1. Preliminaries and terminology

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1 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Prelmres ermology Le egrl s geomercl meg Fg he plr cure s escrbe by he fuco y g(), or by s erse g(y) I I I y f (, y) s ee Fg f, y) b ( f (, g( )) Proeco o (z) ple f, y) y ( f ( g( y), y) y Proeco o (yz) ple c Emple how relos bewee he boe egrls Fg

2 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Le he fuco f (, y) s efe s follows z f (, y) by c From he boury coos we ge A : c B : b c c ; b ; C : c Proeco I o ( ), z : I f (, y ) I I see Fg F (, y) f (, g( )), I b f, y) f (, g( )) I ( Emple Clcule he egrls for he followg fucos f (, y) ( y ), g ( ), F( ) Fg Gre Coser wo-mesol uy Φ cosug sclr fel epeg o Φ Φ (, y), y

3 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc he ol fferel shows how Φ ffers wh poso Φ Φ Φ y y Le's rouce colum ecors Φ { } Φ Φ, { } y y he former ecor s clle he gre of sclr fel Φ (, y) he, he ol fferel c be rewre he form of sclr prouc { } { } Φ Φ, for whch we c wre { Φ} { } cosθ Φ he gre s orhogol o coour les sce f Φ cos, he Φ { } { } cosθ Φ oly f cos Θ, whch we he for π Θ Fg 4 Guss ergece heorem For fucos Φ Φ(, y ), Ψ Ψ (, y ) efe oer he re A, coser he egrls A Ψ (,y) A, A Φ y (,y) A e he seco oe frs he couerclocwse reco log boh boures

$elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Φ y Φ y A y A A b f ( ) Φ y f ( ) y [ Φ (,y) ] f ( ) f( ) b b [ (, f ( ) ) (, f$ s efe by wo fucos f f respecely mlrly for he oher coore for he oher fuco Ψ (, y) A Ψ (, y) A Ψ c (, y) Ψ we ge y () Ag he

s efe by wo fucos f f respecely mlrly for he oher coore for he oher fuco Ψ (, y) A Ψ (, y) A Ψ c (, y) Ψ we ge y () Ag he

4 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Φ y Φ y A y A A b f ( ) Φ y f ( ) y [ Φ (,y) ] f ( ) f( ) b b [ (, f ( ) ) (, f ( ) )] Φ b Φ b Φ Φ (, f ( ) ) (, f ( ) ) (, f ( ) ) (, f ( ) ) b b Φ Fg 5 We c coclue h A Φ y (, y) A Φ c (, y) () Le he cure c s efe by wo fucos f f respecely mlrly for he oher coore for he oher fuco Ψ (, y) A Ψ (, y) A Ψ c (, y) Ψ we ge y () Ag he couerclocwse reco he euos (), () form he Gree heorem he smples cse omemes lso clle he Gree-Osrogrsy heorem or he heorem of Gree-Guss 4

5 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Fg 6 Le us rewre he euos (), () by roucg ecors c r r, such such wy h r, r y y r c y, y c r y f or y y Obserg rgles he preous fgure oe c wre y sθ r r (ue o couerclocwse oreo of he cure), c y cosθ r r, where r c From follows r c, y r c () y ubsug () o (), (), we ge A A Ψ Φ y (, y) (, y) A Ψ A c c Φ (, y) (, y) Now efe rbrry ecor by {} c, (4) y c (5) Ψ (6) y Φ Ag (4), (5) usg (6) we ge 5

6 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 6 ( ) c y y A y c A y he ergece s efe {} (7) he sclr prouc c be epresse {}{} o we c flly wre {} {}{ } A c c A (8) hs s so clle ergece heorem of Guss he Guss-Gree heorem c be see s wo-mesol couerpr of he egro by prs u u u / / (9) he Guss ergece heorem coul be fou lerure ffere forms {} {} {} () he eule oos re s follows { }{} {}{}, { } (),, () f f he Guss ergece heorem for esor uy s efe s follows () o he Gree heorem represes he rsformo of olume egrl o surfce egrl (or ce ers) for ues ssoce wh cosere boy hg he olume, boue by he surfce he ouwr orml s efe ech po of he surfce he

$elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc fuco pperg uer he egrl sg s rel lue wh he frs couous ere wh he boy 4 he geerlzo of per pres egro (egro by prs) Accorg o Gree ergece heorem we c wre ($

7 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc fuco pperg uer he egrl sg s rel lue wh he frs couous ere wh he boy 4 he geerlzo of per pres egro (egro by prs) Accorg o Gree ergece heorem we c wre ( u) ( u) (4) he lef h se coul be rewre ( u) u u (5) Eullg he rgh h ses of he ls wo euos rerrgg ges formul u, (6) u ( u) whch rems he egro by prs b b u [ u ] u b (7) I s of eres o rem he oes heorem whch rsforms he egrl oer he close cure spce o he surfce egrl ( ) c, (8) curl ro y z c y z Fg 7 7

$elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 5 Flu Imge surfce spce he couum flowg by he elocy r$ s][ m ] m/s m (9),[ g / s] () Kec eergy flu correspos o mesol chec ( ) m, so Fg 8, () m s Geerlly g

s][ m ] m/s m (9),[ g / s] () Kec eergy flu correspos o mesol chec ( ) m, so Fg 8, () m s Geerlly g

8 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 5 Flu Imge surfce spce he couum flowg by he elocy r hrough he olume of merl flowg hrough, [ ] he olume flu s efe, [ /s] mlrly he mss flu s me s m [ ][ ][ s][ m ] m/s m (9),[ g / s] () Kec eergy flu correspos o mesol chec ( ) m, so Fg 8, () m s Geerlly g gm s s gm m s s Nm s s he flu of Φ hrough s Φ, () where Φ s uy (efe per u mss) whch s ssoce wh prcles 8

9 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Remr Ofe we ecouer so clle oree surfce efe by r r r Α or A () 6 Merl ere Imge h he moo of prcle s efe by he merl escrpo he form (,) (4) he prcle elocy coul smply be clcule by g he prl me ere wh rble hel cos (,) (,) (5) mlrly, he ccelero s z (,) (,) & (6) hese re emples of merl me eres merl escrpo Noce he ffere ypes of oo he merl me ere my be hough of s me re of chge h woul be msure by obserer rellg wh he specfc prcle uer suy he sme physcl pheomeo coul be escrbe by he spl escrpo he elocy fels s (,) (7) Noe he sue pheomeo s suppose o be he sme regrless of he formulo beg pple, so we re empe o use he sme symbol for he rble escrbg, ee f s efe by ffere fuco here re uhors usg ffere symbols for he sme rbles merl spl escrpos respecely he ere, wh spl coore hel cos (,) (8) s clle he locl re of chge of I s he re of chge of el elocy meer loce he fe plce hs s o he sme hg s he ccelero of he prcle pssg he plce us ow 9

10 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Remr For emple, sey se flow he locl re of chge s zero eerywhere hs oes o mply, howeer, h he ccelero of ll prcles s zero eerywhere Ee sey se flow he elocy res geerl from po o po prcle chges s elocy s moes from oe plce of cos elocy o oher plce, hg ffere cos elocy If we w o clcule he prcle ccelero from owlege of spl elocy escrpo ( ), we he o employ he ch rule of clculus (9) ce we c flly wre z () Aoher me for he merl ere s he subsl ere here re oher oos use eboos refereces, s & () I ecor oo we c wre z gr, () where gr z y For y sclr Φ, ecor u or esor ues, he formuls of merl eres re s follows f Φ Φ Φ Φ Φ &, {} u u u u u u gr &, u u u u &, &

