3 Stress and the Balance Principles

Transcription

1 Stre a the Balace Prcple hree bac law of phyc are cue th Chapter: () he Law of Coerato of Ma () he Balace of Lear Mometum () he Balace of Agular Mometum together wth the coerato of mechacal eergy a the prcple of rtual work, whch are fferet ero of (). () a () ole the cocept of tre, whch allow oe to ecrbe the acto of force materal. 5

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3 Secto.. Coerato of Ma.. Ma a Dety Ma a o-egate calar meaure of a boy teecy to ret a chage moto. Coer a mall olume elemet th olume elemet by the rato Δ whoe ma Δ m. Defe the aerage ety of Δm ρ AE (..) Δ If p ome pot wth the olume elemet, the efe the patal ma ety at p to be the lmtg alue of th rato a the olume hrk ow to the pot, Δm ρ (, lm Δ Spatal Dety (..) Δ I a real materal, the cremetal olume elemet Δ mut ot actually get too mall ce the the lmt ρ woul epe o the atomtc tructure of the materal; the olume oly allowe to ecreae to ome mmum alue whch cota a large umber of molecule. he patal ma ety a repreetate aerage obtae by hag Δ large compare to the atomc cale, but mall compare to a typcal legth cale of the problem uer coerato. he ety, a wth placemet, elocty, a other quatte, efe for pecfc partcle of a cotuum, a a cotuou fucto of coorate a tme, ρ ρ(,. Howeer, the ma ot efe th way oe wrte for the ma of a ftemal olume of materal a ma elemet, or, for the ma of a olume of materal at tme t, m ρ(, (..) ( ) m ρ,t (..4).. Coerato of Ma he law of coerato of ma tate that ma ca ether be create or etroye. Coer a collecto of matter locate omewhere pace. h quatty of matter wth well-efe bouare terme a ytem. he law of coerato of ma the mple that the ma of th ge ytem rema cotat, Sol Mechac Part III 7

4 Secto. Dm Dt Coerato of Ma (..5) he olume occupe by the matter may be chagg a the ety of the matter wth the ytem may be chagg, but the ma rema cotat. Coerg a fferetal ma elemet at poto X the referece cofgurato a at the curret cofgurato, Eq...5 ca be rewrtte a m( X ) m(, (..6) he coerato of ma equato ca be epree term of ete. Frt, trouce ρ, the referece ma ety (or mply the ety), efe through Δm ρ (X) lm Δ Dety (..7) Δ Note that the ety ρ a the patal ma ety ρ are ot the ame quatte. hu the local (or fferetal) form of the coerato of ma ca be epree a (ee Fg...) m ρ ( X) ρ(, cot (..8), ρ X referece cofgurato, ρ curret cofgurato Fgure..: Coerato of Ma for a eformg ma elemet Itegrato oer a fte rego of materal ge the global (or tegral) form, or m ρ ( X) ρ(, cot (..9) m m& ρ(, (..) they ot oly are fucto of fferet arable, but alo hae fferet alue; they are ot fferet repreetato of the ame thg, a were, for eample, the elocte a. Oe coul trouce a materal ma ety, Ρ ( X, ρ( ( X,,, but uch a quatty ot ueful aaly Sol Mechac Part III 8

5 Secto... Cotrol Ma a Cotrol olume A cotrol ma a fe ma of materal whoe olume a ety may chage, a whch may moe through pace, Fg.... here o ma traport through the mog urface of the cotrol ma. For uch a ytem, Eq... hol. m ρ ( ( t ), t ), ( ) m ρ( ( t ), t ), ( ), t, t Fgure..: Cotrol Ma By efto, the erate.. the tme erate of a property ( th cae ma) of a collecto of materal partcle a they moe through pace, a whe they tataeouly occupy the olume, Fg..., or ρ lm Δt (, Δ ) (, ) Δ ρ t t ρ t (..) t ( tδ ( Alterately, oe ca take the materal erate e the tegral g: m [ (, ] ρ (..) h ow equalet to the um of the rate of chage of ma of the ma elemet occupyg the olume. tme t tme t Δt Fgure..: Cotrol Ma occupyg fferet olume at fferet tme A cotrol olume, o the other ha, a fe olume (rego) of pace through whch materal may flow, Fg...4, a for whch the ma may chage. For uch a ytem, oe ha Sol Mechac Part III 9

6 Secto. m t t ρ (, [ ρ(, ] (..) t m (, ρ(,, Fgure..4: Cotrol olume..4 he Cotuty Equato (Spatal Form) A coequece of the law of coerato of ma the cotuty equato, whch ( the patal form) relate the ety a elocty of ay materal partcle urg moto. h equato ca be ere a umber of way: Derato of the Cotuty Equato ug a Cotrol olume (Global Form) he cotuty equato ca be ere rectly by coerg a cotrol olume - th the erato approprate to flu mechac. Ma e th fe olume caot be create or etroye, o that the rate of creae of ma the olume mut equal the rate at whch ma flowg to the olume through t boug urface. he rate of creae of ma e the fe olume m t t ρ ρ (, (..4) t he ma flu (rate of flow of ma) out through the urface ge by Eq..7.9, ρ, ρ where the ut outwar ormal to the urface a the elocty. It follow that ρ t Ue of the ergece theorem.7. lea to ρ ρ, ρ (..5) t ( ρ ) ρ ρ ( ρ ), (..6) t t Sol Mechac Part III

7 Secto. leag to the cotuty equato, ρ ( ) t ρ ρ ρ ρ graρ ρ t ( ρ ) ρ t ρ ρ ρ ρ ρ t Cotuty Equato (..7) h (thee are) the cotuty equato patal form. he eco a thr form of the equato are obtae by re-wrtg the local erate term of the materal erate.4.7 (ee alo.6.b). If the materal compreble, o the ety rema cotat the eghbourhoo of a partcle a t moe, the the cotuty equato reuce to, Cotuty Eq. for Icompreble Materal (..8) Derato of the Cotuty Equato ug a Cotrol Ma Here follow two way to ere the cotuty equato ug a cotrol ma.. Derato ug the Formal Defto From.., ag a ubtractg a term: ρ lm Δt Δt ρ(, t Δ ( tδt ) ( ρ(, t Δ ρ(, t Δ ( ( ρ(, (..9) he term the eco quare bracket correpo to holg the olume fe a eetly equal the local rate of chage: ρ ρ lm Δ ρ(, t Δ (..) t t Δt ( tδ ( he rego ( t Δ ( wept out tme Δ t. Supermpog the olume ( a ( t Δ, Fg...5, t ca be ee that a mall elemet Δ of ( t Δ ( ge by (ee the eample aocate wth Fg..7.7) Δ Δt Δ (..) Sol Mechac Part III

8 Secto. where the urface. hu lm Δt Δt ( tδ ( ρ (, t Δ lm Δtρ(, t Δ ρ(, (..) Δt Δt a..5 aga obtae, from whch the cotuty equato reult from ue of the ergece theorem. ( ( t Δ ( ( t Δ Δ Δ Fgure..5: Ealuato of Eq.... Derato by Coertg to Ma Elemet h erato requre the kematc relato for the materal tme erate of a olume elemet,.5.: ( ) /. Oe ha m. ρ(, ( ρ) & ρ ρ (& ρ ρ ) (..) he cotuty equato the follow, ce th mut hol for ay arbtrary rego of the olume. Derato of the Cotuty Equato ug a Cotrol olume (Local Form) he cotuty equato ca alo be ere ug a fferetal cotrol olume elemet. h calculato mlar to that ge.6.6, wth the elocty replace by ρ...5 he Cotuty Equato (Materal Form) From..9, a ug..5, J, [ ( X) ρ( χ( X,, J ( X, ] ρ (..4) Sol Mechac Part III

