8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY ON THE PREFRACTURE ZONE MODEL IN ELASTIC BODY AT THE CRAC TIP ON THE INTERFACE OF MEDIA Anatoly AMINSY a Lonid IPNIS b Michal DUDI b and nady HAZIN b a SP Timoshnko Institut of Mchanics National Acadmy of Scincs Datmnt of Factu Mchanics yiv Ukain b Uman Pdagogical Univsity Physical and Mathmatical Datmnt Uman Ukain ABSTRACT Th lan symmtical oblm on calculation of th factu zon at th ti of a cack aching th intfac of isotoic lastic mdia is considd Th factu zon is modlld by lins of utu of nomal dislacmnt locatd on th intfac An xact solution of th cosonding static oblm of th thoy of lasticity fo ic-homognous lan with th mdia-saating bounday in th fom of th sids of angl which contains a smi-infinit cack and two staight lins of utu mging fom th con oint is constuctd by th Win-Hof mthod Th lngth of th factu zon is dtd on th bas of this solution Th stss na th con oint is invstigatd It is shown that in dfinit intvals of aamt vaiation th con oint is th singula oint of th abov oblm of th thoy of lasticity It snts th stss concntato itslf Th xonnt of singulaity of stsss dnds on th angl Young s modulus atio and on th Poisson s atios Dndnc of th xonnt of singulaity of stsss on th angl fo vaious intvals of Young s modulus atio vaiation has bn invstigatd Paamt vaiation intvals wh th con oint is not th stss concntato hav bn dtd y wods: Dislacmnt utu lin intfac aching cack factu zon thoy of lasticity Win- Hof mthod
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY INTRODUCTION Th is a lot of scific litatu aimd at lan oblms of factu zon dvlomnt na th cack ti in a ic-homognous body locatd on th intfac of two diffnt mdia [-5 Pfactu zons a modlld by dislacmnt utu lins mging fom th cack tis Considabl intst in connction with its ossibl usag whn th oblm of baking of th cack in comosit matials a solutions of such oblms in an oth cas of mutual osition of th intfac of mdia and th cack in that cas whn on that intfac th is only th cack ti th cack aching th intfac of mdia Fo isotoic lastolastic body lan symmtic oblms on th calculation of th factu zon at th cack ti aching th ough intfac of mdia in th modl fam with a dislacmnt utu lins hav bn solvd in [6 Blow is givn th solution of an analogical oblm fo th lastic body und th condition that factu zon is modlld by a nomal dislacmnt utu lins locatd on th intfac of mdia FORMULATION OF THE PROBLEM Lt a ic-homognous isotoic lastic body bing und th conditions of lan stain b comosd fom diffnt homognous ats connctd in btwn thmslvs by a thin conncting lay th matial of which is mo bittl than th matials of th givn ats Lt s consid that th staight cack achs th ough intfac of two mdia and its ti coincids with th con oint of that intfac Th gion und considation is considd to b symmtical in lation with th staight lin on which th cack is ositiond Pfactu zon aas and dvlos with th incas of th out loading na th cack ti W will study only th initial stag of its dvlomnt considing th out loading bing small nough Thn th siz of th factu zon will b considably small than th cack lngth and all th oth body sizs Taking into account th small otion of th factu zon and th gal bing to mak calculations w com to th lan static symmtical oblm of th thoy of lasticity fo th ic-homognous isotoic lan with th intfac of mdia in th fom of angl sids having a smi-infinit cack mging fom th vtx whn th factu zon is availabl Asymtotic is bing alizd at infinity It is solution of th analogical oblm without th factu zon bon th smallst at th intval -;[ oot of its chaactistic quation That oblm oblm was scutinizd in [6 Mntiond solution contains abitay constant that is considd to b givn It chaactizs th intnsity of th xtnal fild and must b dtd fom th solution of th xtnal oblm In accodanc with th hyothsis of localization th initial factu zons na th cack ti and oth con oints concntatos of stsss a in thmslvs thin lays of matial naow stis mging fom concntatos Following th hyothsis of localization and taking into considation otis of th binding matial w ll consid that th initial factu zon snts a ai of naow stis mging fom th cack ti and locatd on th intfac of mdia It s suosd that in th oblm of th thoy of lasticity fo th finit body which cosonds to th stag of dfomation ocss whn th factu zon has not aad yt oblm I on th intfac of mdia na th cack ti th nomal stss is sttching