11 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Cosero lws Cosero of mss Assume h couum wh esy flls he olume, boue by surfce he ol mss coe s m () I s ssume h he esy (,) s couous fuco of spce me coores h here s o flu hrough he surfce he mss of he cosere boy cofguro C s eul o h cofguro C, e, () (, ) (, ) Relzg h (,) () h se of () he form subsug fer he egrl sg llows rewrg he rgh (, ) ( (, ) J, (4) where he Jcob of he rsformo s he eerm of he eformo gre J e, F F (5) Euos () (4), wre shor, ge J (6) ce he ls euo mus be l for rbrry olume we c wre J (7) Bu J J, sce J > Proof he couum compleely flls he spce he l esy s > A he l cofguro C, here s o eformo, so F I coseuely J whch s greer h zero he lue of J < he process of eformo woul me h cer me (,) he lue of he Jcob woul become J For such cse here woul be o oeo-oe correspoece (,) (,) (8)

12 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc whch s corco wh l ssumpo bou he physcl ccepbly of eformo escrpo he coseuece of he cosero of mss s ow s he couy euo Lgrg (merl) form of he couy euo c be wre ffere forms J cos, where J e F, or ( J ) or F, (9), (9b) J cos, (9c) or J e F (9) Remr Remember h he coo J s ecessry for he eulece of merl spl escrpos (,) (,) If he Jcob of he boe rsformos J e F he he erse fuco of (,) oes o es Also, f e F, he, eher, or Remr Remember h F, F If e F, he F co be compue Euler (spl) form of he couy euo Ag he ol mss of couous meum of esy fllg he olume me s M he me re of crese of he ol mss he olume s M

13 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc We ssume h o mss s cree se he olume he he me re of mss mus be eul o he re of flu of he mss hrough he surfce flu re of mss ouflow re of mss flow ( ) o ( ), or { } { }, where { } From follows ( ) hs euo mus be l for y olume, so ( ) or { } { } or ( ), y z, () here re ffere forms of couy euo spl escrpo he ls euo coul be rewre usg he rule for he ere of prouc Relzg h he merl ere of esy s he euo () coul be smplfe o () () he eule formuls re, (b)

14 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc y y z, (c) z () Remr he elocy gre s L ( L L ) From follows h L, he sr re s s symmercl pr Remr If he merl s compressble, he cos y prcle so he compressbly coo s 4 Cosero of ler momeum For prcle of mss m we sy h he re of chge of ler momeum s eul o he resul force pple o prcle ( m) F () he ly of hs prcple s posule couum mechcs Couum form Assume h me ge mou of mss s olume, boue by surfce eoe b [N/g] - boy force (per u mss) [N/m ] - surfce rco, efe per u surfce Bse upo Newo's seco lw he re of chge momeum of ge mou of mss s 4

15 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc g m m m s gm N s or b b, (4) N m m m g N m g (4b) he relo bewee he sress ecor sress compoes s ge by so clle Cuchy relo, where re he sress ecor Cuchy (rue) sress esor compoes respecely he symbol ss for he compoe of orml ubsug he Cuchy relo o he surfce egrl (4b) usg he ergece heorem ges (5) he former euly s ue o Cuchy relo whle he ler s ue o ergece heorem of Guss 5 Ierlue Wh s he merl me ere of olume egrl (4b)? I c be proe h (6) Proof lerure s bse o Reyol's rspor heorem Guss ergece heorem couy euo efo of merl ere of Usg (5), (6) (4b) we he b (7) From he coo h mus hol for rbrrly chose olume we ge b (8) whch s clle he Cuchy euo of moo 5

16 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 6 Cuchy euo of moo oher form s b && (8b) If we rouce boy forces per u olume g N [ ] N/m f b (9) m g we ob sll oher form of Cuchy relo of moo he form f && (8c) Remrs Cuchy euos of moo represe PE for 6 uow compoes of sress Noce h s symmerc hese euos re wre for ge spl om, for colleco of cosere merl prcles - fllg olume, boue by surfce, cosere cofguro C eres re wh respec o spl coores I specl cses he ccelero coul be eglece euos (8) reuce o he euos of eulbrum f () hese euos o o co y emc rbles hey o o geerlly suffce o eerme he sress srbuo sce hey re oly hree prl fferel euos for s epee uow sress compoes Aol euos mus be cosere, e ) splceme s sr relos - emc relos b) sress s, sr relos - cosue euo 7 Euo of moo he referece se I ws lrey meoe h he Cuchy euos of moo pply o he curre eforme cofguro C he euos of moo coul be rsforme o referel cofguro C by mes of he frs seco Pol-Krchhoff sress esors 6