9 Secto. Sce a arbtrary rego, the tegra mut ah eerywhere, o that ρ ( X) ρ( χ( X,, J ( X, ) Cotuty Equato (Materal Form) (..5) t h kow a the cotuty (ma) equato the materal ecrpto. Sce ρ&, the rate form of th equato mply ( ρ J ) (..6) he materal form of the cotuty equato, ρ ρ J, a algebrac equato, ulke the partal fferetal equato the patal form. Howeer, the two mut be equalet, a ee the patal form ca be ere rectly from th materal form: ug.5., J / J, ( ρj ) & ρ J ρ J& J (& ρ ρ ) (..7) h zero, a J >, a the patal cotuty equato follow. Eample (of Coerato of Ma) Coer a bar of materal of legth l, wth ety the ueforme cofgurato ρ a patal ma ety ρ (,, uergog the -D moto X /( A, X AtX. he olume rato (takg ut cro-ectoal area) J At. he cotuty equato the materal form..5 pecfe that Suppoe ow that ρ ρ( A ρ ( ) o X m X l l o o that the total ma of the bar ρ ( X X m. It follow that the patal ma ety ) ρ m X m ρ ( At ) l At l ( A Ealuatg the total ma of the bar at tme t lea to lo ( A m ρ (, l ( A l ( A o Sol Mechac Part III

10 Secto. whch aga m, a requre. e of bar ( X l ) at t e of bar ( l ( A, X l ) at tme t Fgure..6: a tretchg bar he ety coul hae bee ere from the equato of cotuty the patal form: ce the elocty ( X, ( X, AX, (, ( χ A (,, At oe ha ρ ρ ρ t ρ A t At ρ A ρ At Wthout attemptg to ole th frt orer partal fferetal equato, t ca be ee by ubttuto that the alue for ρ obtae preouly atfe the equato...6 Materal Derate of Itegral Reyol raport heorem I the aboe, the materal erate of the total ma carre by a cotrol ma, ρ (,, wa coere. It qute ofte that oe ee to ealuate materal tme erate of mlar olume (a le a urface) tegral, olg other properte, for eample mometum or eergy. hu, uppoe that A (, the trbuto of ome property (per ut olume) throughout a olume (A take to be a eco orer teor, but what follow apple alo to ector a calar). he the rate of chage of the total amout of the property carre by the ma ytem Sol Mechac Part III 4

11 Secto. A (, Aga, th tegral ca be ealuate a umber of way. For eample, oe coul ealuate t ug the formal efto of the materal erate, a oe aboe for A ρ. Alterately, oe ca ealuate t ug the relato.5., ( ) /, through. [ A(, ] A A [ A& A] & (..8) A(, hu oe arre at Reyol traport theorem A(, A A A graa A t A ( A ) t A ( ) A t A A t A t A t k A k A k k ( Ak ) k A k k k k A Reyol raport heorem (..9) he e otato how for the cae whe A a eco orer teor. I the lat of thee form (obtae by applcato of the ergece theorem), the frt term repreet the amout (of A) create wth the olume wherea the eco term (the flu term) repreet the (olume) rate of flow of the property through the urface. I the lat three ero, Reyol traport theorem ge the materal erate of the mog cotrol ma term of the erate of the tataeou fe olume pace (the frt term). Of coure whe A ρ, the cotuty equato recoere. Aother way to ere th reult to frt coert to the referece cofgurato, o that tegrato a fferetato commute (ce epeet of tme): A(, ( A( X, J ) A( X, J ( AJ AJ& ) ( A& A ) ( A& (, A(, ) & J (..) alo kow a the Lebz formula Sol Mechac Part III 5

12 Secto. Reyol raport heorem for Specfc Properte A property that ge per ut ma calle a pecfc property. For eample, pecfc heat the heat per ut ma. Coer the a property B, a calar, ector or teor, whch efe per ut ma through a olume. he the rate of chage of the total amout of the property carre by the ma ytem mply ρ B, [ Bρ] [ Bm] m B B ( ρ (..) Materal Derate of Le a Surface Itegral Materal erate of le a urface tegral ca alo be ealuate. From.5.8, ( ) / l, ˆ a, ug.5., ( ) ( ) / [ A& Al] A(, (..) l ˆ, A(, ˆ [ A A( l )] & ˆ (..)..7 Problem. A moto ge by the equato X X t, X t X ( t ), X (a) Calculate the patal ma ety ρ term of the ety ρ (b) Dere a frt orer orary fferetal equato for the ety ρ ( term of a t oly) aumg that t epeet of poto Sol Mechac Part III 6

13 Secto.. he Mometum Prcple I Part I a II, the bac yamc prcple ue were Newto Law, a thee are equalet to force equlbrum a momet equlbrum. For eample, they were ue to ere the tre traformato equato Part I,.4 a the Equato of Moto Part II,.. Newto law there were apple to fferetal materal elemet. A alterate but completely equalet et of yamc law are Euler Law; thee are more approprate for fte-ze collecto of mog partcle, a ca be ue to epre the force a momet equlbrum term of tegral. Euler Law are alo calle the Mometum Prcple: the prcple of lear mometum (Euler frt law) a the prcple of agular mometum (Euler eco law)... he Prcple of Lear Mometum Mometum a meaure of the teecy of a obect to keep mog oce t et moto. Coer frt the partcle of rg boy yamc: the (lear) mometum p efe to be t ma tme elocty, p m. he rate of chage of mometum p& p ( m) m ma (..) a ue ha bee mae of the fact that m /. hu Newto eco law, F ma, ca be rewrtte a F ( m) (..) h equato, formulate by Euler, tate that the rate of chage of mometum equal to the apple force. It calle the prcple of lear mometum, or balace of lear mometum. If there are o force apple to a ytem, the total mometum of the ytem rema cotat; the law th cae kow a the law of coerato of (lear) mometum. Eq... a apple to a partcle ca be geeralze to the mechac of a cotuum oe of two way. Oe coul coer a fferetal elemet of materal, of ma m a elocty. Alterately, oe ca coer a fte porto of materal, a cotrol ma the curret cofgurato wth patal ma ety ρ (, a patal elocty fel (,. he total lear mometum of th ma of materal he prcple of lear mometum tate that ( L( t ) ρ (,, Lear Mometum (..) L & ( (, (, F( ρ (..4) Sol Mechac Part III 7

14 Secto. where F ( the reultat of the force actg o the porto of materal. Note that the olume oer whch the tegrato Eq...4 take place ot fe; the tegral take oer a fe porto of materal partcle, a the pace occupe by th matter may chage oer tme. By rtue of the raport theorem relato.., th ca be wrtte a (, F( ) L & ( ρ (, & t (..5) he reultat force actg o a boy ue to the urface tracto t actg oer urface elemet a boy force b actg o olume elemet, Fg...: F ( t b, F t b Reultat Force (..6) a o the prcple of lear mometum ca be epree a t b ρ & Prcple of Lear Mometum (..7) t b Fgure..: urface a boy force actg o a fte olume of materal he prcple of lear mometum, Eq...7, wll be ue to proe Cauchy Lemma a Cauchy Law the et ecto a,.6, to ere the Equato of Moto... he Prcple of Agular Mometum Coerg aga the mechac of a gle partcle: the agular mometum the momet of mometum about a a, other wor, t the prouct of the lear mometum of the partcle a the perpecular tace from the a of t le of acto. I the otato of Fg..., the agular mometum h whch the ector wth magtue h r m (..8) m a perpecular to the plae how. Sol Mechac Part III 8

