βα 3 αβ 3 βα 3 αβ3 K Equtons (3) nd (3) re reduced to the followng sngulr ntegrl equtons for the determnton of unnown functons f (x): (33)

7 THERMAL SCIENCE: Yer 8, Vol., No. B, pp where φ () t c φ( x) d t g( x) πε r x (34) ( tx) πr x nd f (x) re stsfyng the condtons: φ ( x) f ( x) r c d f ( x),, ε cd, ( ),, g( x) [ / ` p ( x) r d / c` p ( x)] π r f ( t)dt,,, The soluton of the prevous ntegrl equtons n (34) my e ssumed: where ω (x) ( x) α ( x) β ( α, β) ω n n n φ ( x) ( x) c P ( x),, (35) α ω, β ω, ω rω nd wth c n s unnown constnts. Now, usng eq. (33), we get: whch mples C,,. From eqs. (3) nd (3), we get: ω φ ( t)dt,, ln π ( α, β ) ω () t cnpn () t ( αβ ) n c d ( x) cn Pn ( x) d t g ( x) n π ε ( ) π r t x r c x (36) Multplyng the prevous equton y P j (α, β ) (x) nd ntegrtng from to, we get: where P ( x) c c L F dx ( α, β ) ( α, ) d β c j l jθ j n nj j πε r n πr c x θ (37) α β ( α, ) β Γ j α Γ j β j j α β Γ ( j α β ) j! L nj nd the prncpl vlue of ( α, β ) ( α, β ) ω tpn j ( t x) () () t P () x dd t x d x/ xs consdered s zero. F g xp x x ( α, ) β j j d

8 43 THERMAL SCIENCE: Yer 8, Vol., No. B, pp Fnlly, the stress ntensty fctors t the crc tps x nd x re clculted: / K r d / c K lm ( x ) ( x ) [ / ( x,) r d / c ( x,)] α β () () I II yy xy x α ( α, β) cnpn (),, (38) n ( ) π / K r d / c K lm ( x ) ( x ) [ / ( x,) r d / c ( x,)] α β () () I II yy xy x The crc energy s clculted: Results nd dscusson α ( α, β) cnpn (),, (39) n ( ) π () () [ (,) (,)]d (4) W p x v x v x x In ths secton, the numercl computtons hve een done to fnd physcl qunttes vz., SIF nd the crc energy for two collner crcs stuted t the nterfce of two prs of orthotropc mterls wth frst one s α-urnum nd epoxy oron, nd the second one s eryllum nd epoxy oron. In ech cse frst type of mterl s ten s hlf plne- nd second type of mterl s hlf plne-. Durng the computtons the crc length s consdered s nd. (.).9 nd lso the lodngs re consdered s p (x) p, p (x). The rtos of the stress temperture coeffcents β () y / β () x nd β () y / β () x re ten s.67 nd.5, respectvely, for the frst pr of mterls, nd.7 nd.5, respectvely, for second pr of mterls. The elstc constnts of the orthotropc mterl α-urnum hve een ten s C.47 6 ps (48.3 GP), C ps (3.6 GP), C ps (33.48 GP), C ps (5. GP), Ds nd Ptr [8].The elstc constnts of the other consdered orthotropc mterl oron-epoxy hs een ten s C ps (8.9 GP), C ps (6.6 GP), C ps (7.85 GP), C ps (7.79 GP), (Sh nd Chen [9]), nd those of orthotropc mterl eryllum re ten s C ps (6. GP), C ps (5.58 GP), C ps (77.4 GP), Ds nd Ptr [8]. For the frst nd second pr of mterls, the SIF t the tp x re descred through fg. nd 3, respectvely, for dfferent vlues of /, wheres the physcl qunttes t the tp x for oth the pr of mterls re depcted through fgs. 4 nd 5 for vrous /. The numercl vlues of crc energes for the two prs of mterls re shown through fgs. 6 nd 7 for dfferent vlues of /. It s seen from fg. tht s the length of the crc decreses, oth K I nd K decrese. II Sme nture s followed for the second pr of mterls, fg. 3, wth only dfference s tht the vlues of SIF chnge s t completely depends on mterl constnts. As the lengths of the crcs decrese, fgs. 4 nd 5,. e., crcs seprton dstnce ncreses, then KI decreses, K II ncreses under thermo-mechncl lodng for oth prs of mterls. Ths shows tht there s lest posslty of crc propgton t x, even when the tps of the crcs come very close to ech other. The decreses of Mode II stress ntensty fctor justfes tht s the dstnce etween two crcs decreses, the effect of ther propgton tendency n sldng mode wll e decresed.