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY condition T Limitation on th oblm aamt nabling th solving of th dfind condition is givn blow Du to th fact that th binding matial is lastic main dfomations in th factu zon a dvloing on th mchanism of nomal utu Consquntly th sti-zon will b modlld by th nomal dislacmnt utu lin on which th nomal stss is qual to th givn constant of th binding matial Thus th bounday conditions of th oblm of th thoy of lasticity which modl th ocss und considation a as follows Figu : l E E Figu Nomal dislacmnt utus lins at th ti of a cack aching th intfac of mdia ; - u ; < > < > < u >; <l ; >l< u >; g o/ 3 Wh ; a is th jum of valu a; C > a givn constants; is th smallst on th intval -;[ oot of quation z b z b z x z b 4 sin b z sin z zsin { 4[ sin z z b z sin z 4 sinz zsin { sin z zsin [sin 4[ b z 4 sin z z sin [sin z z E 3 4 E sin sin 3 z z z z sin } sin } E E a Poisson s atios; g is th known function Th valus of oot of th chaactistic quation 4 fo som valus of and und 3 a shown in th Tabl As shown in th calculation sults function g und fixd is ositiv if < < 5 > is th oint wh th function is zo Considing that satisfis th 5 Thn in th oblm I mntiond abov o if and th condition T ointd abov will b fulfilld Condition T is fulfilld und all if is lss than th smallst valu of th function In aticula
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY und 3 fo th fulfillmnt of th condition T it is sufficint that dos not xcd th Tabl Som valus of th oot of th chaactistic quation 3 5 3 5 5-4398 -4679-4793 -4868-583 -5338-56 -637 3-434 -4483-46 -474-5383 -5688-66 -69 45-45 -435-4476 -46-5588 -6-664 -744 6-489 -488-438 -456-574 -644-688 -7665 75-386 -397-473 -449-5795 -635-6957 -775 9-3467 -3757-466 -437-5737 -6-679 -7546 5-34 -389-45 -445-5594 -5969-6475 -79-479 -46-445 -4636-548 -567-645 -6647 35-46 -4679-4748 -484-57 -538-563 -599 5-489 -496-494 -4945-587 -553-554 -546 65-4987 -4988-499 -4993-53 -55-547 -589 Na th ti of th sid utu lin taking into account gnal things about th stss bhavio in th vicinity of con oints of lastic body th asymtotic is alizd It snts th solution of a homognous oblm of th thoy of lasticity fo ichomognous lan which contains on th staightlind intfac of mdia a smi-infinit nomal dislacmnt utu lin bon th oot / of its chaactistic quation In aticula ki l ~ 6 l Wh k I is th stss intnsity facto at th ti of th nomal dislacmnt utu lin which is to b dfind Th task is to dfin th lngth l of utu lins which modl th factu zon and to invstigat th stss na th ti O of th cack Th lngth l of th utu lin is dfind fom th condition of th stss limitation na its ti 3 SOLUTION OF WINER-HOPF EQUATION AND DEFININ THE STRESS INTENSITY FACTOR Th solution of th fomulatd oblm of th thoy of lasticity with bounday conditions -3 Figu snts th sum of solutions of th following two oblms Th fist oblm I diffs fom it in th following way: instad of th fist condition w hav 4
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY 5 Cg l < 7 and at infinity th stsss dcas as / / o o in 3 Th scond oblm is oblm Using Mllin s intgal tansfom with a comlx aamt to th quilibium quations comatibility quation Hook s law conditions and considing th scond condition and condition 7 w com to th following Win-Hof functional quation of oblm : tg A 8 [ [ A a a cos [ [ b b b sin [ [ sin [sin sin s in a sin [sin cos cos a 4 d u E Ô d l Ô Cgl l Wh - R ε ε <ε < a sufficintly small ositiv numbs Simsla quations a solvd in [5-7 Solution of quation 8 is as follows: R } [ [ { < Ô R [ > A Ô 9 R R ln x[ dz z z i i i m m Γ Γ > < ± z Γ is gamma function Using 69 w can find th stsss and stss intnsity facto Stss intnsity facto at th ti of th utu lin is sntd by th fomula
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY gγ k I l Cl 3 Γ 4 RESULTS Equalling k I to zo s w gt th following fomula aimd at dfining th lngth of th factu zon: C gγ I l L L [ Γ 3 I ln it ln it I x[ dt I x[ dt t t Du to th calculation sults if 7 45 3 function L IN quals dcasing L and if < and < 3 - has th only xtmum maximum at th oint m f is qual to ;;3;5;;4; ; 3 thn is qual to ;73 ;5 ;47 ;3 ; 8 m 7 ;9 4 and L m is qual to 34;564;677;785;9;66; 33; 6 Thus in cas whn C / / wakly changs with th changing and if < and < 3 th lngth of th factu zon will b th most if m n th abov discussd cas th lngth of th factu zon incass with a dcasing if t is known that at of th factu zon locatd at th cack ti snts th dstuction of matial gion wh th lvl of stsss is xtmly high Th givn gion diffs at its highst maximum lvl of dfomations sulting in availability of os and micocacks n that connctionaticula intst lis in th analysis of stss bhaviou na th con oint O iin th study of influnc of changing th oblm aamts on th xonnt of stss singulaity at th con oint With aim of alization th abov mntiond invstigation using 9 and Mllin s fomula w dfin th stsss in th oblm n aticula w gt τ F M i D γ M D a a M Γ M d [ m m Γ l gγ m Cl m Γ 3 6
