1.9 Cartesian Tensors

Transcription

1 Scton.9.9 Crtsn nsors s th th ctor, hghr ordr) tnsor s mthmtc obct hch rprsnts mny physc phnomn nd hch xsts ndpndnty of ny coordnt systm. In ht foos, Crtsn coordnt systm s sd to dscrb tnsors..9. Crtsn nsors scond ordr tnsor nd th ctor t oprts on cn b dscrbd n trms of Crtsn componnts. For xmp, b) c, th, b nd c, s b) c b c) 4 Exmp h Unt Dydc or Idntty nsor) h dntty tnsor, or nt tnsor, I, hch mps ry ctor onto tsf, hs bn ntrodcd n th pros scton. h Crtsn rprsntton of I s hs foos from.9.) Not so tht th dntty tnsor cn b rttn s I, n othr ords th Kroncr dt gs th componnts of th dntty tnsor n Crtsn coordnt systm. Scond Ordr nsor s Dydc In ht foos, t b shon tht scond ordr tnsor cn ys b rttn s dydc nong th Crtsn bs ctors. Consdr n rbtrry scond-ordr tnsor hch oprts on to prodc b, or ) b. From th nrty of, ) b, ths cn b gnrsd to th cs of non-crtsn bs ctors, hch mght not b orthogon nor of nt mgntd s.6) Sod Mchncs Prt III 75 Ky

2 Scton.9 Sod Mchncs Prt III Ky 76 b ) ) ) Jst s trnsforms nto b, t trnsforms th bs ctors nto som othr ctors; sppos tht ), ), ), thn b nd so.9.) hch s ndd dydc. Crtsn componnts of Scond Ordr nsor h scond ordr tnsor cn b rttn n trms of componnts nd bs ctors s foos: rt th ctors, nd n.9.) n componnt form, so tht Introdc nn scrs by ttng,,, so tht Scond-ordr Crtsn nsor.9.) hs nn scrs r th componnts of th scond ordr tnsor n th Crtsn coordnt systm. In ndx notton, hs hrs ctor hs thr componnts, scond ordr tnsor hs nn componnts. Smry, hrs th thr ctors form bss for th spc of ctors, th nn dyds from bss for th spc of tnsors,.. scond ordr tnsors cn b xprssd s nr combnton of ths bss tnsors. It cn b shon tht th componnts of scond-ordr tnsor cn b obtnd drcty from { Probm } Componnts of nsor.9.4)

3 Scton.9 hch s th tnsor xprsson nogos to th ctor xprsson. Not tht, n Eqn..9.4, th componnts cn b rttn smpy s thot dot ), snc. Exmp h Strss nsor) Dfn th trcton ctor t ctng on srfc mnt thn mtr to b th forc ctng on tht mnt ddd by th r of th mnt, Fg..9.. Lt n b ctor norm to th srfc. h strss σ s dfnd to b tht scond ordr tnsor hch mps n onto t, ccordng to t σn h Strss nsor.9.5) x n t t x x Fgr.9.: strss ctng on pn If on no consdrs coordnt systm th bs ctors, thn for xmp, σ σ nd, hs th componnts, nd of th strss tnsor r th thr componnts of th trcton ctor hch cts on th pn th norm. gstn-los Cchy s th frst to rgrd strss s nr mp of th norm ctor onto th trcton ctor; hnc th nm tnsor, from th Frnch for strss, tnson. ths forc od b d, for xmp, to ntrmocr forcs thn th mtr: th prtcs on on sd of th srfc mnt xrt forc on th prtcs on th othr sd Sod Mchncs Prt III 77 Ky

4 Scton.9 Hghr Ordr nsors h bo cn b gnrsd to tnsors of ordr thr nd hghr. h foong notton b sd:,, 0th-ordr tnsors scrs ), b, c st-ordr tnsors ctors ), B, C nd-ordr tnsors dydcs ), B, C rd-ordr tnsors trdcs ), B, C 4th-ordr tnsors ttrdcs ) n mportnt thrd-ordr tnsor s th prmtton tnsor, dfnd by E.9.6) hos componnts r thos of th prmtton symbo, Eqns forth-ordr tnsor cn b rttn s.9.7) It cn b sn tht zroth-ordr tnsor scr) hs 0 componnt, frst-ordr tnsor hs componnts, scond-ordr tnsor hs 9 componnts, so hs 7 componnts nd hs 8 componnts..9. Smp Contrcton nsor/ctor oprtons cn b rttn n componnt form, for xmp,.9.8) hs oprton s cd smp contrcton, bcs th ordr of th tnsors s contrctd to bgn thr s tnsor of ordr nd tnsor of ordr, nd to nd thr s tnsor of ordr t s cd smp to dstngsh t from dob contrcton s bo). hs s ys th cs hn tnsor oprts on nothr n ths y, th ordr of th rst b to ss thn th sm of th orgn ordrs. n xmp of smp contrcton of to scond ordr tnsors hs rdy bn sn n Eqn..8.4; th tnsors thr r smp tnsors dyds). Hr s nothr xmp: Sod Mchncs Prt III 78 Ky

