3.4 Properties of the Stress Tensor

Transcription

1 cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato rul..5: Q Q (.4.) p qj pq whr Q ar th drcto coss, Q. j Isotropc tat of trss uppos th stat of strss a body s [ ] O fds that th applcato of th tsor trasformato rul ylds th vry sam compots o mattr what th coordat systm. hs s trmd a sotropc stat of strss, or a sphrcal stat of strss (s..). O xampl of sotropc strss s th strss arsg flud at rst, whch caot support shar strss, whch cas pi (.4.) whr th scalar p s th flud hydrostatc prssur. For ths raso, a sotropc stat of strss s also rfrrd to as a hydrostatc stat of strss. A ot o th rasformato Formula Usg th vctor trasformato rul.5.5, th tracto ad ormal trasform accordg to [ t ] [ Q ][ t], [ ] [ Q ][ ]. Also, Cauchy s law trasforms accordg to [ t ] [ ][ ] whch ca b wrtt as [ Q ][ t] [ ][ Q ][ ], so that, pr-multplyg by [ Q ], ad sc [ Q ] s orthogoal, [ t] {[ Q][ ][ Q ]}[ ], so [ ] [ Q][ ][ Q ], whch s th vrs tsor trasformato rul..6a, showg th tral cosstcy of th thory. I Part I, Nwto s law was appld to a matral lmt to drv th two-dmsoal strss trasformato quatos, Eq..4.7 of Part I. Cauchy s law was provd a smlar way, usg th prcpl of momtum. I fact, Cauchy s law ad th strss trasformato quatos ar quvalt. Gv th strss compots o coordat systm, th strss trasformato quatos gv th compots a w coordat systm; partcularsg ths, thy gv th strss compots, ad thus th tracto vctor, old Mchacs Part III 5

2 cto.4 actg o w surfacs, ortd som way wth rspct to th orgal axs, whch s what Cauchy s law dos..4. Prcpal trsss c th strss s a symmtrc tsor, t has thr ral gvalus,,, calld prcpal strsss, ad thr corrspodg orthoormal gvctors calld prcpal drctos. h gvalu problm ca b wrtt as ( t ) (.4.) whr s a prcpal drcto ad s a scalar prcpal strss. c th tracto vctor s a multpl of th ut ormal, s a ormal strss compot. hus a prcpal strss s a strss whch acts o a pla of zro shar strss, Fg..4.. ( ) t t t t o shar strss oly a ormal compot to th tracto Fgur.4.: tracto actg o a pla of zro shar strss h prcpal strsss ar th roots of th charactrstc quato..5, whr, Eq...6-7,..7, I I I (.4.4) I I I tr [( tr) tr ] [ tr trtr ( tr) ] dt (.4.5) old Mchacs Part III 6

3 cto.4 h prcpal strsss ad prcpal drctos ar proprts of th strss tsor, ad do ot dpd o th partcular axs chos to dscrb th stat of strss., ad th strss varats I, I, I ar varat udr coordat trasformato. c.f.... If o chooss a coordat systm to cocd wth th thr gvctors, o has th spctral dcomposto.. ad th strss matrx taks th smpl form.., [ ] ˆ ˆ, (.4.6) Not that wh two of th prcpal strsss ar qual, o of th prcpal drctos wll b uqu, but th othr two wll b arbtrary o ca choos ay two prcpal drctos th pla prpdcular to th uquly dtrmd drcto, so that th thr form a orthoormal st. hs strss stat s calld ax-symmtrc. Wh all thr prcpal strsss ar qual, o has a sotropc stat of strss, ad all drctos ar prcpal drctos..4. Maxmum trsss Drctly from.., th thr prcpal strsss clud th maxmum ad mmum ormal strss compots actg at a pot. hs rsult s r-drvd hr, togthr wth rsults for th maxmum shar strss Normal trsss Lt,, b ut vctors th prcpal drctos ad cosdr a arbtrary ut ormal vctor, Fg..4.. From..8 ad Cauchy s law, th ormal strss actg o th pla wth ormal s ( ) ( t ) (.4.7) N N ( ) t prcpal drctos Fgur.4.: ormal strss actg o a pla dfd by th ut ormal old Mchacs Part III 7

4 cto.4 old Mchacs Part III 8 Wth rspct to th prcpal strsss, usg.4.6, ) ( t (.4.8) ad th ormal strss s N (.4.9) c ad, wthout loss of gralty, takg, o has ( ) N (.4.) mlarly, ( ) N (.4.) hus th maxmum ormal strss actg at a pot s th maxmum prcpal strss ad th mmum ormal strss actg at a pot s th mmum prcpal strss. har trsss Nxt, t wll b show that th maxmum sharg strsss at a pot act o plas ortd at 45 o to th prcpal plas ad that thy hav magtud qual to half th dffrc btw th prcpal strsss. From..9,.4.8 ad.4.9, th shar strss o th pla s ( ) ( ) (.4.) Usg th codto to lmat lads to ( ) ( ) ( ) ( ) [ ] (.4.) h statoary pots ar ow obtad by quatg th partal drvatvs wth rspct to th two varabls ad to zro: ( ) ( ) ( ) ( ) [ ] { } ( ) ( ) ( ) ( ) [ ] { } (.4.4) O ss mmdatly that (so that ± ) s a soluto; ths s th prcpal drcto ad th shar strss s by dfto zro o th pla wth ths ormal. I

5 cto.4 ths calculato, th compot was lmatd ad was tratd as a fucto of th varabls (, ). mlarly, ca b lmatd wth (, ) tratd as th varabls, ladg to th soluto, ad ca b lmatd wth (, ) tratd as th varabls, ladg to th soluto. hus ths solutos lad to th mmum shar strss valu. A scod soluto to Eq..4.4 ca b s to b, / (so that ± ±/ ) wth corrspodg shar strss valus ( ) 4. wo othr solutos ca b obtad as dscrbd arlr, by lmatg ad by lmatg. h full soluto s lstd blow, ad ths ar vdtly th maxmum (absolut valu of th) shar strsss actg at a pot:, ±, ±, ±,, ±, ±, ±,, (.4.5) akg, th maxmum shar strss at a pot s τ max ( ) (.4.6) ad acts o a pla wth ormal ortd at 45 o to th ad prcpal drctos. hs s llustratd Fg..4.. τ max prcpal drctos τ max Fgur.4.: maxmum shar strss at apot Exampl (maxmum shar strss) Cosdr th strss stat old Mchacs Part III 9

6 cto.4 [ ] 5 6 hs s th sam tsor cosdrd th xampl of... Usg th rsults of that xampl, th prcpal strsss ar, 5, 5 ad so th maxmum shar strss at that pot s 5 τ max ( ) h plas ad drcto upo whch thy act ar show Fg x τ max ˆ ˆ o 7 x x ˆ Fgur.4.4: maxmum shar strss old Mchacs Part III 4