9 THERMAL SCIENCE: Yer 8, Vol., No. B, pp Normlzed SIF Normlzed SIF Fgure. Plots of K I / / p nd K II / / p vs. / for the frst pr of mterls / / Fgure 3. Plots of K I / / p nd K II / / p vs. / for the second pr of mterls Normlzed SIF Normlzed SIF Fgure 4. Plots of K I / / p nd K II / / p vs. / for the frst pr of mterls / / Fgure 5. Plots of K I / / p nd K II / / p vs. / for the second pr of mterls Normlzed crc energy Normlzed crc energy Fgure 6. Plot of W/p vs. / for the frst pr of mterls / / Fgure 7. Plot of W/p vs. / for the second pr of mterls The nture of ehvor of crc energy for frst pr of mterls s sme s the second pr of mterls wth the dfference s tht n frst cse the nture of the decrese s very fst s compred to the grdully decrese of the second cse. In the numercl computton t s lso gven specl emphss to determne other physcl quntty crc energy, W, to determne the energy requred y the crc per unt ncrese n re. Fgures 6 nd 7 show tht the crc energy ncreses wth the ncrese of crc length. The ncrement of crc energy represents tht s crc dvnces then plstc zone sze ecomes lrge due to whch more energy wll e requred for the crc propgton fter ttnng ts crtcl vlue. It s seen from the fgs. -5 tht frst pr of mterls cn sustn more stress ntensty compred to second pr of mterls wthout frcture nd t s lso justfed from fgs. 6 nd 7 tht for the frst pr of mterls the crc energy s hgher compred to the second pr of mterls due to formton of lrge plstc zone t the crc tps wth ncrese of crc length.