5 Scton.9 S S S S S.9.9) From th bo, th smp contrcton of to scond ordr tnsors rsts n nothr scond ordr tnsor. If on rts S, thn th componnts of th n tnsor r rtd to thos of th orgn tnsors throgh S. Not tht, n gnr, B B BC BC ssoct.9.0) B C B dstrbt C h ssoct nd dstrbt proprts foo from th fct tht tnsor s by dfnton nr oprtor,.8.; thy ppy to tnsors of ny ordr, for xmp, B B.9.) o d th tnsors of ny ordr, on hs to rmmbr s ho smp tnsors oprt on ch othr th to ctors hch r bsd ch othr r th ons hch r dottd togthr: bc b c bc d b c d bc d b c d b cd f c d b f.9.) n xmp nong hghr ordr tnsor s E E E mn n n m n nd B C b C B C Not th rton { Probm 0} Sod Mchncs Prt III 79 Ky

6 Scton.9 B CD B CD.9.) Pors of nsors Intgr pors of tnsors r dfnd ndcty by xmp, 0 I, n n, so, for h Sqr of nsor.9.4), tc..9. Dob Contrcton Dob contrcton, s th nm mps, contrcts th tnsors tc s mch smp contrcton. hs, hr th sm of th ordrs of to tnsors s rdcd by to n th smp contrcton, th sm of th ordrs s rdcd by for n dob contrcton. h dob contrcton s dnotd by coon :),.g. : S. Frst, dfn th dob contrcton of smp tnsors dyds) throgh b c d cb d :.9.5) So n dob contrcton, on ts th scr prodct of for ctors hch r dcnt to ch othr, ccordng to th foong r: For xmp, b c: d f b dc f S : S : S S.9.6) hch s, s xpctd, scr. Hr s nothr xmp, th contrcton of th to scond ordr tnsors I s Eqn..9.) nd, I : :.9.7) Sod Mchncs Prt III 80 Ky

7 Scton.9 so tht th scr prodct of to ctors cn b rttn n th form of dob contrcton nong th Idntty nsor. n xmp of dob contrcton nong th prmtton tnsor.9.6 s { Probm } E :.9.8) It cn b shon tht th componnts of forth ordr tnsor r gn by compr th Eqn..9.4) : :.9.9) In smmry thn, : B : b : B c : B C Not th foong dntts: B: C B : C B : C : B C C : B : BC B: C D B : C D DB : C.9.0) Not: hr r mny oprtons tht cn b dfnd nd prformd th tnsors. h to most mportnt oprtons, th ons hch rs most n prctc, r th smp nd dob contrctons dfnd bo. Othr possbts r: ) dob contrcton th to horzont dots, S, b, tc., hch s bsd on th dfnton of th foong oprton s ppd to smp tnsors: bcdf bcdf b) oprtons nong on cross : b cd d bc c) dob oprtons nong th cross nd dot: b cdcbd b cdcbd b cdcbd.9.4 Indx Notton h ndx notton for sng nd dob contrcton of tnsors of ny ordr cn sy b rmmbrd. From th bo, sng contrcton of to tnsors mps tht th ndcs Sod Mchncs Prt III 8 Ky

8 Scton.9 bsd ch othr r th sm, nd dob contrcton mps tht pr of ndcs s rptd. hs, for xmp, n both symboc nd ndx notton: B C : B c m B B m C c.9.).9.5 Mtrx Notton Hr th mtrx notton of.4 s xtndd to ncd scond-ordr tnsors 4. h Crtsn componnts of scond-ordr tnsor cn connnty b rttn s mtrx, h oprtons nong ctors nd scond-ordr tnsors cn no b rttn n trms of mtrcs, for xmp, symboc notton short mtrx notton f mtrx notton h tnsor prodct cn b rttn s s.4.).9.) hch s consstnt th th dfnton of th dydc trnsformton, Eqn..8.. compr th th bsd ch othr r for mtrx mtpcton gn n.4. 4 th mtrx notton cnnot b sd for hghr-ordr tnsors Sod Mchncs Prt III 8 Ky

9 Scton Probms. Us Eqn..9. to sho tht th componnt of tnsor cn b td from, nd tht nd so on, so tht ).. Et sng th ndx notton for Crtsn bss). Wht s ths oprton cd? Is yor rst q to, n othr ords s ths oprton commtt? No crry ot ths oprton for to ctors,.. b. Is t commtt n ths cs?. Et th smp contrctons b nd B, th rspct to Crtsn coordnt systm s ndx notton). 4. Et th dob contrcton : B s ndx notton). 5. Sho tht, sng Crtsn coordnt systm nd th ndx notton, tht th dob contrcton : b s scr. Wrt ths scr ot n f n trms of th componnts of nd b. 6. Consdr th scond-ordr tnsors D 5 F 4 6 Compt DF nd F : D. 7. Consdr th scond-ordr tnsor D 4. Dtrmn th mg of th ctor r 4 5 hn D oprts on t. 8. Wrt th foong ot n f r ths th componnts of scrs, ctors or scond ordr tnsors? ) B b) C c) B mn d) b 9. Wrt b: c d n trms of th componnts of th for ctors. Wht s th ordr of th rstng tnsor? 0. Vrfy Eqn Sho tht E : s.9.6,.9.8). [Hnt: s th dfnton of th cross prodct n trms of th prmtton symbo,..4), nd th fct tht.] Sod Mchncs Prt III 8 Ky