15 Secto. o r Fgure..: urface a boy force actg o a fte olume of materal Coer ow a collecto of partcle. he prcple of agular mometum tate that the reultat momet of the eteral force actg o the ytem of partcle, M, equal the rate of chage of the total agular mometum of the partcle: Geeralg to a cotuum, the agular mometum h M r F (..9) H r ρ Agular Mometum (..) m a the prcple of agular mometum ε k r t t ( ) k ( ) ε k r b b k r ρ ε ρ k k Prcple of Agular Mometum (..) he prcple of agular mometum,.., wll be ue,.6, to euce the ymmetry of the Cauchy tre. Sol Mechac Part III 9

16 Secto.. he Cauchy Stre eor.. he racto ector he tracto ector wa trouce Part I,.. o recall, t the lmtg alue of the rato of force oer area; for Force Δ F actg o a urface elemet of area Δ S, t ( ) ΔF t lm (..) Δ S ΔS a eote the ormal to the urface elemet. A fte umber of tracto ector act at a pot, each actg o fferet urface through the pot, efe by fferet ormal... Cauchy Lemma Cauchy lemma tate that tracto ector actg o oppote e of a urface are equal a oppote. h ca be epree ector form: t t Cauchy Lemma (..) ( ) ( ) h ca be proe by applyg the prcple of lear mometum to a collecto of partcle of ma Δ m tataeouly occupyg a mall bo wth parallel urface of area Δ, thcke δ a olume Δ δ Δ, Fg.... he reultat urface force ( ) ( ) actg o th matter t Δ t Δ. () t Δ thcke δ ( ) t Fgure..: tracto actg o a mall porto of materal partcle he total lear mometum of the matter ρ m. By the mea alue Δ Δ m theorem (ee Appe A to Chapter,.B.), th equal Δm, where the elocty at ome teror pot. Smlarly, the boy force actg o the matter b bδ, where b the boy force (per ut olume) actg at ome teror pot. he total ma Δ th equalet to Newto (thr) law of acto a reacto t eem lke a lot of work to proe th eemgly obou reult but, to be cotet, t uppoe that the oly fuametal yamc law aalable here are the prcple of lear a agular mometum, a ot ay of Newto law Sol Mechac Part III

17 Secto. ca alo be wrtte a Eq...7, a ce Δm ρ ρδ. From the prcple of lear mometum, Δ Δ m oe ot chage wth tme, t ( ) Δ t ( ) Δ bδ [ Δm] Δm ρδ ρδδ (..) ( ) ( ) Dg through by Δ a takg the lmt a δ, oe f that t t. ( ) ( ) Note that the alue of t, t actg o the bo wth fte thcke are ot the ame a the fal alue, but approach the fal alue at the urface a δ... Stre I Part I, the compoet of the tracto ector were calle tre compoet, a t wa llutrate how there were e tre compoet aocate wth each materal partcle. Here, the tre efe more formally, Cauchy Law Cauchy Law tate that there et a Cauchy tre teor whch map the ormal to a urface to the tracto ector actg o that urface, accorg to t, t Cauchy Law (..4) or, full, t t t (..5) Note: may author efe the tre teor a t. h amout to the efto ue here ce, a metoe Part I, a a wll be (re-)proe below, the tre teor ymmetrc,, the Cauchy tre refer to the curret cofgurato, that, t a meaure of force per ut area actg o a urface the curret cofgurato. Stre Compoet akg Cauchy law to be true (t proe below), the compoet of the tre teor wth repect to a Cartea coorate ytem are, from.9.4 a..4, ( e ) e e e t (..6) whch the th compoet of the tracto ector actg o a urface wth ormal e. Note that th efto cotet wth that ge Part I,. there, the frt Sol Mechac Part III

18 Secto. ubcrpt eote the recto of the ormal but, aga, the two efto are equalet becaue of the ymmetry of the tre teor. he three tracto ector actg o the urface elemet whoe outwar ormal pot the recto of the three bae ector e are ( ) t e e, t t t ( ) e ( e ) ( e ) e e e e e e e e e (..7) Eq are llutrate Fg.... e ( ) e t e ( e ) t ( ) e t e (a) (b) Fgure..: tracto actg o urface wth ormal the coorate recto; (a) tracto ector, (b) tre compoet Proof of Cauchy Law he proof of Cauchy law eetally follow the ame metho a ue the proof of Cauchy lemma. Coer a mall tetraheral free-boy, wth erte at the org, Fg.... It requre to eterme the tracto t term of the e tre compoet (whch are all how pote the agram). Let the area of the bae of the tetrahera, wth ormal, be Δ. he area the Δ coα, where α the agle betwee the plae, a how Fg...b; th agle the ame a that betwee the ector a e, o Δ ( e ) Δ Δ, a mlarly for the other urface: Δ Δ a Δ Δ. Sol Mechac Part III

19 Secto. ( ) t Δ Δ α e Δ (a) (b) Fgure..: free boy agram of a tetraheral porto of materal; (a) tracto actg o the materal, (b) relatohp betwee urface area a ormal compoet he reultat urface force o the boy, actg the recto, t Δ Δ Δ Δ Aga, the mometum ΔM, the boy force b Δ a the ma Δ m ρ Δ ρ ( h / ) Δ, where h the perpecular tace from the org (erte) to the bae. he prcple of lear mometum the tate that t Δ Δ Δ Δ b ( h / ) Δ ρ ( h / ) Δ Aga, the alue of the tracto a tre compoet o the face wll geeral ary oer the face, o the alue ue th equato are aerage alue oer the face. Dg through by Δ, a takg the lmt a h, oe f that t a ow thee quatte, t,,,, are the alue at the org. he equato for the other two tracto compoet ca be ere a mlar way. Normal a Shear Stre he tre actg ormal to a urface ge by () N t (..8) he hear tre actg o the urface ca the be obtae from Sol Mechac Part III

20 Secto. ( ) S N t (..9) Eample he tate of tre at a pot ge the matr form [ ] Determe (a) the tracto ector actg o a plae through the pot whoe ut ormal ˆ (/ )ˆ e ˆ ( / )ˆ e ( / ) e (b) the compoet of th tracto actg perpecular to the plae (c) the hear compoet of tracto. Soluto (a) he tracto t t t ( ˆ ) ( ˆ ) ( ˆ ) 9 ( ) or t ˆ ( / )ˆ e ˆ ˆ e e. (b) he compoet ormal to the plae the proecto of ( ˆ ) t the recto of ˆ,.e. N ( ) t ˆ ˆ ( / )(/ ) ( / ) ( / ) / 9.4. (c) he hearg compoet of tracto S ( ˆ ) t ( / 9)ˆ [ ( / ) ( / 7) ] eˆ [ (44 / 7) ] eˆ [ (44 / 7) ] eˆ ] [( 4 / 7)ˆ e (7 / 7)ˆ e (7 / 7)ˆ e ].e. of magtue ( 4 / 7) (7 / 7) (7 / 7)., whch equal ˆ ( ˆ ) t. N Sol Mechac Part III 4