17 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc I shoul be reme h Pol-Krchhoff esors re useful sress mesures whch reclcule he cul sress C cofguro C I ws lrey show h s P-K: τ P-K: F F he erse relos re for s P-K: o he referece se, e o he o-eforme or τ Fr () r F or Fs sr Fr () F τ or Fr τ r () for P-K: F F or Fs sr Fr, (4) where F, F (5) o he euo (5) b &, where coul be rsforme o he referel cofguro followgly b where b b (,) usg (,) τ &, (6) Noce h for he rgh-h se we coul wre sce J (see 9c)), where J s he Jcob of he rsformo (,) & merl ere of & merl ere of Usg he ergece heorem τ τ epresse spl coores (,) epresse merl coores (,) 7

18 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc relzg h e (6) mus hol for y olume we flly ge τ b && (7) hs s he Cuchy euo of moo epresse referel coores by mes of he frs Pol-Krchhoff sr esor Usg he relo bewee he frs Pol-Krchhoff he seco Pol-Krchhoff τ F or τ r F (8) r we coul wre he Cuchy euo of moo by mes of he seco Pol-Krchhoff sress esor he form ( ) rfr b && (9) 8 Cosero of gulr momeum he lw of cosero of gulr momeum for prcle of he mss m s ( m( r ) r F, where For couum we coul smlrly wre ( r ) ( r ) ( r b) r () () For rewrg o cl oo we he o relze h he eule of ecor prouc c b s c e b Noe, where e s he C-Le permuo symbol e for ee permuo of ces, e:,,,,,,, e for o permuo of ces, e:,,,,,,, e for repeg ces:,, ec o, erm m erm m erm m b () ubsug he Cuchy relo, usg he ergece heorem for he surfce egrl he cocluso (6) cocerg he merl me ere of olume egrl, e 8

19 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 9 we coul rewre e () o he form ( ) ( ) b e e m m rm m rm Relzg h m m, m m δ, ( ) m m m m m δ, we he m m rm m m rm b e e, euo of moo, see e (8) lso m rm e, sce rm e s sew-symmerc m, Emple of ouble prouc eluo shows he rc : e e e e e e e e e e m m : A smlrly for oher ces o wh rems of e () s m rm e for rbrry olume () m rm e oly f m s symmerc, e m m (4)

20 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc hs esblshes he symmery of sress mr whou y ssumpos of eulbrum or of uformy of he sress srbuo he symmery of he sress mr s so clle Cuchy's seco lw of moo 9 Cosero of eergy If mechcl ues oly re cosere he prcple of cosero of eergy for he couum my be ere recly from he euo of moo Power pu Assume frs h oly eerl surfce rco per u re boy forces b per u mss re og wor o he mss seously occupyg olume, boue by he power pu s where Ppu b, (5) re compoes of he elocy fel As before we epress he compoes of rcos by mes of sress compoes use he Guss ergece heorem for he rsformo of he surfce egrl o olume egrl ge ( ) (6) Relzg h he elocy gre f efe by L we ob (7) bu P pu b L b by (8) (8)

