INTERFACE CORNERS IN ANISOTROPIC/PIEZOELECTRIC/ VISCOELASTIC MATERIALS

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1 INERFAE ORNERS IN ANISOROPI/PIEZOELERI/ VISOELASI MAERIALS hyanbn Hwu a-lang uo Insttut of Aronautcs and Astronautcs Natonal hng ung Unvrsty anan AIWAN R.O.. Ansotropc matrals bhav dffrntly n dffrnt drctons. Pzolctrc matrals produc an lctrc fld whn dformd and undrgo dformaton whn subctd to an lctrc fld. Vscolastc matrals xhbt a tm and rat dpndnc that s compltly absnt n th lastc matrals. Du to th dffrnc of th consttutv rlatons for ths thr dffrnt nds of matrals thr mchancal analyss ar usually mad by dffrnt approachs. Among svral dffrnt approachs thr conncton can b mad by th ways such as th xpanson of th lastc proprts to nclud th pzolctrc ffcts and th us of th Laplac transform to ma th vscolastc strss-stran rlatons loo l lnar lastc matrals. h ntrfac cornrs nclud a crac or cornr n homognous matrals as wll as an ntrfac crac or cornr btwn two dssmlar matrals. Intrfac cornrs appar commonly wthn ngnrng obcts and lots of falur may ntat from ths crtcal rgons du to th dscontnuts of gomtry and matral proprts. Ordr of strss sngularty and strss ntnsty factor ar two mportant paramtrs usually mployd to valuat th dangrous dgr and th mod of fractur wthn th catgory of lnar lastc fractur mchancs. In ordr to buld a drct conncton btwn svral smlar but dffrnt problms n ths papr a unfd approach s proposd to dal wth th ntrfac cornrs n ansotropc lastc pzolctrc and/or vscolastc matrals. h unfd approach for svral dffrnt nds of matrals s mad by xtndng th Stroh s complx varabl formalsm for ansotropc lastcty to pzolctrc matrals and vscolastc matrals through propr corrspondnc prncpls. By ths approach th sngular ordrs and strss ntnsty factors prsntd n ths papr wll all prsrv th sam matrx form xprssons for dffrnt nds of cracs/cornrs n dffrnt nds of matrals and th only dffrnc s th contnts and dmnsons of th matrcs. ywords: ntrfac cornrs ordrs of strss sngularty strss ntnsty factors ansotropc lastcty pzolctrc lastcty vscolastcty Stroh formalsm. Introducton In ngnrng applcatons ansotropc lastc matrals play vry mportant rols such as th fbrrnforcd lamna usd n compost matrals and th sngl-crystal fans usd n th gas turbn of powr plant or powr vhcl. Pzolctrc matrals ar usually adoptd as th consttunts of snsors and actuators n advancd control systm du to thr lctrcmchancal couplng ffcts whl vscolastc matrals ar vastly mployd as th undrfll and moldng compound to protct th 4 bump or chp wthn th lctronc pacag from th harmful nvronmnt. hs thr dffrnt nds of matrals ar also appld to many othr trrtors such as mdcal ngnrng aronautcal and astronautcal ndustrs and so on. Du to th dffrnc of th consttutv rlatons for ansotropc lastc pzolctrc and vscolastc matrals th two mportant paramtrs ordr of strss sngularty and strss ntnsty factors ar usually calculatd by dffrnt approachs. Nvrthlss by mans of th bauty of Stroh formalsm [] th gnral solutons satsfyng th basc quatons of ansotropc/pzolctrc/vscolastc matrals can b arrangd nto a sam matrx form xprsson and ths hlps us succssfully buld a unfd soluton tchnqu for th calculaton of ths two mportant paramtrs by xtndng th authors prvous and rcnt studs [3-7]. A numrcal xampl dscussng a cntr ntrfac crac mbddd btwn two dssmlar ansotropc/pzolctrc/vscolastc matrals wll b analyzd to valdat th proposd approach. Profssor chwu@mal.ncu.du.tw Ph. D. Studnt p4895@mal.ncu.du.tw

