2.12 Pull Back, Push Forward and Lie Time Derivatives

Transcription

1 Secton Pull Bck Push Forwrd nd e me Dertes hs secton s n the mn concerned wth the follown ssue: n oserer ttched to fxed sy Crtesn coordnte system wll see mterl moe nd deform oer tme nd wll osere rous ectorl nd tensorl qunttes to chne lso. Howeer hypothetcl oserer ttched to the deformn mterl nd mon nd deformn wth the mterl wll see somethn dfferent. he queston s: wht qunttes wll e seen to chne from ths emedded oserer s ewpont? 2.2. me Dertes of Sptl Felds In terms of the sptl ss sptl ector cn e expressed n terms of the cornt components nd contrrnt components (2.2.) We wnt to dstnush etween two qunttes. he frst s the mterl tme derte of the ector : (2.2.2) he second s the tme derte holdn the se ectors fxed (2.2.3) hs ltter s clled the conected derte nd s the rte of the chne s seen y n oserer ttched to the deformn ses. From Eqn the components of cn e expressed s (2.2.4) kn the mterl tme derte nd usn Eqns l l (2.2.5) hus there re two conected dertes of ector dependn on whether one s usn cornt or contrrnt components: Sold Mechncs Prt III 307

2 Secton 2.2 l l (2.2.6) s wll e seen elow these qunttes re e dertes of the ector. he tme derte of the components cn e expressed n n lternte wy y expressn the sptl se ectors n terms of the mterl se ectors ; usn Eqns : F F F F (2.2.7) So s n lternte to Eqns F F (2.2.8) s wll e seen further elow the qunttes on the rht re the mterl tme dertes of the pull-ck of the ector. epetn the oe now for sptl tensor : n terms of the sptl ss cn e expressed n terms of the cornt components nd contrrnt components s he mterl tme derte of the tensor s (2.2.9) (2.2.0) nd the conected derte s the frst term: he components of cn e expressed s (2.2.) Sold Mechncs Prt III 308

3 Secton 2.2 (2.2.2) kn the mterl tme derte nd n usn Eqns l l he conected dertes re thus l l l l (2.2.3) ll ll (2.2.4) s wll e seen elow these qunttes re e dertes of the tensor. he tme derte of the components cn e expressed n n lternte wy y expressn the sptl se ectors n terms of the mterl se ectors ; usn Eqns : FF FF F F F F (2.2.5) So s n lternte to Eqns FF F F (2.2.6) s wll e seen next the qunttes on the rht re the mterl tme dertes of the pullck of the tensor. Exmple Consdern n Exmple whch ws worked throuh n detl n 2.0 suppose we he shern deformton s shown n F (ths s F ). Sold Mechncs Prt III 309

4 Secton X x E2 e2 E e X x Fure 2.2.: pure sher deformton of one prllelorm nto nother et the sher nle n F eole oer tme ccordn to t (2.2.7) From Eqns the rtes of chne of the se ectors re d d d d e 0 e e e 2 dt dt dt dt tn t sn t 2 2 d d d 2 d e e2 e 2 2 e20 dt dt tn t sn t dt dt (2.2.8) he elocty rdent s from Eqn l 2 2 e e 2 sn e (2.2.9) where s en y Eqn nd d t (2.2.20) 2 dt tn t tn sn t Sold Mechncs Prt III 30

5 Secton 2.2 Consdern n the ector of Eqn x y deformed ector of Eqn e E nd ts correspondn x y y y 0 e (2.2.2) he contrrnt nd cornt components of re y y 0 (2.2.22) y tn he ht on the s to emphsse tht (see Eqns ) (2.2.23) From Eqns the conected dertes re y 0 x y l y 0 0 y 0 0x y l 0 0 y y x y (2.2.23) hus l 0 whch from Eqn mples tht 0. hs s the expected result: the contrrnt components do not chne oer tme. hey re lwys x y tn y s en y Eqn Consder now n exmple tensor xx xy yx E (2.2.24) yy he cornt nd contrrnt components re Sold Mechncs Prt III 3

