Derivation of the Differential Forms of the Conservation Laws Momentum

Transcription

1 Deatn f the Dffeental Fms f the Cnseatn Las Mmentm Aach t Deng the Dffeental Fms f the Cnseatn Las. Wte t the la f a sstem f atcles DNss D ηd Dt Dt. Rete the la n tems f a cntl lme sng the R.T.T. and Lebn s Theem DN Dt ss C ( η ) d η( nˆ ) t CS 3. Use Gass s Theem t tansfm aea ntegals nt lme ntegals s that the la ma be tten n the fm: C { } d 0 da

2 Deng the Dffeental Fms f the Cnseatn Las 4. F an abta cntl lme the ntegand mst be e eldng the dffeental eqatn: { } 0 5. Smlf and Recast eqatns nt the mst sable fm Cach s Eqatn f Mtn Stesses n a Fld Element Stess Tens Smmetc Nmal stesses alng the dagnal Incldes sface & bd fces actng n a fld element Al Netn s nd La f a dffeental fld element Catesan Cdnate Sstem

3 Stesses n a Fld Element Pf f Smmet Cnsde a Tqe n the element bel abt an as alng the centd aallel t the -as Tqe d d d d [ dd] d [ dd] d [ dd] d [ dd] d Tqe ( ) ddd Iα d Mment f Ineta I ddd ( d d ) / ( ) ( d d ) α d centd as d As d d g t e d Cach s Eqatn f Mtn t f D Dt The ttal acceleatn at nt Stess gadents stess degence Bd fce e lme Gd bt f e nclde cnseatn f mass e stll hae 9 nnns and nl 4 eqatns! Catesan Cdnate Sstem 3

4 Knematcs f Fld Mtn - A a t elate stesses t elctes and edce nmbe f nnns Tanslatn Rtatn Defmatn Dlatatn Hstcall: Cach (84) fm sld bd defmatn analss Stes (845) scs flds & the elatnsh beteen stess & stan Helmlt (858) analed the defmatn f deal flds - tct elct Cmstn Path f fld atcle Cnsde a atcle f fld mng alng a ath. P ( ) 4

5 5 elct Cmstn HOT HOT HOT : elct cmnent - : - elct cmnent : elct cmnent - P Q Tal Sees Eansn f the elct ect abt the nt P. - If ( ) s small e can neglect the HOT elct Cmstn P Q N a bt f Mathematcal Jggle:. Bea the atal dffeental tems -

6 6 elct Cmstn P Q N a bt f Mathematcal Jggle:. Bea the atal dffeental tems - elct Cmstn P Q N a bt f Mathematcal Jggle:. Bea the atal dffeental tems. Add and sbtact le dffeentals -

7 7 elct Cmstn P Q We n hae a ne eessn f the elct ect - S & & Ω Stan Rate elated t the defmatn and dlatatn f fld elements Rtatn Rate Ω F e tatn & tanslatn elct Cmstn S & & Ω Stan Rate Rtatn Rate S & Ω & Smmetc Asmmetc Snce the stess tens s als smmetc can nl be elated t the Stan Rate

8 elct Cmstn Defmatn Rate Tens e Rtatn Rate Tens Ω Asmmetc Stan Rate Tens S Smmetc e Ω S Cnsttte Eqatns Need t elate the stan ate tens t the stess tens:. If the fld s a est the stess s hdstatc and the esse eeted n the fld s themdnamc esse. δ τ Shea Stess Tens. Netnan Fld Amatn shea stess s lneal elated t the defmatn-ate tens. 3. Snce thee s n sheang actn n a sld-bd tatn f a fld n shea stesses ll act dng sld bd tatn bt Ω s nne n sld bd tatn hence τ S Leads t 8 cnstants 8

9 9 Cnsttte Eqatns Need t elate the stan ate tens t the stess tens: 4. Fld etes ae stc Wth these smlfcatns the cnsttte elatn f stess tens becmes: Whee the scs stess tens s: F ncmessble flds: λ and ae scst Ceffcents that mst be detemned emcall λδ δ nd de smmetc tens δ τ λδ τ Nae-Stes Eqatns λδ δ f t f t λ f t F an ncmessble fld: 0 Dt D δ δ 3 Stes Assmtn λ 3

10 0 Nae-Stes Eqatns (ncmessble fl) 0 f t 4 Eqatns & 4 nnns and P We n hae a clsed sstem f Eqatns!!!