2 Stroh Formalsm In a fxd rctangular coordnat systm x 3 lt u D and E b rspctvly th dsplacmnt strss stran lctrc dsplacmnt (or calld nducton and lctrc fld. h consttutv laws stran-dsplacmnt quatons and th qulbrum quatons for ansotropc pzolctrc and vscolastc matrals can b wrttn as follows. Ansotropc matrals: ll ( u u l 3. Pzolctrc matrals: E ll E ( u u D ll E l 3. D Vscolastc matrals: t l ( ) ( l t d u u l 3. In th abov t s th tm varabl rpatd ndcs mply summaton and a comma stands for dffrntaton. l and ar rspctvly th lastc stffnss tnsor pzolctrc strss tnsor and dlctrc prmttvty tnsor. h suprscrpts E and dnot rspctvly th valus at constant lctrc fld and constant stran. hs tnsors hav th followng symmtry proprts () () (3). (4) l l l For pzolctrc matrals lt D 4 E u4 4 3 E l l l 3 4l l 4 l l l 3 l 3 l 3 44l l (5) th basc quatons () can b rwrttn n an xpandd tnsor notaton as For vscolastc matrals f th boundary of a body s nvarant wth tm th Laplac transformaton of Eq. (3) gvs sll ( u u whr th notaton brv dnots th Laplac transform.g. f () s s th Laplac transform of f () t n whch s s th transform varabl. By xpandng th rang of th ndcs l from 3 to 4 for pzolctrc matrals and rplacng th lastc tnsor l by s l for vscolastc matrals t can b obsrvd that th transformd basc quatons (6) and (7) for pzolctrc and vscolastc matrals can b mad to b dntcal to thos of ansotropc matrals shown n Eq. (). Bcaus th basc quatons ( (6) and (7) ar quvalnt th gnral solutons satsfyng ths quatons can b organzd nto th sam matrx form. hus th gnral solutons vald for ansotropc matrals may now b applcabl for th pzolctrc and vscolastc matrals. Wth ths undrstandng th gnral solutons gvn n th Stroh formalsm for twodmnsonal lnar ansotropc lastcty [] can now b appld to th ansotropc pzolctrc and vscolastc matrals [36-8] z z (7) u R Af( ) R Bf( ) (8) whr R stands for th ral part of a complx numbr u and ar rspctvly th dsplacmnt vctor and strss functon vctor; f ( z) s a vctor contanng holomorphc functons of complx varabls z ( x x ) ; A and B ar th matral gnvctor matrcs assocatd wth th matral gnvalus. h rang of th subscrpt s to 3 for ansotropc and vscolastc matrals and s to 4 for pzolctrc matrals and hnc th dmnson of th assocatd vctors and matrcs ar as follows: u f :3 A B:33 for ansotropc and vscolastc matrals; u f :4 A B:44 for pzolctrc matrals. (9) h strsss and lctrc dsplacmnts ar rlatd to th strss functon by IJ IJLL IJ ( ui J uj I I JL 34 IJ J whr th xpandd lastc stffnss tnsor posssss th fully symmtrc proprts. (6) IJL 3 and D D from whch t has bn provd that () t s ()

3 whr t s th gnralzd surfac tracton vctor and s s th arc lngth masurd along a curvd boundary of th body. Not that for vscolastc matrals u and n Eqs. (8) and () ar th dsplacmnts strss functons and strsss of Laplac doman. o gt thr valus n tm doman th Laplac nvrson should b prformd. x Mat. r x Mat. 3 Intrfac ornrs onsdr an ntrfac cornr btwn two dssmlar matrals (Fg. ). Assum prfct bond along th ntrfac. h tracton and dsplacmnt contnuty across th ntrfac can b wrttn as u () u ( () ( () whr th subscrpts and dnot th valus rlatd to matrals and and th argumnt dnots th valus on th ntrfac whos. h cornr flans can b fr-fr fxd-fxd fr-fxd or fxd-fr. Hr fr mans mchancally fr and lctrcally opn (or lctrcally mprmabl and fxd mans mchancally clampd and lctrcally closd. Snc th mathmatcal form of th gnral solutons (8) and th boundary condtons () ar xactly th sam for ansotropc pzolctrc and vscolastc matrals th solutons obtand for th ntrfac cornrs n ansotropc matrals [3] can now b xtndd to svral dffrnt nds of combnatons wth ansotropc pzolctrc and/or vscolastc matrals. Followngs ar th nar tp solutons obtand n [378]. R u r V( ) (R ) ( r/ ) Λ ( ) ( ) ( / ) R σ r Ω( ) Λ( ) ( r/ ) Λ R r Λ R r Λ (3) n whch ( r ) s th polar coordnat wth orgn locatd on th cornr tp (Fg. ); th angular bract <> dnots a dagonal matrx n whch