6 Secton 2.2 yx yy yy tn xx xy xx tn tn tn tn xx xy yx 2 yy xy yy tn tn tn yx xx yy xy yx 2 xx hs deforms to (wth F en y Eqn ) (2.2.25) xx xy xx yx xy yy 0 yx yy yx e (2.2.25) yy Now F mn m n (2.2.26) Conertn etween the rous conected se ectors usn Eqns the contrrnt nd cornt nd components re : lso xx xy yx yy xy yy tn tn tn tn tn yx yy yy tn xx yx xy yy xx yx tn yx xx yx yy yx xy yy 2 xx yx tn tn tn tn (2.2.27) nd the contrrnt nd cornt components re yx yy 0 0 e (2.2.28) Sold Mechncs Prt III 32

7 Secton 2.2 yx yy yy tn 0 0 yx yx yy tn yx yx yy tn tn tn (2.2.29) n the ht emphsses tht (see Eqns ) Now (2.2.30) l l 0 xy yy yy 0 l l yx xx yx yy xx yx xy yx yy (2.2.3) hus ll 0.e. 0 only when 0 whch s consstent wth Eqn (only constnt terms ndependent of remn n tht cse). xy yy Push-Forwrd nd Pull-Bck Next re defned the push-forwrd nd pull-ck of ectors nd tensors whch wll led nto the concept of e dertes whch relte ck to wht ws ust dscussed oe rerdn conected dertes. ectors Consder ector en n terms of the reference confurton se ectors: (2.2.32) he push-forwrd symolsed y s defned to e the ector wth the sme components ut wth respect to the current confurton se ectors. here re 2 push- Sold Mechncs Prt III 33

8 Secton 2.2 forwrd opertons dependn on the type of components used; the symol s used for cornt components nd the symol for contrrnt components ; usn F F F F Push-forwrd of ector (2.2.33) Eqn sys tht the push forwrd of the contrrnt form of s smply F. In other words the push forwrd here s the ctul correspondn ector n the deformed F nd s consequence of the defntons s confurton llustrted n F F 2 Fure 2.2.2: he push-forwrd of ector specl cse of Eqn s the push forwrd of lne element n the reference confurton n the correspondn lne element n the current confurton: dx d dx. (2.2.34) Smlrly consder ector en n terms of the current confurton ss: he pull-ck of respect to the reference confurton se ectors (2.2.35) s defned to e the ector wth components (or (or ). sn ) wth F F - F F Pull-ck of ector (2.2.36) nd for lne element n the current confurton dx dx F dx dx. (2.2.37) Sold Mechncs Prt III 34

9 Secton 2.2 Note tht push-forwrd nd pull-ck ppled successely to ector wth the sme component type wll result n the ntl ector. From the oe for two mterl ectors nd nd two sptl ectors u nd u u u (2.2.38) For exmple s specl cse of ths n the reference confurton nd 2 re 2 perpendculr:. Pushn forwrd these ectors we et from Eqn : F nd F nd n 0. ensors Consder mterl tensor : (2.2.39) s for the ector the push-forwrd of s defned to e the tensor wth the sme components ut wth respect to the deformed se ectors. hus usn F F F F F F FF F F FF F F F F Push-forwrd of ensor (2.2.40) Smlrly consder sptl tensor : (2.2.4) he pull-ck s Sold Mechncs Prt III 35

10 Secton F F FF F F F F F F F F F F FF Pull-ck of ensor (2.2.42) he frst of these FF s clled the cornt pull-ck wheres the second clled the contrrnt pull-ck. F F s me Dertes It wll e reconsed tht the expressons for the pull cks of sptl cornt tensor nd sptl contrrnt tensor n Eqns re those ppern n Eqns Keepn n mnd Eqn one sees tht for sptl tensor n terms of cornt components nd contrrnt components FF ll F F l l (2.2.43) Other Push-Forwrd nd Pull-Bck reltons for ectors nd ensors Here follow some reltons noln the push-forwrd nd pull-cks of tensors. For two mterl tensors nd B nd two sptl tensors nd the sclr product s : B : B B B B (2.2.44) hs sclr product then push-forwrds nd pull-cks s { Prolem } : B : : B : B : B : B : : : : (2.2.45) Sold Mechncs Prt III 36