10 43 THERMAL SCIENCE: Yer 8, Vol., No. B, pp Concluson In the present rtcle the uthors hve cheved four mportnt gols. The frst one s the nvestgton of pr of collner Grffth crcs t the nterfce of two orthotropc med under thermo mechncl lodng. Second one s fndng the nlytcl form of the stress ntensty fctors t the vcnty of the crc tps. Thrd one s the successful presentton of vrtons of the SIF wth crc seprton dstnce. Fourth one s the ncrese of crc energy due to ncrese of length of the crcs showng the posslty of the formton of lrge plstc zone t the vcnty of the crc tp. Acnowledgment The uthors of the rtcle express ther hertfelt thns to the revered revewers for ther vlule suggestons for the mprovement of the rtcle. The frst uthor cnowledges the fnncl support from the CSIR, New Delh, Ind, under the SRF scheme. Nomenclture, oundry ponts of the crc, [m] K I / / p, K I / / p normlzed SIF of Mode I type t x nd x, respectvely, [ ] K II / / p, K II / / p normlzed SIF of Mode II type t x nd x, respectvely, [ ] K () the rto of therml conductve coeffcents long the y- nd x-drectons, respectvely, [ ] T () temperture dstruton functon, [ o C] References u (), ν () dsplcement components, [m] W/p normlzed crc energy, [ ] Gree symols β () x, β () () xx, () ν stress temperture coeffcents, [GP o C ] νν stress tensors long the x nd y xs, respectvely, [GP] sher stress tensor, [GP] [] Sh, G. C., On the Sngulr Chrcter of Therml Stresses Ner Crc, ASME Journl of Appled Mechncs, 9 (96), 3, pp [] Sene, H., Therml Stresses Ner Tps of n Insulted Lne Crc n Sem-Infnte Medum under Unform Het Flow, Engneerng Frcture Mechncs, 9 (977),, pp [3] Sene, H., Thermoelstc Intercton etween Two Neghorng Crcs (n Jpnese), Trnscton of Jpn Socety of Mechncl Engneers, 45 (979), pp [4] Chen, B., Zhng, X., Thermoelstcty Prolem of n Orthotropc Plte wth Two Collner Crcs, Interntonl Journl of Frcture, 38 (988), 3, pp. 6-9 [5] Itou, S., Rengen, Q., Therml Stresses round Two Prllel Crcs n Two Bonded Dssmlr Elstc Hlf-Plnes, Archve of Appled Mechncs, 63 (993), 6, pp [6] Chen, B., Zhng, X., On Plne Thermoelstcty Prolem of n Orthotropc Strp wth Two Collner Crcs, Journl of Northwestern Polytechncl Unversty, (993),, pp. -6 [7] Itou, S., Therml Stresses round Two Prllel Crcs n n Infnte Orthotropc Plte under Unform Het Flow, Journl of Therml Stresses, 4 (), 7, pp [8] Bs, A., et l., Note on Thermo-Elstc Prolem of two Collner Grffth Crcs n n Orthotropc Medum, Int. J. of Pure nd Appled Mthemtcs, 36 (7), 4, pp [9] Zhong, X. C., et l., Thermlly Conductng Collner Crcs Engulfed y Thermo-Mechncl Feld n Mterl wth Orthotropy, Theoretcl nd Appled Frcture Mechncs, 65 (3),, pp [] Thur, P., Stedy Therml Stress nd Strn Rtes n Rottng Crculr Cylnder under Stedy-Stte Temperture, Therml Scence, 8 (4), Suppl., pp. S93-S6 [] Zhu, Z.-W., et l., Evluton of Therml Effects nd Strn Rte Senstvty n Frozen Sol, Therml Scence, 8 (4), 5, pp [] Thur, P., Stedy Therml Stress nd Strn Rtes n Crculr Cylnder wth Homogeneous Compresslty Sujected to Therml Lod, Therml Scence, 8 (4), Suppl., pp. S8-S9 [3] Slls, L. B., Dolev, O., The Conservtve M-Integrl for Therml-Elstc Prolems, Interntonl Journl of Frcture, 5 (4),, pp [4] Itou S., Therml Stress Intensty Fctors of n Infnte Orthotropc Lyer wth Crc, Interntonl Journl of Frcture, 3 (), 3, pp () xν

11 THERMAL SCIENCE: Yer 8, Vol., No. B, pp [5] Itou S., Therml Stresses round Crc n n Adhesve Lyer etween Two Dssmlr Elstc Hlf- Plnes, Journl of Therml Stresses, 6 (993), 4, pp [6] Clements, D. L., Tuchert, T. R., A Thermoelstc Crc Prolem for n Ansotropc Sl, J. Austrl. Mth. Soc. (Seres B), (979),, pp [7] Shrm, B., Therml Stresses n Trnsversely Isotropc Sem-Infnte Elstc Solds, J. Appl. Mech., (958), p. 86 [8] Ds, S., Ptr, B., Stress Intensty Fctors for Movng Interfcl Crc etween Bonded Dssmlr Fxed Orthotropc Lyers, Computers & Structures, 69 (998), 4, pp [9] Sh, G. C., Chen, E. P., Crcs n Composte Mterls, Mrtnus Njhoff Pulshers, Hgue, The Netherlnd, 98 Pper sumtted: Octoer 7, 5 Pper revsed: Mrch 6, 6 Pper ccepted: August 5, 6 7 Socety of Therml Engneers of Ser Pulshed y the Vnč Insttute of Nucler Scences, Belgrde, Ser. Ths s n open ccess rtcle dstruted under the CC BY-NC-ND 4. terms nd condtons