ach componnt s vard accordng to th subscrpt ; R and ar rspctvly th ral and magnary part of th most crtcal sngular ordr c whch s locatd n th rang of R and somtms s also calld oscllatory ndx; s a lngth paramtr whch may b chosn arbtrarly as long as t s hld fxd whn spcmns of a gvn matral par ar compard; V ( ) and Λ ( ) ar gnfuncton matrcs of dsplacmnt and strss functon and Λ Λ () whos dtald xprssons can b found n [37]. σ and Ω ( ) dnot rspctvly th strss vctor and transformaton matrx and s a vctor contanng dffrnt mods of strss ntnsty factors. Fgur : An ntrfac cornr h ordrs of strss sngularty can b dtrmnd by th followng gn-rlaton [79] whch s vald for cracs ntrfac cracs cornrs and ntrfac cornrs and th matrals can b any nds of lnar lastc ansotropc matrals or pzolctrc matrals or vscolastc matrals. bondd: I (3) () fr-fr: fxd-fxd: () (4) fr-fxd: fxd-fr: (4) whr () 34 ar th submatrcs of dfnd by whr () () (3) (4) EE ˆ (. (5a) E N (5b) ar th angular locatons of th two sds of th th matral and ˆN s th y matrx dfnd by N ˆ ( ) cos( ) Isn( ) N ( (6) whr I s a unt matrx and N ( ) s th gnralzd fundamntal lastcty matrx of N []. h R appard n th powr of y matrx ˆN n Eq. (5a) s th sngular ordr to b dtrmnd whch may b postv or ngatv or zro ral or complx rpatd or dstnct. Not that th gnralzd fundamntal lastcty matrx N( ) s a 6 6 matrx for ansotropc or vscolastc matrals and s an 8 8 matrx for pzolctrc matrals. o provd a stabl and ffcnt computng approach for th gnral mxd-mod strss ntnsty factors th path-ndpndnt H-ntgral basd on th rcprocal thorm of Btt and Raylgh was stablshd n Rf. [3] for D problms n Rf. [5] for 3D problms and n Rf. [] for thrmolastc problms..

4 D: 3D: ( ˆ H utut ˆ )d ( ˆ H ut ut ˆ )d (7a) ( ˆ u ˆ u uˆ uˆ )d S S (7b) thrmal: H ( ˆ utut ˆ )d ( ˆ ˆ )d S (7c) S n whch u and t ar th dsplacmnt and tracton vctors of th actual systm that can b obtand from any approprat mthod such as fnt lmnt or boundary lmnt or xprmntal tstng and û and ˆt ar thos of th complmntary systm whch s th nar tp soluton wth sngular ordr rplacd by ; th path manats from th lowr cornr flan and trmnats on th uppr cornr flan n countrclocws drcton. For pzolctrc matrals th fourth componnt of tracton vctor t s surfac lctrc dsplacmnt and th fourth componnt of dsplacmnt vctor u s th lctrc potntal. For vscolastc matrals u and t ar th quantts of Laplac doman and so s H. By usng th nar tp solutons (3 t has bn provd that th strss ntnsty factor s rlatd to th path-ndpndnt H-ntgral by whr Λ ( R ) H h (8a) * ˆ ˆ H [ Λ ( ) V( ) V ( ) Λ ( )]d (8b) and h s a vctor consstng th valu of H-ntgral calculatd wth crtan spcfd complmntary solutons. For vscolastc matrals th valus of strss ntnsty factors s now a functon of th transform varabl s and should b nvrsly Laplac transformd nto th valu of tm doman. 4 Numrcal Exampls Bcaus th ntrfac crac s a spcal cas of th ntrfac cornr an ntrfac crac mbddd n an nfnt plat mad up of two dssmlar ansotropc/pzolctrc/vscolastc matrals wll b dscussd to valdat th proposd soluton tchnqu. hs nfnt plat s smulatd by lttng ts wdth and hght b 3 tms of th crac lngth and th half crac lngth s.m. A unform rmot tnson MPa prpndcular to th crac s appld on th two horzontal dgs of ths plat. All xampls ar undr gnralzd plan stran assumpton. 4. Ansotropc Matrals o s th pzo ffcts th ansotropc matrals ar smulatd by xcludng th pzolctrc constants of PZ-7A and PZ-5H whos matral proprts ar lstd n abl. h sngular ordrs calculatd from Eq. (4) and th strss ntnsty factors calculatd from th H-ntgral.. (7a) and (8 ar obtand as follows. c =.77 MPa m =.3 MPa m I whch agr vry wll wth th analytcal solutons provdd n th ltratur []. 