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY Wh - < < ; F is th know nti function of having zo of fist od in th oint ; γ is th aalll to th imaginay axis lin that lis in th sti - ε < R < Function D has zos of fisty od at th oints and wh is th only on th intval -;[ oot of quation D x Function D has not oth zos in th sti < R < Thus in th sti < R < th intgand in has th singulaitis siml ols at th oints ; ; Using th data about th singula oints of th intgand in alying to th intgal th sidu thom and adding th solutions of oblms and w find th incial tms of th xansion of stss τ in asymtotic sis if in th oblm of th thoy of lasticity und considation with bounday conditions 3 Figu Th blow givn fomula taks its lac: τ f C f f l C 3 C ϕ l ϕ Cl f ϕ f ϕ a th known functions; f if ; < R < Und th valus of th oblm aamts wh th quation D x on th intval -;[ has not oot in 3 addndum containing is not availabl Th sult analysis divs us to th following conclusions n dfinit intvals of changing of th aamts th con oint O snts in itslf th singula oint of th considd oblm of th thoy of lasticity t itslf snts th stss concntato Th xonnt of singulaity of stsss ;[ dnds on th angl Young s modulus atio E / E and on th Po sson s at os f 75 function is dcasing / and thus incasing th angl th stss concntation in th gion of th dstuction of matial incass Lt 394 with th angl incas fom zo to max th stss concntation in th gion of dstuction of matial fist incass thn wakns n this th stss concntation will b maximum if With th dcas of th angl and dcas and angl max and max incas f thn max 4 Whn angl max aoximatly quals to 653 and max f / thn 47 ; 54 and max 553 ; 54 With th incas of angl fom to th stss max max concntation in th gion og matial dstuction incass / Suos < with th incas of angl fom zo to th stss concntation in th dstuction gion incas and with its incas fom to dscass 7
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY f < < th con oint O is not th concntato of stsss With th dcas of angl incass and tnds to / if and angl dcass and if tnds to th uniqu oot of quation cos sin aoximatly qual to 5 3 Angl and dcas with th dcas of bsids and / if f is qual to 3; ; ; ; thn is qual to 6 ;548 ;545 ;59 ;56 ; is qual to 69 ;844 ; 85 ; 885 ; 894 ; is qual to 8 6 ; 48 ; 4 ;3 ; 3 and is qual to 73;36; 36; 4589; 4938 with th incas of angl fom to th stss concntationin th dstuction gion of matial incass / 3 3 f < 37 thn with th angl incas th stss concntation in th dstuction gion incass / 3 4 4 Lt < 639 With th incas of angl concntation of ctsss in th dstuction gion fist incass and thn dcass f concntation of ctss will b th lagst With th incas of angl thn Suos 4 3 and dcass f > f < < th con oint O is not th concntato of 3 3 stsss With th incas of angl 3 incass and tnds to / 4 if With th incas of angl fom to concntation of stsss in th dstuction 3 gion of matial incass and with its incas fom to - dcass Angl and dcas with th incas of ; / and if To th valus of qual to and 3 cosond th valus 3 qul to 4 4 ; 96 ;57 and 43 ; 985 ; 5834 f 47 thn with th incas of < < concntation of stsss in th gion of matial dstuction dcass and if 49 - incass 5 CONCLUSIONS Thus th lan symmtical oblm on calculation of th factu zon at th ti of a cack aching th intfac of isotoic lastic mdia is invstigatd Th factu zon is modlld by lins of utu of nomal dislacmnt An xact solution of th cosonding oblm of th thoy of lasticity is constuctd It is shown that in dfinit intvals of aamt vaiation th con oint is th singula oint of this oblm It snts th stss concntato itslf Th tnd of th stsss to infinity cosonding to that at of th factu zon locatd na th cack ti is th gion of matial dstuction wh th stss lvl is xtmly high 8
8 Ulusla Aas lma onfans Bildiil itab 7 9 as m 7 Poocdings of 8th Intnational Factu Confnc 7 9 Novmb 7 Istanbul/TUREY REFERENCES Comninou M Th Intfas Cack Jounal of Alid Mchanics Vol44 4 63-636 977 Comninou M Schmus D Th Intfac Cack in a Combind Tnsion-Comssion and Sha Fild Jounal of Alid Mchanics Vol46 345-348 979 3 autsn A Dunds J Th Intfac Cack in a Tnsion Fild Jounal of Alid Mchanics Vol54 93-98 987 4 oldstin RV Plmut M Modling of Bonding at an Intfac Cack Intnational Jonal of Factu Vol99 53-79 999 5 am nsky AA Dudik MV n s LA On th Diction of Dvlomnt of a Thin Factu Pocss Zon at th Ti of an Intfacial Cack Btwn Dissimila Mdia Intnational Alid Mchanics Vol4 36-44 6 6 am nsky AA n s LA Dudik MV Initial Dvlomnt of th Pfactu Zon Na th Ti of a Cack Raching th Intfac Btwn Dissimila Mdia Intnational Alid Mchanics Vol4 76-8 4 7 am nsky AA n s LA hazin A On Cition of th Stat of Two Sha Cacks in an Elastic Body und th Plan Stain Intnational Alid Mchanics Vol4 4 8-87 6 9