21 Secto.4.4 Properte of the Stre eor.4. Stre raformato Let the compoet of the Cauchy tre teor a coorate ytem wth bae ector e be. he compoet a eco coorate ytem wth bae ector e,, are ge by the teor traformato rule..5: Q Q (.4.) p q pq where Q are the recto coe, Q e e. Iotropc State of Stre Suppoe the tate of tre a boy [ ] Oe f that the applcato of the teor traformato rule yel the ery ame compoet o matter what the coorate ytem. h terme a otropc tate of tre, or a phercal tate of tre (ee..). Oe eample of otropc tre the tre arg flu at ret, whch caot upport hear tre, whch cae pi (.4.) where the calar p the flu hyrotatc preure. For th reao, a otropc tate of tre alo referre to a a hyrotatc tate of tre. A ote o the raformato Formula Ug the ector traformato rule.5.5, the tracto a ormal traform accorg to [ t ] [ Q ][ t], [ ] [ Q ][ ]. Alo, Cauchy law traform accorg to [ t ] [ ][ ] whch ca be wrtte a [ Q ][ t] [ ][ Q ][ ], o that, pre-multplyg by [ Q ], a ce [ Q ] orthogoal, [ t] {[ Q][ ][ Q ]}[ ], o [ ] [ Q][ ][ Q ], whch the ere teor traformato rule..6a, howg the teral cotecy of the theory. I Part I, Newto law wa apple to a materal elemet to ere the two-meoal tre traformato equato, Eq..4.7 of Part I. Cauchy law wa proe a mlar way, ug the prcple of mometum. I fact, Cauchy law a the tre traformato equato are equalet. Ge the tre compoet oe coorate ytem, the tre traformato equato ge the compoet a ew coorate ytem; partcularg th, they ge the tre compoet, a thu the tracto ector, Sol Mechac Part III 5

22 Secto.4 actg o ew urface, orete ome way wth repect to the orgal ae, whch what Cauchy law oe..4. Prcpal Stree Sce the tre a ymmetrc teor, t ha three real egealue,,, calle prcpal tree, a three correpog orthoormal egeector calle prcpal recto. he egealue problem ca be wrtte a ( t ) (.4.) where a prcpal recto a a calar prcpal tre. Sce the tracto ector a multple of the ut ormal, a ormal tre compoet. hu a prcpal tre a tre whch act o a plae of zero hear tre, Fg..4.. ( ) t t e t e t e o hear tre oly a ormal compoet to the tracto Fgure.4.: tracto actg o a plae of zero hear tre he prcpal tree are the root of the charactertc equato..5, where, Eq...6-7,..7, I I I (.4.4) I I I tr [( tr) tr ] [ tr trtr ( tr) ] et (.4.5) Sol Mechac Part III 6

23 Secto.4 he prcpal tree a prcpal recto are properte of the tre teor, a o ot epe o the partcular ae choe to ecrbe the tate of tre., a the tre arat I, I, I are arat uer coorate traformato. c.f.... If oe chooe a coorate ytem to coce wth the three egeector, oe ha the pectral ecompoto.. a the tre matr take the mple form.., [ ] ˆ ˆ, (.4.6) Note that whe two of the prcpal tree are equal, oe of the prcpal recto wll be uque, but the other two wll be arbtrary oe ca chooe ay two prcpal recto the plae perpecular to the uquely eterme recto, o that the three form a orthoormal et. h tre tate calle a-ymmetrc. Whe all three prcpal tree are equal, oe ha a otropc tate of tre, a all recto are prcpal recto..4. Mamum Stree Drectly from.., the three prcpal tree clue the mamum a mmum ormal tre compoet actg at a pot. h reult re-ere here, together wth reult for the mamum hear tre Normal Stree Let e, e, e be ut ector the prcpal recto a coer a arbtrary ut ormal ector e e e, Fg..4.. From..8 a Cauchy law, the ormal tre actg o the plae wth ormal ( ) ( t ) (.4.7) N N ( ) t prcpal recto Fgure.4.: ormal tre actg o a plae efe by the ut ormal Sol Mechac Part III 7

24 Secto.4 Sol Mechac Part III 8 Wth repect to the prcpal tree, ug.4.6, ) ( e e e t (.4.8) a the ormal tre N (.4.9) Sce a, wthout lo of geeralty, takg, oe ha ( ) N (.4.) Smlarly, ( ) N (.4.) hu the mamum ormal tre actg at a pot the mamum prcpal tre a the mmum ormal tre actg at a pot the mmum prcpal tre. Shear Stree Net, t wll be how that the mamum hearg tree at a pot act o plae orete at 45 o to the prcpal plae a that they hae magtue equal to half the fferece betwee the prcpal tree. From..9,.4.8 a.4.9, the hear tre o the plae ( ) ( ) S (.4.) Ug the coo to elmate lea to ( ) ( ) ( ) ( ) [ ] S (.4.) he tatoary pot are ow obtae by equatg the partal erate wth repect to the two arable a to zero: ( ) ( ) ( ) ( ) [ ] { } ( ) ( ) ( ) ( ) [ ] { } S S (.4.4) Oe ee mmeately that (o that ± ) a oluto; th the prcpal recto e a the hear tre by efto zero o the plae wth th ormal. I

25 Secto.4 th calculato, the compoet wa elmate a S wa treate a a fucto of the arable (, ). Smlarly, ca be elmate wth (, ) treate a the arable, leag to the oluto e, a ca be elmate wth (, ) treate a the arable, leag to the oluto e. hu thee oluto lea to the mmum hear tre alue. S A eco oluto to Eq..4.4 ca be ee to be, / (o that ± ±/ ) wth correpog hear tre alue ( ) S 4. wo other oluto ca be obtae a ecrbe earler, by elmatg a by elmatg. he full oluto lte below, a thee are eetly the mamum (abolute alue of the) hear tree actg at a pot:, ±, ±, ±,, ±, ±, ±,, S S S (.4.5) akg, the mamum hear tre at a pot τ ma ( ) (.4.6) a act o a plae wth ormal orete at 45 o to the a prcpal recto. h llutrate Fg..4.. τ ma prcpal recto τ ma Fgure.4.: mamum hear tre at apot Eample (mamum hear tre) Coer the tre tate Sol Mechac Part III 9

26 Secto.4 [ ] 5 6 h the ame teor coere the eample of... Ug the reult of that eample, the prcpal tree are, 5, 5 a o the mamum hear tre at that pot 5 τ ma ( ) he plae a recto upo whch they act are how Fg τ ma ˆ ˆ o 7 ˆ Fgure.4.4: mamum hear tre Sol Mechac Part III 4

27 Secto.5.5 Stre Meaure for Large Deformato hu far, the urface force actg wth a materal hae bee ecrbe term of the Cauchy tre teor. he Cauchy tre alo calle the true tre, to tguh t from other tre teor, ome of whch wll be cue below. It calle the true tre becaue t a true meaure of the force per ut area the curret, eforme, cofgurato. Whe the eformato are mall, there o tcto to be mae betwee th eforme cofgurato a ome referece, or ueforme, cofgurato, a the Cauchy tre the eble way of ecrbg the acto of urface force. Whe the eformato are large, howeer, oe ee to refer to ome referece cofgurato. I th cae, there are a umber of fferet poble way of efg the acto of urface force; ome of thee tre meaure ofte o ot hae a clear a phycal meag a the Cauchy tre, but are ueful oethele..5. he Frt Pola Krchhoff Stre eor Coer two cofgurato of a materal, the referece a curret cofgurato. Coer ow a ector elemet of urface the referece cofgurato, N S, where S the area of the elemet a N the ut ormal. After eformato, the materal partcle makg up th area elemet ow occupy the elemet efe by, where the area a the ormal the curret cofgurato. Suppoe that a force f act o the urface elemet ( the curret cofgurato). he by efto of the Cauchy tre f (.5.) he frt Pola-Krchhoff tre teor P (whch wll be calle the PK tre for brety) efe by f P N S (.5.) he PK tre relate the force actg the curret cofgurato to the urface elemet the referece cofgurato. Sce t relate to both cofgurato, t a two-pot teor. he (Cauchy) tracto ector wa efe a f t, t (.5.) Smlarly, oe ca trouce a PK tracto ector uch that f, P N (.5.4) S Wherea the Cauchy tracto the actual phycal force per area o the elemet the curret cofgurato, the PK tracto a fcttou quatty the force actg o a elemet the curret cofgurato e by the area of the correpog elemet Sol Mechac Part III 4