4. Pzolctrc Matrals Exampl 4. s rsolvd hr by consdrng th pzolctrc constants of PZ-7A and PZ-5H. In addton to th rmot tnson a unform lctrc flux dnsty D. / m and an lctrc constrant u4 ar rspctvly spcfd on th bottom dg and top dg of th plat. By usng th sam formula and soluton tchnqu as xampl 4. th sngular ordrs and strss/lctrc ntnsty factors of ths problm ar calculatd as follows. c I.77 MPa m II.6 MPa m 3.77 (/m ) m IV whch agr vry wll wth th analytcal solutons provdd n th ltratur [7]. h rsults of xampls 4. and 4. show that th pzolctrc constants nflunc mostly on th apparanc of th lctrc ntnsty factor and hav nsgnfcant ffct on th magntud of sngular ordrs and ntnsty factors of strsss. 4.3 Vscolastc Matrals o mphasz that th proposd soluton tchnqu also appls to vscolastc matrals th rlaxaton modul modl usd for th two matrals ar assumd as t / () t ( ). for mat..5 for mat. n whch ~6 ar th sam as th n abl ; th Posson s ratos ar assumd to b constant; s th rlaxaton constant usd to normalz th opraton tm t. h sngular ordr calculatd from Eq. (4) has a constant ral part.5 and a tm-varant magnary part () t that s plottd n Fg. n whch our numrcal rsult agrs xcllntly wth th analytcal soluton provdd n th Rf. []. Not that ths analytcal soluton s drvd for ansotropc II

5 matrals and transformd nto th soluton for vscolastc matrals by usng th corrspondnc prncpl btwn lnar lastcty and lnar vscolastcty [3]. In Fg. th oscllatory ndx at th ntal stag s.8 that s consstnt wth th rsult of xampl 4.. h rsults for th tm-varant strss ntnsty factors ar undr worng and not shown n ths papr. (t) abl : Matral constants of PZ-5H and PZ-7A PZ-5H PZ-7A 33 [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [/m ] [/m ] [/m ] [/m ] [ -9 /(V m)] [ -9 /(V m)) t/ Analytcal soluton [] Prsntd rsults Fgur : Oscllatory ndx of xampl oncludng Rmars A unfd approach s proposd to dal wth th ntrfac cornrs n ansotropc lastc pzolctrc and/or vscolastc matrals. hrough th valdaton n numrcal xampls w s that th prsnt approach ndd offrs an accurat ffcnt and systmatc soluton tchnqu for fractur analyss no matr what th cornr gomtrs and th consttunt matral modls ar ncountrd. Acnowldgmnts h authors would l to than Natonal Scnc ouncl AIWAN R.O.. for support through Grants NS 97--E-6-36-MY3 and NS 98--E- 6- -MY3. Rfrncs [] Stroh A.N. Dslocatons and racs n Ansotropc Elastcty Phlosophcal Magazn Vol. 3 p [] ng... Ansotropc Elastcty: hory and Applcatons Oxford Scnc Publcatons Nw Yor 996. [3] Hwu. and uo.l. A Unfd Dfnton for Strss Intnsty Factors of Intrfac ornrs and racs Intrnatonal Journal of Solds Structurs Vol. 44 p [4] Hwu uo.l. and hn Y.. Intrfacs Btwn two Dssmlar Elastc Matrals M: omputrs Matrals and ontnua Vol. p [5] uo. L. and Hwu. Mult-ordr Strss Intnsty Factors Along hr-dmnsonal Intrfac ornrs Journal of Appld Mchancs Vol. 77 (3 (March 5 publshd onln). [6] Hwu. and uo.l. Intrfac ornrs n Pzolctrc Matrals accptd for publcaton n Acta Mchanca. [7] Hwu. and Ida. Elctromchancal Fractur Analyss for ornrs and racs n Pzolctrc Matrals. Intrnatonal Journal of Solds Structurs Vol. 45 p [8] uo.l. and Hwu. Intrfac ornrs n Vscolastc Matrals prpard for publcaton. [9] Hwu. Omya M. and shmoto. A y Matrx N for th Strss Sngularty of th Ansotropc Elastc ompost Wdgs JSME Intrnatonal Journal Srs A Vol. 46 p [] Hwu. and L W.J. hrmal Effct on th Sngular Bhavor of Mult-bondd Ansotropc Wdgs Journal of hrmal Strsss Vol. 7 p. 4. [] Hwu. uo.l. and Huang.. Pathndpndnt H-ntgral for ntrfac ornrs undr hrmal Loadngs submttd for publcaton. [] Hwu. Explct Solutons for th ollnar Intrfac racs Problms. Intrnatonal Journal of Solds Structurs Vol. 3 p [3] hrstnsn R.M. hory of Vscolastcty nd Ed. Acadmc Prss Nw Yor 98.