Theory of Spatial Problems

Transcription

1 Chpt 7 ho of Sptil Polms 7. Diffntil tions of iliim (-D) Z Y X Inol si nknon stss componnts:. 7-

2 7. Stt of Stss t Point t n sfc ith otd noml N th sfc componnts ltd to (dtmind ) th 6 stss componnts X N l m n Y N m n l > Z N n l m Bond condition: X N Y N Z N plcd th cosponding ond sfc foc componnts X Y Z h 6 stss componnts compltl dfin th stss stt t point. Pincipl Stsss nd Pincipl Diction Whn Mimm nd Minimm Stsss 7-

3 7. omticl tions displcmnt fo o stin igid-od displcmnts lt s tk h cosponding displcmnt componnts cn pssd s ω ω ω ω ω ω o o o th igid-od tnsltions nd ω ω ω th igid-od ottions ot th coodint s. Volm stin chng of olm p nit of th oiginl olm 7-

4 i.. V V ( )( )( ) 7.5 Phsicl tions fo n isotopic nd pfctl lstic od Hook s l [ ( )] ( ) [ ( )] [ ( )] ( ) ( ) if thn h i.. th coincidnc of Pincipl dictions of stss ith Pincipl dictions of stin (thml stss o thml stin contition). m m ( ) ( ) ( ) ( ) K m K lk modls (lts olm stin nd olm stss) o sm p th 5 nknon fnctions in sptil polm: stss(6) stin(6) displcmnt(). h shold stisf 5 tions: iliim() gomticl(6) phsicl(6). 7-4

5 displcmnt ond conditions on th sfc: 7.6 il smmt nd sphicl smmt. il smmt: if th gomticl shp of th od concnd th condition of constint nd th tnl lods ll smmticl ith spct to n pln pssing thogh ctin is thn th stss stin nd displcmnt componnts ill h th sm condition of smmt il smmt polm in spc. clindicl coodints th stss stin nd displcmnt componnts ill fnctions of onl to coodints: nd. <> stss componnts: θ. <> od focs K nd Z in nd dictions <> stin componnts: θ nd <4> displcmnts: θ. 7-5

6 iliim tions: Z K θ omticl tions θ Phsicl tions [ ] [ ] [ ]. θ θ θ θ Bond conditions <> displcmnt <> stss 7-6

7 Sphicl smmt (o point smmt): shp of th od th condition of constint nd tnl lod ll smmticl ith spct to n pln pssing thogh ctin point. h stss stin nd displcmnts ill fnctions of singl il th distnc fom th point of smmt. idntl sch condition ists onl in solid o hollo sphs. d <> tion of iliim: ( ) K d <> gomticl tions: d d <> phsicl tions: [ ( )] ( ) [ ( )] [( ) ] h cosponding stss componnts ( ) d d d. d 7-7

8 7.7 Soltion in tms of displcmnts In tms of displcmnt th diffntil tions nd to sol :. Z Y X h Bond conditions: Fo th spcil cs of ismmtic polms. Z K h. Fo th spcil cs of sphicl smmt onl on nknon fnction K d d d d 7-8

9 7.8 Infinit lstic L Und it nd Unifom Pss Consid n lstic l of infinit tnt nd nifom thicknss hich is fid t its lo sfc nd sjctd to nifom pss of intnsit on its pp sfc ith pln in th pp sfc. h od foc componnts X Y Zρg ρ is th dnsit of th l. W s th smi-ins mthod nd ssm tht: cs of smmt th displcmnt t n point in th l is ticl nd dpnds on onl: () hs h d d d d h diffntil tions fo displcmnts coms: ( ) d d d ρg d Fom hich h d d ( )( ) ( ) ρg 7-9

10 intgtion h B g g d d ρ ρ B it constnts. g g ρ ρ Stss ond condition: (- ) /(ρg) Displcmnt ond condition: h g h g B ρ ρ So finll h:. h g h g g ρ ρ ρ W cn s: m nd ls h th ltion - cofficint of ltl pss 7-

11 ht is th slts in pln stss condition? Pln stin? 7-

12 7.9 Hollo Sph Sjctd to Unifom Pss Consid hollo sph ith inn dis nd ot dis sjctd to nifom pss of intnsit nd on th inn nd ot sfcs spctil h od focs not considd th iliim tion coms d d d d fom hich h B stss componnts: B B ond conditions B hs:. 7-

13 If nd (onl ot pss ll) If (onl intnl pss) ht is th mimm tnsil stss? If th ond condition is gin inn nd ot displcmnt dtmin th stss fild. 7. Displcmnt Fnctions nd Displcmnt Potntil Intodc displcmnt fnctions to pss th displcmnt componnts Whn od focs nglctd th sic iliim tions in tms of displcmnts com No ssm tht th displcmnt might h potntil i.. th displcmnt componnt in n diction is popotionl to th diti of ctin potntil fnction ( ) ths fth h Sstitting nd o into iliim tions cn s tht mst stisf: 7-

14 his is tht C. C is n it constnt. hs n fnction stisfing o tion m tkn s displcmnt potntil fom hich displcmnt componnts otind. Whn tk C h thn coms hmonic fnction. In this cs h nd th stss componnts in tms of :. h displcmnt componnts nd stss componnts shold stisf ll th ond conditions. In th spcil cs of il smmt hn od focs nglctd th iliim tions com h Simill tk [ ] thn h 7-4

15 mst stisf: hich ild C gin. h cosponding stss componnts :. θ W not tht is constnt thoghot th od concnd so th ppliction is limitd sinc constnt l occ. Lo s Displcmnt Fnction Fo th soltion of ismmtic sptil polms.. H. Lo intodcd displcmnt fnction ζ nd pssd th displcmnt componnts s: ζ ζ h nd olm stin is ζ ith nd th fist tion of iliim is idnticll stisfid nd th scond tion is tht 4 ζi.. ζ mst ihmonic fnction. h cosponding stss componnts. ζ ζ ζ ζ θ 7-5

16 hs n ismmtic polm cn sold if sccd in finding pop ihmonic fnction ζ so tht th otind displcmnt componnts nd stss componnts stisf th ond condition. Concnttd Noml Lod on Bond of Smi-infinit Bod. Consid concnttd noml lod P on th ond pln. idntl this is n ismmtic polm. h ond conditions :. t n hoiontl pln sction t distnc fom th ond pln h dditionll ( πd) P. nd const. W t to s Lo s displcmnt fnction. ccoding to dimnsionl nlsis th pssion fo stss mst P diidd dtic tms of th lin ntitis nd. h displcmnt fnction ζ mst P mltiplid lin tms of nd. No W ssm ζ to podct of constnt (ith th dimnsion of foc) nd th ihmonic fnction : ζ ths h 7-6

17 θ 7-7

18 h ond condition of is stisfid t tht of is not sinc ( ) dos not nish fo ll ls of. hs t to find noth hmonic fnction s displcmnt potntil (not th Lo s fnction) hich ill gi nd cncl ot th o non-o sh stss on th ond. ft sl tils find tht th hmonic fnction ln( ) flfills th imnt tk ln( ) h ln() is non-dimnsionl fnction is n it constnt (ith dimnsion of foc) th cosponding displcmnt nd stss componnts : ( Z) θ ( ) ( ). ft spposition ith th fist soltion cn s tht is stisfid nd th condition coms: ( ) i.. ( ). (*) h condition ( πd) P coms: Fom (*) nd (**) otin 4π(-) π P (**) P ( ) P π π 7-8

19 h finl soltions ( J. Bonssins (878)) :. P ) ( ) ( ) ( ) ( 5 5 P P P P P π π π π π π θ 7-9