28 Secto.5 the referece cofgurato. Note that, ce f t S, t follow that a t act the ame recto (but hae fferet magtue), Fg..5.. N t f t S S referece cofgurato curret cofgurato Fgure.5.: racto ector Uaal eo Coer a uaal tele tet whereby a pecme tretche uformly by a cotat force f, Fg..5.. he tal cro-ectoal area of the pecme A a the croectoal area of the pecme at tme t A (. he Cauchy (true) tre a the PK tre f ( (.5.5) A( f P (.5.6) A h tre meaure, force oer area of the ueforme pecme, a ue the uaal tele tet, alo calle the egeerg tre. f curret cofgurato Fgure.5.: Uaal teo of a bar he Nomal Stre he PK tre teor alo calle the omal tre teor. Note that may author ue a fferet efto for the omal tre, amely N P, a the efe the PK tre to be the trapoe of th P. hu all author ue the ame efto for the PK tre, but a lghtly fferet efto for the omal tre. Sol Mechac Part III 4

29 Secto.5 Relato betwee the Cauchy a PK Stree From the aboe efto, P NS (.5.7) Ug Nao formula,..59, J F NS, P J F J P F PK tre (.5.8) he Cauchy tre ymmetrc, but the eformato graet ot. Hece the PK tre teor ot ymmetrc, a th retrct t ue a a alterate tre meaure to the Cauchy tre meaure. I fact, th lack of ymmetry a lack of a clear phycal meag make t ucommo for the PK tre to be ue the moelg of materal. It, howeer, ueful the ecrpto of the mometum balace law the materal ecrpto, where P play a aalogou role to that playe by the Cauchy tre the equato of moto (ee later)..5. he Seco Pola Krchhoff Stre eor he eco Pola Krchhoff tre teor, or the PK tre, S, efe by S J F F PK tre (.5.9) Ee though the PK oe ot amt a phycal terpretato (ecept the mplet of cae, but ee the terpretato below), there are three goo reao for ug t a a meaure of the force actg a materal. Frt, oe ca ee that ( F F ) ( F ) ( F ) F F a ce the Cauchy tre ymmetrc, o the PK tre: S S (.5.) A eco reao for ug the PK tre that, together wth the Euler-Lagrage tra E, t ge the power of a eformg materal (ee later). hr, t parameterze by materal coorate oly, that, t a materal teor fel, the ame way a the Cauchy tre a patal teor fel. Note that the PK a PK tree are relate through P FS, S F P (.5.) Sol Mechac Part III 4

30 Secto.5 he PK tre ca be terprete a follow: take the force ector the curret cofgurato f a locate a correpog ector the ueforme cofgurato accorg to f F f. he PK tre teor th fcttou force e by the correpog area elemet the referece cofgurato: f SNS, a.5.9 follow from.5.,.5.8: f P NS J F NS.5. Alterate Stre eor Some other ueful tre meaure are ecrbe here. he Krchhoff Stre he Krchhoff tre teor τ efe a τ J Krchhoff Stre (.5.) It a patal teor fel parameterze by patal coorate. Oe reao for t ue that, may equato, the Cauchy tre appear together wth the Jacoba a the ue of τ mplfe formulae. Note that the Krchhoff tre the puh forwar of the PK tre; from..9b,..b, τ χ* S χ * # ( S) FSF # ( τ) F τf (.5.) he Corotatoal Cauchy Stre he corotatoal tre ˆ efe a ˆ R R Corotatoal Stre (.5.4) where R the orthogoal rotato teor. Wherea the Cauchy tre relate to the PK tre through J FSF, the corotatoal tre relate to the PK tre through (wth F replace by the rght (ymmetrc) tretch teor U): ( JF F ) U ( UF ) ( F U) R R ˆ J USU J U (.5.5) he corotatoal tre efe o the termeate cofgurato of Fg...8. It ca be regare a the puh forwar of the PK tre from the referece cofgurato through the tretch U, cale by J (Eq...8b): # ( S) U( G ) J S gˆ g J S ( UG UG ) J USU USU ˆ J χ J (.5.6) ˆ * Sol Mechac Part III 44

31 Secto.5 or a the pull-back of the Cauchy tre wth repect to R (Eq...7f): he Bot Stre he Bot (or Jauma) tre teor # ( ) R( g) gˆ g R R ˆ χ (.5.7) ˆ * B efe a R B P US Bot Stre (.5.8) From.5., t mlar to the PK tre, oly wth F replace by U. Eample Coer a pre-tree th plate wth,, that, t ha a o-zero tre although o force are actg, Fg..5.. I th tal tate, F I a, coerg a two-meoal tate of tre, P S ˆ τ B o he materal ow rotate a a rg boy 45 couterclockwe the tre-tate froze wth the materal a rotate wth t. he / F R / / / * he tre compoet wth repect to the rotate ae how Fg..5.b are *, etc.; the compoet wth repect to the patal ae ca be fou from the * * tre traformato rule [ ] [ Q ][ ][ Q] [ R][ ][ R ], a o ( ) ( ) ( ) ( ) Note that the Cauchy tre chage wth th rg boy rotato. Further, wth J, / / τ, P, S ˆ B / / * Note that the PK tre ot ymmetrc. Now attach ae to the materal a rotate thee ae wth the pecme a t rotate, a Fg..5.b. he compoet wth repect for eample a pece of metal ca be eforme; whe the loa remoe t ofte pre-tree there a o-zero tate of tre the materal Sol Mechac Part III 45

32 Secto.5 to thee rotate ae ge the corotatoal tre; the corotatoal tre the tre a boy, takg out the tre chage caue by rg boy rotato oe ay that the corotatoal tre (a PK tre) rotate wth the boy. * * referece cofgurato (a) rotate cofgurato (b) Fgure.5.: Pre-tree materal; (a) orgal poto, (b) rotate cofgurato.5.4 Small eformato From.7, whe the eformato are mall, eglectg term olg prouct of placemet graet, F I grau O(grau) I O(grau) (.5.9) Here, O(grau ) mea term of the orer of placemet graet (a hgher) hae bee eglecte a O(grau ) mea term of the orer of prouct of placemet graet (a hgher) hae bee eglecte. Alo, J et F et ( I grau O(grau) ) u O(grau) O(grau) (.5.) From.5.8 a.5.9, ug.5.9-, oe ha J P F J FS F O(grau) P O(grau) O(grau) S O(grau) (.5.) I the lear theory the, wth O(grau ), the tre meaure ecoutere th ecto are all equalet. Sol Mechac Part III 46