11 Secton 2.2 Sold Mechncs Prt III 37 For mterl tensor nd mterl ectors nd sptl tensor nd sptl ectors u u u u u u (2.2.46) hen u u u u u (2.2.47) For mterl tensor nd mterl ector nd sptl tensor nd sptl ector the contrctons nd re (2.2.48) nd so trnsform s (2.2.49) Fnlly for mterl tensors B nd sptl tensors k k k k k k k k k k k k k k k k B B B B B (2.2.50) nd so

12 Secton 2.2 B B B B B B (2.2.5) Push-Forwrd nd Pull-Bck opertons for Strn ensors he push-forwrd of the cornt rht Cuchy-reen strn nd ts contrrnt nerse re C C F CF C C FCF. (2.2.52) From C the cornt components of the dentty tensor expressed n terms of the conected se ectors n the current confurton.e. the sptl metrc tensor nd C the contrrnt components of. hus the push-forwrd of cornt C s nd the pull-ck of cornt s C nd the push-forwrd of contrrnt C s nd the pull-ck of contrrnt s C : C C C C (2.2.53) Push-forwrd of the rht Cuchy-reen strn Smlrly the pull-ck of cornt s nd the push-forwrd of cornt s nd the pull-ck of contrrnt s nd the push-forwrd of contrrnt s. (2.2.54) Pull-ck of the left Cuchy-reen strn For the cornt form of the reen-rne strn the push-forwrd s E E F EF. (2.2.55) From E e the cornt components of the Euler-lmns strn tensor nd so the push-forwrd of cornt E s e nd the pull-ck of cornt e s E. Sold Mechncs Prt III 38

13 Secton 2.2 Sold Mechncs Prt III 39 E e e E. (2.2.56) Push-forwrd of the reen-rne strn Pull-ck of the Euler-lmns strn Push-Forwrd nd Pull-Bck wth Polr Decomposton Intermedte Confurtons Pull cks nd push-forwrds cn e defned relte to ny two confurtons. Consder the polr decomposton nd the ntermedte confurtons dscussed n 2.0 (see F. 2.0.). Effectely we re replcn F wth : pushn forwrd mterl tensor from the reference confurton to the confurton Ĝ leds to. (2.2.57) Note tht the result s the sme rerdless of whether one s usn the cornt contrrnt or mxed forms. Smlrly the pull ck of tensor  from the ntermedte confurton Ĝ to the reference confurton s (2.2.58) he push-forwrd of tensor â from ĝ to nd the correspondn pull-ck of sptl tensor s (2.2.59)

14 Secton 2.2 Sold Mechncs Prt III 320 he push-forwrds nd pull-cks due to the stretch tensors re. (2.2.60) (2.2.6) nd (2.2.62) Push-forwrds nd pull-cks cn lso e defned usn F (n the plce of F) nd these moe etween the ntermedte confurtons. ecll Eqn whch stte tht the cornt components of wth respect to the ses respectely re equl. hs cn e explned lso n terms of push-forwrds nd pull-cks. For exmple wth nd one cn wrte (n fct these reltons re ld for ll component types) (2.2.63) he frst of these shows tht the components of wth respect to re the sme s those of wth respect to Ĝ (for ll component types). he second shows tht the components of wth respect to ĝ re the sme s those of wth respect to. s nother exmple wth 2 C C C (2.2.64)

15 Secton he e me Derte he e (tme) derte s concept of tensor nlyss whch s used to dstnush etween the chne n some quntty nd the chne n tht quntty excludn chnes due to the motonconfurton chnes. s mentoned n the ntroducton to ths secton we cn mne hypothetcl oserer ttched to the deformn mterl who moes nd deforms wth the mterl. hs oserer wll see no chne n the confurton tself 0. Howeer they wll stll see chnes to ectors nd tensors. hese chnes re mesured usn the e Derte whch wll e seen to e none other thn the conected derte dscussed oe. ectors Frst the e (tme) derte of ector s the mterl derte holdn the deformed ss constnt tht s Eqns : (2.2.65) Formlly t s defned n terms of the pull-ck nd push-forwrd d dt he e me Derte (2.2.66) hs s llustrted n the F he sptl ector s frst pulled ck to the reference confurton there the dfferentton s crred out where the se ectors re constnt then the ector s pushed forwrd n to the sptl descrpton. d dt ( ) d dt ( ) ( ) Fure 2.2.3: he e Derte Sold Mechncs Prt III 32