21 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc he frs erm of e (8) o he rgh h se coul be rewre s (9) Remember merl ere Kec eergy of olume egrl of he sysem see e (6) represes he me re of he ec eergy of he sysem Relzg h he sress esor s symmerc usg (9) we coul rewre (8) o he form Ppu L (4) he ls erm of (4) s ouble o prouc of sress elocy gre esors he elocy gre esor c be ecompose o symmerc sew-symmerc prs L W, where, s eple before, respecely I coul be show esly h W represe re of eformo sp esors W, ( - symmerc, W - sew-symmerc) From follows h L (4) he fl form for he power pu epresso s Ppu (4) We c coclue h he power pu s he sum of wo olume egrls he frs oe s he merl me ere of he ec eergy of he sysem, whle he seco oe corbues o he erl eergy he sclr : L euls o : s clle sress power per u olume ress power oes o corbue o he ec eergy of he sysem hs resul s ue o oes (85) If boh mechcl o-mechcl eerges re o be cosere he prcple of cosero of eergy s mos geerl form mus be use

22 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc I hs form he cosero prcple ses h he me re of chge of ec erl eergy s eul o sum of he re of wor ll oher eerges supple o he couum per u me uch eerges my clue herml, chemcl, elecromgec eerges I he followg oly mechcl herml eerges re cosere he he eergy prcple es o he form of he frs lw of hermoymcs For our purposes we wll coser hermoymc sysem chose s close sysem o erchgg mer wh surrougs he frs lw of hermoymcs reles he wor oe o he sysem he he rsfer o he sysem o he chge eergy of he sysem I s ssume h oly eergy rsfers o he sysem re by ) mechc wor oe o he sysem by surfce rcos boy forces, b) he rsfer hrough he boury, c) srbue erl he sources urfce rcos boy forces her corbuo o power pu o he sysem were lrey ree by preous prgrph re summrze by e (4)

23 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Fe eformos, cremel ecomposo fe eleme screzo Oerew Prcple of rul wor reles he wor oe by erl eerl forces ue o prescrbe rul splcemes δ U δ W () O he lef h se we he he rul sr eergy of sysem δ U δ ε C δ () E Usg he ol Lgrg pproch, he cremel ecomposos for he seco Pol- Krchhoff sress esor he Gree-Lgrge sr esor re E E, () E (4) I ws lrey show h he creme of Gree-Lgrge sr esor s E ( Z Z ) ( Z Z Z Z) ( Z Z) where u Z L Z s he merl splceme gre We c rouce he followg oo whch wll cossely be use ler E E E L L E E L N E L ( Z Z ) L E (6) E E L N ( Z Z Z Z) ( Z Z) ro of e (4) ges δ E δ E (7) ubsug es (7) () o () we ge δ U ( ) E δ N N ( )( δ E δ E )

24 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc L N L N ( E δ E δ E δ E ) δ (8) he creme of he seco Pol-Krchhoff sress esor pperg he hr erm of he egr coul be lerze C E (9) L l he ls erm (8) coul be eglece sce s oe orer less h oher erms, so he rul sr eergy coul be pprome by where δ U δu δu L U I E δ I II δu III () δ, () δ, (b) N U II E δ L L U III Cl El E δ δ (c) hs ppromo mplcly ssumes h he chges bewee he cofguros C C re smll For fe eleme mplemeo of hese es s coee o swch from he esor o mr oo he process coul be summrze four seps ) Ise of esor we wll use colum rry { E L } L E L L L L L L L ( ) { E E E E E E } efe by E () b) Ise of esor we wll use colum rry { E ~ N} N E ~ N N N N N N N N N N ( ) { E E E E E E E E E } efe by E () c) he seco Pol-Krchhoff sress esor wll he wo pperces Ise of we wll use eher { } efe by { } { } or wo-mesol rry [ ~ ] [ ] [ ~ ] [ ] [ ] 9 9 () efe by, (4) 4