33 Secto Obecte Stre eor I orer to acerta the obectty of the tre teor, frt ote that, by efto, force a obecte ector, a therefore o alo the tracto ector. Smlarly for the ormal ector. he ormal a tracto ector traform uer a oberer * * traformato accorg to.8., Q a t Qt. he t Q t * Q * t * ( QQ ) * (.5.) * a o QQ ; accorg to.8., the Cauchy tre obecte. he PK tre S obecte, ce t a materal teor uaffecte by a oberer traformato. For the PK tre, ug.8., P * ( F ) JQQ ( QF) Q( J ) J F (.5.) * * * a o, accorg to.8.6, P obecte (traformg lke a ector, beg a two-pot teor)..5.6 Obecte Stre Rate Oe ee to corporate tre rate moel of materal where the repoe epe o the rate of treg, for eample wth coelatc materal. A cue.8.5, the rate of obecte teor are ot ecearly obecte. A cue.., the Le erate of a patal eco orer teor obecte. For the Cauchy tre, there are a umber of fferet obecte rate oe ca ue, bae o the Le erate (ee Eq ,..4,..44): Cotter-Rl tre rate Jauma tre rate Olroy tre rate & l l w w L b b # & ( L L ) & l l L # (.5.4) Stre rate of other patal tre teor ca be efe the ame way, for eample the Olroy rate of the Krchhoff tre teor τ& lτ τl. he materal erate of the materal PK tre teor, S &, obecte. he puh forwar of S &, from..9b, # ( S & ) FS& χ (.5.5) * F th ometme calle the cotraarat Olroy tre rate, to tguh t from the Cotter-Rl rate, whch alo ometme calle the coarat Olroy tre rate Sol Mechac Part III 47

34 Secto.5 h puh forwar, cale by the ere of the Jacoba, J F SF & calle the rueell tre rate. h ca be epree term of the Cauchy tre by ug.5.9, a the.5.,.5.5: J F ( JF F ) F. F J J& F F J F & l l tr( ) F JF & F JF F. F (.5.6) hu far, obecte rate hae bee cotructe by pullg back, takg erate a puhg forwar. Oe ca cotruct obecte rate alo by pullg back a puhg forwar wth the rotato teor R oly, ce t the rotato whch caue the tre rate to be o-obecte. For eample, L #, ettg F R,, from.5.7 a..7b, χ [ χ ] # ( ) R ( g ) ( gˆ ) χ* ˆ * * R R R [ ] R( gˆ ) (& R R & R R & ) & Ω R # Ω R R R (.5.7) where Ω RR & R the kew-ymmetrc agular elocty teor.6.. he tre rate.5.7 calle the Gree-Nagh tre rate. From the aboe, the Gree-Nagh rate the puh forwar of the tme erate of the corotatoal tre. Eample Coer aga the eample cue at the e of.5., oly let the plate rotate at cotat agular elocty ω, o co F R ( ω ( ω ( ω co( ω, F& R& ω co ( ω co( ω ( ω ( ω * * Aga, ug the tre traformato rule [ ] [ Q ][ ][ Q] [ R][ ][ R ] a, wth J, Alo, co co, ( ω ( ( ( ( ) ( ) ( )( ) ( ) ( ) ω co ω ω ωt ωt ωt co ωt ( ω ( ω ( ω co( ω co τ, P, S ˆ B Sol Mechac Part III 48

35 Secto.5 l w FF & RR & Ω R ω he a & ω ( ω co( ω ( ω ( ω P & ω, S& ˆ & B co ( ω co( ω( ) ( ( ) ( ))( ) ( ( ) ( ))( ) ( ) ( )( ) co ωt ωt co ωt ωt ωt co ωt For a rg boy rotato, t ca be ee that the efto of the Cotter-Rl, Jauma, Olroy, rueell a Gree-Nagh rate are equalet, a they are all zero: & w w h a epecte ce obecte tre rate for two cofgurato whch ffer by a rg boy rotato wll, by efto, be equal (the tre compoet wll ot chage); they are zero the referece cofgurato a o wll be zero the rotate cofgurato..5.7 Problem. Coer the cae of uaal tre, where a materal wth tal meo legth l, breah w a heght h eform to a compoet wth meo legth l, breah w a heght h. he oly o-zero Cauchy tre compoet, actg the recto of the legth of the compoet. (a) wrte ow the moto equato the materal ecrpto, χ(x) (b) calculate the eformato graet F a cofrm that J et F the rato of the olume the curret cofgurato to that the tal cofgurato (c) Calculate the PK tre. How t relate to the Cauchy tre for th uaal tre-tate? () calculate the PK tre. A materal uergoe the eformato X t, X t X, X he Cauchy tre at a pot the materal t t [ ] t t (a) Calculate the PK a PK tree at the pot (check that PK ymmetrc) Sol Mechac Part III 49

36 Secto.5 (b) Calculate the epreo P : F&, J :, S : E& (for E &, ue the epreo.5.8b, E& F F ). I thee epreo, the rate of eformato teor. (You houl get the ame reult for all three cae, ce they all ge the rate of teral work oe by the tree urg the eformato, per ut referece olume ee later). Show that the Olroy rate of the Krchhoff tre, τ& lτ τl, equal to the Jacoba tme the rueell tre rate of the Cauchy tre,.5.6. Sol Mechac Part III 5

37 Secto.6.6 he Equato of Moto a Symmetry of Stre I Part II,., the Equato of Moto were ere ug Newto Law apple to a fferetal materal elemet. Here, they are ere ug the prcple of lear mometum..6. he Equato of Moto (Spatal Form) Applcato of Cauchy law t a the ergece theorem.4. to..7 lea rectly to the global form of the equato of moto [ b] ρ &, b ρ & (.6.) he correpog local form the b ρ, b ρ Equato of Moto (.6.) he term o the rght calle the ertal, or ketc, term, repreetg the chage mometum. he materal tme erate of the patal elocty fel ( gra) t o t, etc. a t ca be ee that the equato of moto are o-lear the elocte. Equato of Equlbrum Whe the accelerato zero, the equato reuce to the equato of equlbrum, Flow b Equato of Equlbrum (.6.) A flow a et of quatte aocate wth the ytem of force t a b, for eample the quatte,, ρ. A flow teay f the aocate patal quatte are epeet of tme. A potetal flow oe for whch the elocty fel ca be wrtte a the graet of a calar fucto, graφ. A rrotatoal flow oe for whch curl. Sol Mechac Part III 5

38 Secto.6.6. he Equato of Moto (Materal Form) I the patal form, the lear mometum of a ma elemet ρ. I the materal form t ρ. Here, the ame elocty a, oly t ow epree term of the materal coorate X, a ρ ρ. he lear mometum of a collecto of materal partcle occupyg the olume the curret cofgurato ca thu be epree term of a tegral oer the correpog olume the referece cofgurato: ( X) ( ) L( t ) ρ X,t Lear Mometum (Materal Form) (.6.4) a the prcple of lear mometum ow, ug.., ρ ( X) ( X ρ F( ) (.6.5), t he eteral force F to be coere are thoe actg o the curret cofgurato. Suppoe that the urface force actg o a urface elemet the curret cofgurato f urf t S, where t a are, repectely, the Cauchy tracto ector a the PK tracto ector (Eq..5.-4). Alo, ut a the PK tre meaure the actual force the curret cofgurato, but per ut urface area the referece cofgurato, oe ca trouce the referece boy force B: th the actual boy force actg the curret cofgurato, per ut olume the referece cofgurato. hu f the boy force actg o a olume elemet the curret cofgurato f, the he reultat force actg o the boy the boy f boy b B (.6.6) F ( S B, F S B (.6.7) S Ug Cauchy law, PN, where P the PK tre, a the ergece theorem..,.6.5 a.6.7 lea to a the correpog local form [ DP B] ρ (.6.8) P DP B ρ, B ρ X S Equato of Moto (Materal Form) (.6.9) Sol Mechac Part III 5

