3.10 Convected Coordinates

Transcription

1 Seco.0.0 Coveced Coordaes Some of he mpora resuls from secos.-.9 are ow re-epressed erms of coveced coordaes. As before, ay relaos epressed symbolc form hold also he coveced coordae sysem..0. The Sress Tesors Traco ad Sress Compoes Cosder a dffereal parallelepped eleme he curre cofurao bouded by he coordae curves as..0. (see..6.. The boud vecors are d Θ, ad d Θ. The surface area d S of a face of he elemeal parallelepped o whch Θ s cosa, o whch s ormal, s he ve by Eq..6.5, d S (.0. ad smlarly for he oher surfaces. Θ curve Θ curve Θ curve ure.0.: vecor elemes boud surface elemes The posve sde of a face s defed as ha whose ouward ormal s he dreco of he assocaed coravara base vecor. The u ormal o a posve sde s he same as he u coravara base vecor; as Eq..6.4, Le he force,..0., so ha ˆ ( o sum (.0. d ac o he surface eleme wh ormal s he raco (force per u area. be S d (o sum over Sold Mechacs Par III 8

2 Seco.0 ĝ The compoes of ure.0.: raco ac o a surface eleme alo he u covara base vecors are deoed by : ˆ (.0. wh o sum over he he compoes,..0.. erm; are called he physcal sress Θ Θ Θ Iroduce ow a ew vecor ure.0.: physcal sress compoes defed by (o sum over (.0.4 I wll be show ha hs vecor s coravara, ha s, rasforms bewee coordae sysems accord o.7.a (ad so does o sasfy he vecor rasformao rule, hece he superscrp poed braces. The compoes of alo he covara base vecors are deoed by : (.0.5 Compar.0.-5, Sold Mechacs Par III 8

3 Seco.0 (o sum (.0.6 Cauchy s Law ad he Cauchy Sress Tesor Cauchy s law ca ow be derved he same way as., by cosder a small erahedral free-body, The physcal sress compoes show ac o he eave sdes of he surfaces ad so ac drecos oppose ha of he correspod compoes o he posve sdes (a cosequece of Cauchy s Lemma. I s requred o deerme he raco erms of he physcal sress compoes ad he u ormal o he base area. Θ Θ Θ ure.0.4: free body daram of a erahedral poro of maeral The ormal o he base has compoes (.0.7 Cosder he vecor elemes d a ad d b show Defe he surface area eleme o be he vecor wh maude equal o wce he area of he erahedro base ad he dreco of he ormal o he base, so ( da db ( ( ( + + ( + + (.0.8 where d S,, are he surface eleme areas of he hree coordae sdes of he parallelepped of..0. (wce he area of he coordae sdes of he erahedro; from.0., Sold Mechacs Par III 84

4 Seco (.0.9 wh o sum over he he erm, or (.0.0 da db ure.0.5: vecor eleme of area for he base of he erahedro The prcple of lear momeum, vecor form, s he (cacell ou a facor of ½ rom.0.4, 0 (.0. (.0. ad so (.0. Def he (symmerc Cauchy sress esor hrouh Cauchy Sress Tesor (.0.4 oe arrves a Cauchy s law. The Cauchy sress s aurally a coravara esor because he ormal vecor upo whch operaes o produce he raco s aurally represeed he form of a covara vecor (see.0.. Noe ha he sress ca also be epressed he form Sold Mechacs Par III 85

5 Seco.0 (.0.5 Oher Sress Tesors The PK, PK ad Krchhoff sress esors are P P S S τ τ (.0.6 By defo,.9.8, τ J, ad so τ J. By defo, T S J, ad so, from S J J τ (.0.7 Thus, as see already, he Krchhoff sress s he push-forward of he PK sress. Smlarly, by defo T P J ad so P J J J J m ( m (.0.8 J.0. The Equaos of Moo The Equaos of moo have bee ve he symbolc form by.6. ad.6.9. To epress hese curvlear coordaes, recall he defo of he dverece of a esor,.8.8, dv ( (.0.9 Θ The spaal ad maeral descrpos of he equaos of moo are he P ( v d + b ρ d dv + B ρ0 d Equaos of Moo (.0.0 Sold Mechacs Par III 86