25 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc where [ ] ) C l [ C] (5) I he mr oo, he rul sr eergy compoes, eule o hose pperg e (), re L { δ E } { } ~ N ~ ~ N { δ E } [ ]{ E } δ, (7) U I δ, (7b) U II L L { δ E } [ C]{ E } δ, (7c) U III A ow, he fe eleme screzo eers he sge he geerlze splcemes wh fe eleme re usully epresse by mes of shpe fucos sysemclly rrge A by geerlze ol splcemes he form u A (8) *LMAX LMAX* where LMAX s he umber of OF for prculr eleme he cremes of splcemes re u A (9) Kowg he shpe fucos pperg A we c esly clcule he compoes of he merl splceme gre he o epress s cremes I wll co eres of shpe fucos wll epe o u Z u Z () he resuls for ler pr of sr cremes ccorce wh es (9) () coul be epresse he form L L L ( B B ) B6*LMAX LMAX* E () where obously L L L 6* E E L L L B B B () he lower lef h e zero emphsses h he eres pperg hese mrces re e wh respec o coores 5

26 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc mlrly for he oler pr of sr cremes ~ N N E B () 9* 9*LMAX LMAX* Now, he hree corbuos o he rul sr eergy (7), (7b), (7c) coul be formlly rewre he frs compoe s δ δ F (4) U I where so clle ecor of erl forces oes s L ( B ) F (5) For he seco compoe, e (7b), we c wre N δu δ K, (6) II where so clle o-ler pr of cremel sffess mr s ~ N N N K B B (7) ( ) he hr compoe of he rul sr eergy, e (7c), c be screze usg (9) () Afer some lgebrc mpulos we ge where L δu δ K, (8) III L ( B ) L L C B K (9) s he ler pr of cremel sffess mr Usg es (4), (6) (8) relzg h he rul wor oe by eerl forces { R} s δ W { δ } { R} we c coclue h greeme wh () we ge δ, () L N ( F K K ) δ R () hs euo mus hol for y rul plceme, hece flly we he where K R L N K K K F, () () hs sysem of lgebrc euos cosues he coos of eulbrum olg he sysem ges he uow cremes of ol splcemes he ew splcemes he cofguro C coul be clcule by 6

27 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc (4) he ew splcemes, howeer, he o be e s he frs ppromo oly refe subseuely ere process Iroucg he ero couer () we c rewre e () s follows ( ) ( ) ( ) K R F for K A he begg of he process we se () (), F F,, (5) () K K (6) 7

28 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc he Newo Rphso ero process wh ol Lgrg pproch coul be mplemee s follows Le s ssume h from o m here re m (sme) log seps he mmum force, correspog o fl cofguro me m s m R m R If ler crese of force bewee cosecue log seps s cosere, he he force m correspog o h log sep s R * R/m for o m o % loop for log seps ; f he () () () K K; F F ; ; else K () K ( ls ) ; F () F ( ls ) ; () ; ( ls) e of f ermee lo leel s ssfe flse whle o ssfe o ; R m R */m; % ero loop sole K R F ( ) ( ) ( ) ; ( ) ( ) ( ) ( ) clcule % Noe: ls ; f() ; ( ) ( ) ( ) ( ) ( ); F g( ); ( ) ( ) ; K f, g() re fucos e ( ) (ls) R F ssfe < ε < (ls) ε R e of whle loop of for loop 8

29 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 4 Wh s he merl ere of olume egrl? Le ϕ be fuco h s suffcely smooh ge olume he ϕ ϕ (*) Proof: From lef-h se of he euo (*), we c wre ϕ ϕ ϕ (4) Usg Guss heorem we c rsform surfce egrl o olume egrl ge (4) ( ϕ) ( ϕ) ( ϕ) ϕ ϕ For ergece of prouc of sclr fuco α geerl ecor fuco, we c wre ( α ) gr α α (4) Usg hs epresso we c rewre egr of egrl () we ob ϕ ( ϕ ) ( ϕ ) ϕ ϕ ( ) grϕ ( ) ϕ ϕ grϕ ϕ (44) he seco brce of epresso (44) s from euo of couy eul o zero frs brce from hs epresso s efo relo for egrl (4) we flly ge ϕ ϕ Afer subsug bc o ( ) ϕ grϕ ϕ ϕ ϕ ϕ (45) 9