39 Secto.6 Derato from the Spatal Form he equato of moto ca alo be ere rectly from the patal equato. I orer to o th, oe mut frt how that D( J F ) zero. Oe f that (ug the ergece theorem, Nao formula..59 a the fact that I ) ( JF ) ( JF ) D X S S JF JF NS N S I δ I δ (.6.) h reult kow a the Pola etty. hu, wth the PK tre relate to the Cauchy tre through.5.8, P J F, a ug etty.4.6c, From..8c, DP D D ( ( JF ) ( JF ) Gra : ( JF ) JGra : F (.6.) DP J (.6.) he, wth J a.6.6, the equato of moto the patal form ca ow be traforme accorg to a before. [ b] ρ [ DP B] & ρ &.6. Symmetry of the Cauchy Stre It wll ow be how that the prcple of agular mometum lea to the requremet that the Cauchy tre teor ymmetrc. Applyg Cauchy law to.., k r kl ( ) ε S l ε k r b b k r ρ ε ρ k k (.6.) he urface tegral ca be coerte to a olume tegral ug the ergece theorem. Ug the e otato, a cocetratg o the tegra of the reultg olume tegral, oe ha, ug..4 (the permutato ymbol a cotat here, ε / ), k l Sol Mechac Part III 5

40 Secto.6 ( kl ) kl kl ε k ε k klδ l ε k k r E : (.6.4) l l l where E the thr-orer permutato teor, Eq..9.6, E ε ( e e e ) wth the Reyol traport etty.., { r : } r b ρ ( r ) k k. hu, E (.6.5) he materal erate of th cro prouct a o ( ) r r r r r (.6.6) E : r b ρ (.6.7) From the equato of moto.6., the term e the bracket zero, o that E :, ε k (.6.8) k It follow, from epao of th relato, that the matr of tre compoet mut be ymmetrc:, Symmetry of Stre (.6.9).6.4 Coequece the Materal Form Here, the coequece of.6.9 o the PK a PK tree eame. Ug the reult a.5.8, J PF, o that ( J PF ) FP J PF J (.6.) PF FP, P F F P (.6.) k k k k hee equato are tral whe, ot prog ay cotrat o P. O the other ha, whe oe ha the three equato Sol Mechac Part III 54

41 Secto.6 P P P F F F P P P F F F P P P F F F F F F P P P F F F P P P F F F P P P (.6.) hu agular mometum coerato mpoe thee three cotrat o the PK tre (a they mpoe the three cotrat,, o the Cauchy tre). It ha alreay bee ee that a coequece of the ymmetry of the Cauchy tre the ymmetry of the PK tre S; thu, formally, the ymmetry of S the reult of the agular mometum prcple. Sol Mechac Part III 55

42 Secto.7.7 Bouary Coo a he Bouary alue Problem I orer to ole a mechac problem, oe mut pecfy certa coo arou the bouary of the materal uer coerato. Such bouary coo wll be cue here, together wth the reultg bouary alue problem (BP). (ee Part I,.5., for a cuo of tre bouary coo.).7. Bouary Coo here are two type of bouary coo, thoe o placemet a thoe o tracto. Deote the boy the referece coo by B a the curret cofgurato by B. Deote the bouary of the boy the referece cofgurato by S a the curret cofgurato by, Fg..7.. Dplacemet Bouary Coo he poto of partcle may be pecfe oer ome porto of the bouary the curret cofgurato. hat, χ( X) pecfe to be ay, oer ome porto u of, Fg..7., whch correpo to the porto S u of S. Wth u( ) X( ), or U( X) ( X) X, th ca be epree a u( ) u( ), U( X) U( X), u X S u (.7.) hee are calle placemet bouary coo. he mot commoly ecoutere placemet bouary coo where ome porto of the bouary fe, u. whch cae ( ) o S u u u S u U U S t t X B S B Fgure.7.: Bouary coo Sol Mechac Part III 56

43 Secto.7 racto Bouary Coo racto t t ca be pecfe oer a porto of the bouary, Fg..7.. hee tracto bouary coo are relate to the PK tracto oer the correpog urface S the referece cofgurato, through Eq..5.-4, S PNS t (.7.) Oe uually kow the poto of the bouary S a the ormal N (X) the referece cofgurato. A eformato procee, the PK tracto eelop accorg to PN wth, from.5.8, P J F. he PK tre wll geeral epe o the moto a the eformato graet F, o the tracto bouary coo ca be epree the geeral form Eample: Flu Preure ( X, F), (.7.) Coer the cae of flu preure p arou the bouary, t p, Fg..7.. he Cauchy tracto t epe through the ormal o the ew poto a geometry of the urface. Alo, pj F N, whch of the geeral form.7.. X, h p θ t p X, E,e a E,e Fgure.7.: Flu preure o eformg materal Coer a materal uer water wth part of t urface eformg a how Fg..7.. Referrg to the fgure, N E, coθ e θe, pi, p ρ g( h ) a X a X ta θ X X, taθ F, J et F he tracto ector a PK tre are Sol Mechac Part III 57

44 Secto.7 t ρg coθ ( h ) θ, ρg( h X ) taθ, P ρg( h ) ta θ wth (ote that S / coθ ) t p a p / coθ. he tracto ector clearly epe o both poto, a the eformato through θ. I th eample, grau F I GraU I F taθ e e a θ ( gra u) arcta grau arcta grau : grau Dea Loag A pecal cae of loag that of ea loag, where ( X) (.7.4) Here, the PK tre o the bouary oe ot chage wth the eformato a a tally ormal tracto wll ot rema o a eformato procee. For eample, f oe coer aga the geometry of Fg..7., th tme take he ( X) PN pn ρ g, ρ X ( h X ) P( X) g( h )I t ( θ ) coθ ρg( h ), (, θ ) ρg( h ) ta, θ.7. he Bouary alue Problem he equato of moto.6.,.6.9, are a et of three fferetal equato. I the oluto of ay problem, oe woul hae to upplemet thee equato wth other, for eample a cottute equato epreg a relatohp betwee the tre a the kematc arable (ee Part I). h cottute relato wll typcally relate the tre to the tra, or rate of tra, for eample f ( e, ). Suppoe the that the tree are kow term of the tra a hece the placemet u. he equato of moto are the a et of three eco orer fferetal equato the three ukow u (aumg that the boy force b a precrbe fucto of the problem). hey ee to be ubecte to certa bouary a tal coo. Sol Mechac Part III 58

45 Secto.7 Aume that the bouary coo are uch that the placemet are pecfe oer that part of the urface u a tracto are pecfe oer that part, wth the total urface u, wth. hu u t u t, o u, o u Bouary Coo (.7.5) where the oerbar gfe quatte whch are precrbe. Ital coo are alo requre for the placemet a elocty, o that u(, u u& (, u& ( ), ( ), at t at t Ital Coo (.7.6) a t uually take that that u u, u X at t. Comparg.7.5 a.7.6, oe alo requre & u oer u, o that the bouary a tal coo are compatble. hee equato together, the fferetal equato of moto a the bouary a tal coo, are calle the trog form of the tal bouary alue problem (BP): b ρ& ρu&& t t, o u u, o u(, u u& (, u& ( ), ( ), u at t at t Strog form of the Ital BP (.7.7) Whe the problem qua-tatc, o the accelerato ca be eglecte, the equato of moto reuce to the equato of equlbrum.6.. I that cae oe oe ot ee tal coo a oe ha a bouary alue problem olg.7.5 oly. It oly certa pecal cae a certa mple problem that a eact oluto ca be obtae to thee equato. A alterate oluto trategy to coert thee equato to what kow a the weak form. he weak form, whch the form of tegral rather tha fferetal equato, ca the be ole appromately ug a umercal techque, for eample the Fte Elemet Metho. he weak form cue.9. It poble to pecfy both tracto a placemet oer the ame porto of the bouary, but ot the ame compoet. For eample, f oe pecfe t t e o a bouary, oe coul alo pecfy u u e, but ot u u e. I that cae, oe coul mage the bouary to cot of two eparate bouare, oe wth coo wth repect to e a oe wth repect to e, a tll wrte u. Further, t ofte eaer to proe reult regarg the uquee a tablty of oluto to the problem whe t cat the weak form Sol Mechac Part III 59