16 Secton 2.2 For cornt components one frst pulls ck the ector to the derte s tken nd then t s pushed forwrd to whch s consstent wth the defnton he defnton llows one to clculte the e derte n solute notton: usn d d F dt dt F F F F F F l F l (2.2.67) he e derte for the contrrnt components cn e clculted n smlr wy nd n summry (these re smply Eqns ): { Prolem 2} l l e Dertes of ectors (2.2.68) ensors he mterl tme derte of sptl tensor s (2.2.69) he e (tme) derte s then (2.2.70) For exmple for cornt components one frst pulls ck the tensor the derte s tken hus usn to nd then t s pushed forwrd to. Sold Mechncs Prt III 322

17 Secton 2.2 d dt F F F F F F F F F F l FF F F lf F l l (2.2.7) he e derte for the other components cn e clculted n smlr wy nd n summry (these re Eqns ): { Prolem 3} l l l l l l l l e Dertes of ensors (2.2.72) he frst of these l l s clled the Cotter-ln rte. he second of these ll s lso clled the Oldroyd rte. e Dertes of Strn ensors From d e l e el l l 0 (2.2.73) nd so the e derte of the cornt form of the Euler-lmns strn s the rte of deformton nd the e derte of the contrrnt form of the left Cuchy-reen tensor s zero. Further from the e derte of the metrc tensor s the push forwrd of the mterl tme derte of the rht Cuchy-reen strn: lso drectly from 2..5 C (2.2.74) 2d (2.2.75) Corottonl tes he e dertes n were dered usn pull-cks nd push-forwrds etween the reference confurton nd the current confurton. If nsted we relte qunttes to the Sold Mechncs Prt III 323

18 Secton 2.2 rotted ntermedte confurton n other words use nsted of F n the clcultons we fnd tht usn Eqn Ω d dt d dt Ω Ω (2.2.76) hs s clled the reen-nhd rte. ther thn pulln ck from the ntermedte confurton to the reference confurton we cn choose the current confurton to e the reference confurton. ottn from ths confurton (see secton 2.6.3) Ω w the spn tensor nd one otns the Jumnn rte ww. e Dertes nd Oecte tes he concept of oectty ws dscussed n secton 2.8. Essentlly f two oserers re rottn relte to ech other wth rotton Q t nd oth re osern some sptl tensor s mesured y one oserer nd s mesured y the other then ths tensor s oecte proded QQ for ll Q.e. the mesurement of the deformton would e ndependent of the oserer. One of the most mportnt uses of the e derte s tht e dertes of oecte sptl tensors re oecte sptl tensors. hus the rtes en n re ll oecte. For exmple suppose we he n oecte sptl tensor.e. so tht QQ. he elocty rdent s not oecte nd nsted stsfes the relton : l QlQ QQ. sn the propertes of the trnspose the orthoonlty of Q nd the dentty QQ QQ one hs for Eqns Sold Mechncs Prt III 324

19 Secton 2.2 l l l l QQ QlQ QQ QQ QQ QlQ QQ QQ QQ QQ Ql Q QQ QQ QQ QQ QlQ QQ QQ Q l lq QQ QlQ QQ QQ QQ QlQ QQ QQ QQ QQ QlQ QQ QQ Ql Q QQ QQ Q l l Q QQ QQ (2.2.77) shown tht these rtes re ndeed oecte. Further ny lner comnton of them s oecte for exmple 2 l l l l l l l l w w 2 (2.2.78) s oecte proded s. hs s the Jumnn rte ntroduced n Eqn nd mentoned fter Eqn oe. Further s mentoned fter Eqn the Cotter- ln rte of Eqn s equlent to. he e Derte nd the Drectonl Derte ecll tht the mterl tme derte of tensor cn e wrtten n terms of the drectonl derte Hence the e derte cn lso e expressed s f (2.2.79) nd hence the suscrpt on the. hus one cn sy tht the e derte s the push forwrd of the drectonl derte of the mterl feld n the drecton of the elocty ector Prolems Sold Mechncs Prt III 325

20 Secton 2.2. Eqns follow mmedtely from Howeer use Eqns e. F F etc. drectly to erfy reltons Dere the e dertes of ector Eqns Dere the e dertes of tensor Eqns Sold Mechncs Prt III 326