30 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 5 Couge sress sr mesures A sress s clle couge o he sr f s sclr prouc wh sr ges wor ress sr ues gg mechcl wor s her sclr prouc re eergeclly couge Mechcl wor per u me s power, or re of wor, so we coul lso rele sress sr re ues whose sclr prouc ges power uch ues coul be clle power couge he mechcl wor of surfce rcos boy forces he curre cofguro C W u f u (5) All ues re rele o he curre cofguro Le's om he upper lef e for mome Usg he Cuchy relo he Guss heorem we ge W u f u ( u ) u f u f u (5) he seco erm e (5) s eul o zero, sce s he eulbrum euo Eplog he fc h he sress esor s symmerc, he mechcl wor coul be clcule by ouble o prouc of Cuchy (rue) sr fesml sr he curre cofguro C W u ε :ε (5) We c coclue h he rue sress fesml sr cosues he eergeclly couge rbles s Remr Remember h he fesml sr coul be clcule ecly, olg o ppromo, from ε ( Z Z ) ( F F ) I sce he eformo gre F coul be epresse by mes of he merl splceme gre u Z he form

31 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc F Z I mlrly for he mechcl power, or he re of wor ges P W& u& L ( W ) :, (54) so he power couge ues re he rue sress he elocy gre L Usg he fc h he sp esor W s sew-symmerc s sclr prouc wh symmerc sress esor ges zero, we c se h he rue sress he sr re re lso power couge ues Usg he efo of he frs Pol-Krchhoff sress esor we coul epress he preous euo he referece cofguro C ubsug Bu sce F τ o e (54) we ge τ (55) F τ L LF L F F& (56) F & & F L F o he mechcl power he referece cofguro s epresse by τ & & τ : F & (57) F τ F P whch s ouble o prouc of he frs Pol-Krchoff sress esor he re of eformo gre esor hese esors form oher suble couple of power couge ues mlrly for he seco Pol-Krchhoff sress esor ubsug F F or Fr rs F s o e (54) we ge Fr Fs rs P Fr Fs rs

32 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc Bu F r F s E& rs sce for he Gree-Lgrge sr esor ( F F I) E we coul epress s me re by ( F F) ( F& F F F& ) E & Usg e (56) he preous euo c be rewre o E& ( F L L F LF) ( F ( W) F F ( W) F) ( F ( W ) F F ( W) ) F ( F ( W) F F ( W) F) F F sce s symmerc W sew-symmerc o he sress power referece cofguro c lso be epresse by E P & : E & gg oher couple of suble ues I s obous h erms of mechcl wr we he ε E W Flly, le's f wh role plys he Alms sr esor s hese coseros I c be proe h where Proof P A : : A, s so clle Rl-Ercse re of Alms sr A ) he Alms sr esor s efe by ( s) ( s) A b) he me re of he preous epresso s ( s) ( s) ) ( s) ( A )

33 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc ( A A& & A & ) Bu L, L so sce ( L A A A L) & c) We c lso wre ( s) ( ) L ( W) W ue o he sew-symmery of W he symmery of hs wy we he proe h A L A A& A L 6 ummry for couge sr sress mesures Mesure of sr Mesure of sress her sclr prouc ε - Cuchy (fesml) E - Gree-Lgrge E & - re of Gree-Lgrge F & - re of eformo gre - sr re L - elocy gre - Cuchy (rue) sress - seco Pol-Krchhoff - seco Pol-Krchhoff τ - frs Pol-Krchhoff - Cuchy (rue) sress - Cuchy (rue) sress wor wor power, re of wor power, re of wor power, re of wor power, re of wor A - Rl-Erse re of Alms - Cuchy (rue) sress power, re of wor

34 elemchos c:\euco_er_5\_cm_bcgrou\ll_ogeher 5_coc 4

Stat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty

Stat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty S 6863-Hou 5 Fuels of Ieres July 00, Murce A. Gerghy The pror hous resse beef cl occurreces, ous, ol cls e-ulero s ro rbles. The fl copoe of he curl oel oles he ecooc ssupos such s re of reur o sses flo.