46 Secto.7 I the materal form, the bouary coo are PN, o S U U, o S u Bouary Coo (.7.8) a the tal coo are U( X, U U& ( X, U& a the tal ale problem ( X), ( X), at t at t Ital Coo (.7.9) DP B ρ & ρu&& PN, U U, U( X, U U& ( X, U& ( X), ( X), o S o S u at t at t Strog form of the Ital BP (.7.) Sol Mechac Part III 6

47 Secto.8.8 Balace of Mechacal Eergy.8. he Balace of Mechacal Eergy Frt, from Part I, Chapter 5, recall work a ketc eergy are relate through W et W t ΔK (.8.) where W et the work of the eteral force a W t the work of the teral force. he rate form P P K& et t (.8.) where the eteral a teral power a rate of chage of ketc eergy are P W P W K& et et, t t, ΔK (.8.) h epree the mechacal eergy balace for a materal. Eq..8. equalet to the equato of moto (ee below). he total eteral force actg o the materal ge by..6: F et t b (.8.4) he cremet work oe W whe a elemet ubecte to a boy force (per ut olume) b uergoe a placemet u b u. he rate of workg P b ( u / ). hu, a mlarly for the tracto, the power of the eteral force Pet t b (.8.5) where the elocty. Alo, the total ketc eergy of the matter the olume Ug Reyol traport theorem, K ρ (.8.6) K ( ) ρ ρ (.8.7) hu the epreo.8. become Sol Mechac Part III 6

48 Secto.8 t b Pt ρ (.8.8) power of urface force power of boy force power of teral force rate of chage of ketc eergy It ca be ee that ome of the power eerte by the eteral force alter the ketc eergy of the materal a the remaer chage t teral eergy tate. Coerate Force Sytem I the pecal cae where the teral force are coerate, that, o eergy pate a heat, but all eergy tore a teral eergy, oe ca epre the power of the teral force term of a potetal fucto u (ee Part I, 5.), a rewrte th equato a t b u ρ ρ (.8.9) rate of chage of teral eergy Here, the rate of chage of the teral eergy ha bee wrtte the form U u u ρ ρ (.8.) where u the teral eergy per ut ma, or the pecfc teral eergy..8. he Stre Power o epre the power of the teral force P t term of tree a tra-rate, frt re-wrte the rate of chage of ketc eergy ug the equato of moto, K ρ ( b) (.8.) Alo, ug the prouct rule of fferetato, ( ) ( ) : l, (.8.) Sol Mechac Part III 6

49 Secto.8 where l the patal elocty graet, l /. Decompog l to t ymmetrc part, the rate of eformato, a t atymmetrc part w, the p teor, ge : l : : w :, (.8.) ce the ouble cotracto of ay ymmetrc teor ( ) wth ay kew-ymmetrc teor (w) zero,..c. Alo, ug Cauchy law a the ergece theorem.4., t t k k ( ) ( ) ( k ) k (.8.4) hu, fally, from Eq..8.8, P t P :, t Stre Power (.8.5) he term : calle the tre power; the tre power the (egate of the) rate of workg of the teral force, per ut olume. he complete equato for the coerato of mechacal eergy the t b : ρ Mechacal Eergy Balace (.8.6) he tre power that part of the eterally upple power whch ot coerte to ketc eergy; t coerte to heat a a chage teral eergy. Note that, a wth the law of coerato of mechacal eergy for a partcle, th equato oe ot epre a eparate law of cotuum mechac; t merely a rearragemet of the equato of moto (ee below), whch themele follow from the prcple of lear mometum (Newto eco law). Coerate Force Sytem If the teral force are coerate, oe ha or, local form, u : U ρ (.8.7) Sol Mechac Part III 6

50 Secto.8 u : ρ Mechacal Eergy Balace (Coerate Sytem) (.8.8) h the local form of the eergy equato for the cae of a purely mechacal coerate proce..8. Derato from the Equato of Moto A metoe, the coerato of mechacal eergy equato ca be ere rectly from the equato of moto. he erato mlar to that ue aboe (where the mechacal eergy equato were ue to ere a epreo for the tre power ug the equato of moto). Oe ha, multplyg the equato of moto by a tegratg, ρ ( b) { ( ) : l b} { ( ) : b} : t b (.8.9).8.4 Stre Power a the Cotuum Elemet I the aboe, the tre power wa ere ug a global (tegral) form of the equato. he tre power ca alo be euce by coerg a fferetal ma elemet. For eample, coer uch a elemet whoe bouary partcle are mog wth elocty a whoe bouary ubecte to tree, Fg..8.. Coer frt the compoet of force a elocty actg the recto. he eteral force act o the e. O three of them (the oe that ca be ee the llutrato) the tre a elocty act the ame recto, o the power pote; o the other three they act oppote recto, o there the power egate. Δ Δ (,, ) Δ Sol Mechac Part III 64

51 Secto.8 Sol Mechac Part III 65 Fgure.8.: A fferetal ma elemet ubecte to tree A uual (ee.6.6), the elemet aume to be mall eough o that the prouct of tre a elocty are learly oer the elemet, o that the aerage of th prouct oer a elemet face ca be take to be repreetate of the power of the urface force o that elemet. he power of the eteral urface force actg o the three face to the frot the ( ) ( ) ( ),,,,,, urf P Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ (.8.) Ug a aylor ere epao, a eglectg hgher orer term, the lea to ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ,,,,,, urf P (.8.) he et power per ut olume (ubtractg the power of the tree o the other three urface a g through by the olume) the ( ) ( ) ( ) ( ) P urf (.8.) Aume the boy force b to act at the cetre of the elemet. Neglectg hgher orer term whch ah a the elemet ze allowe to hrk towar zero, the power of the boy force the recto, per ut olume, mply b. he total power of the eteral force the (clug the other two compoet of tre a elocty), ug the equato of moto, ( ) ( ) ( ) ( ) P b b b b P b b l b l b : : : et et ρ ρ ρ ρ (.8.)

52 Secto.8 whch aga equal the tre power term plu the chage ketc eergy. he power of the teral force :, a reult of the force actg e the fferetal elemet, reactg to the apple force a b..8.5 he Balace of Mechacal Eergy (Materal form) he materal form of the power of the eteral force wrtte a a fucto of the PK tracto a the referece boy force B,.6.7, a the ketc eergy a a fucto of the elocty (X) : S B Pt ρ (.8.4) S Net, ug the ette.5.4, & a..h, ( A : ( BC) AC ) : B F lf, ge ( FF & ) ( F ): F& : : l : w : l :, (.8.5) a o a : ( F ) F& : ( F ) P : F& : F& J (.8.6) S S B P : F& ρ Mechacal Eergy Balace (Materal Form) (.8.7) For a coerate ytem, th ca be wrtte term of the teral eergy S u S B ρ ρ (.8.8).8.6 Work Cougate arable Sce the tre power the ouble cotracto of the Cauchy tre a rate-ofeformato, oe ay that the Cauchy tre a rate of eformato are work cougate (or power cougate or eergy cougate). Smlarly, from.8.6, the PK tre P power cougate to F &. It ca alo be how that the PK tre S power cougate to the rate of Euler-Lagrage tra, E & (a hece alo the rght Cauchy-Gree tra) { Problem } : Sol Mechac Part III 66