Engineering Tensors. Friday November 16, h30 -Muddy Charles. A BEH430 review session by Thomas Gervais.

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1 ngneerng Tensors References: BH4 reew sesson b Thoms Gers tgers@mt.ed Long, RR, Mechncs of Solds nd lds, Prentce-Hll, 96, pp - Deen, WD, nlss of trnsport phenomen, Oford, 998, p Goodbod, M, Crtesn tensors: Wth pplctons to Mechncs, ld Mechncs nd lstct, John Wle & Sons, 98 rd Noember 6, 6h -Mdd Chrles

2 Sclrs, Vectors & Tensors Sclr: Qntt tht s nrnt n tself (does not depend on n referentl) lso nown s zeroth order tensor. : mss (non reltstc referentls), Tempertre, nerg, Concentrton Vector: Qntt tht possess both drecton nd mgntde locted somewhere n spce. It s frst order tensor. Note: ector possesses two nrnts wth respect to coordnte spce: Its mgntde,, nd ts drecton. : orce, lectrc feld, l ˆ ˆ ˆ' ˆ' e e ' e ' e ' e' ˆ

3 Hgher order tensors nd order tensor: n ordered set of nne nmbers, ech of whch possessng drectons. n helpfl nlog wold be to mgne ector whose three components ech wold be ector. tensor of order n cn be reprensented b n nmbers n trdmensonl spce. It s mtr of dmenson n. nd possesses nrnts whch re the coeffcents of ts chrcterstc polnoml rsng from det T λ I. The trce nd the determnnt of the tensor re two of ts nrnts. In generl, tensor of order n wll he n nrnts. Phscl pplctons of tensors: Dscplne Phenomenon Qntt mple Phscs: -M Lght trnsmsson n nsotropc med Refrcton nde n Optcll cte polrzers Dnmcs nglr momentm Moment of Inert I Groscope ld mechncs Hndered trnsport n poros med ffecte permeblt K C dffson n mscles Sold mechncs Tensle propertes of nsotropc solds Yong s modls Dct Tpe, Mscles

4 Ddc Prodct (Tensor Prodct) Ths the generl form of tensor prodct: Sclr ( th order tensor) e e e e e e e e e e etc. e e B ddng the n three ector components, we get rd order tensor, nd so on... Vector ( st order tensor) nd order tensor

5 ' ' Notton The proecton of the ector on the s,,, z cn ech be decomposed n the bss (,, z ). The relton between ech of the components s then gen b: z ' ' ' ' ' ' ' ' z ' z z ' ' z z (nsten s notton)

6 Dot nd Cross prodcts Defnton of dot prodct: B B B Where δ s the Kronecer delt, nd order tensor. Does t hold n tensor notton? Let s test t sng chnge of coordnte: B ' l l B' l l ' l B l δ ' B' l δ l B If wht we sd bot the conserton of the mgntde of ector from one crtesn referentl to nother, then B B ' B' or ths to be tre, we need: δ l δ l (nsten notton)

7 l Dot nd Cross prodct (cont.) δ δ epnded elds: l These re the normlzton nd the orthogonlt condtons tht n orthonorml bse respects. Defnton of cross prodct: B B B B Where s clled the Le-Ct denst nd s the cross prodct eqlent of the Kroneer delt, δ, for the dot prodct. It s thrd order tensor. ε ε,,, or,,, -,,, ε Ths s the fmos rght hnd rle

8 Grdents nd Dergence f We now: f nd e ˆ Q: Wht bot the grdent of ector? : The generlzton of ector proded b tensor nlss mples: second order tensor! The grdent of tensor ncreses ts order b one Q: Wht bot the dergence of nd order tensor? : Usng or defnton of the dot prodct: T T δ T The dergence of tensor decreses ts order b one ector!

9 Prctcl problem: How to set p tensor from phscl resonng? Consder the followng problem: We wnt to stretch pece of nsotropc tsse nd fnd the components of the Yong s Modl ppl gen dsplcement nd compte the force s fncton of the ngle (ssmng constnt strn nd Posson coeffcent of ) ' ' e ˆ'

10 Prctcl problem: How to set p tensor from phscl resonng? (cont.) We need to compte the 4 mtr elements: : : : : sn sn sn sn We re mesrng the mgntde of the force reqred to mpose nt dsplcement n the drecton. ' ' eff [ ] ' ' sn ffecte modls ngle (rd) Normlzed effecte modls s fncton of the ngle for doble of

11 Sme problem: other pproch sng tensors ' ' ' ' ' ' ' ' ' sn sn ' Q: re epressons eqlent,.e. '? ' sn sn sn sn : Yes!

12 ld mechncs: The stress nd rte of strn tensors ) Stress tensor z S Stress ector n z' ' ' S S n S n ( S ) B) Rte of strn tensor S σ ( S ) ( r, t) ( r δr, t ) B Tlor epnson to the frst order: t ( r, t) δ δ δr ( r, t) ( ( r, t) ) δr ( r, t)s tensor (grdent of ector)! ( ) t e& ξ e& ξ & [ t ( ) ( )] [( ) t ( )] Vortct tensor

13 Conclson nd order tensors estblsh relton between two sets of ectors. : Vectors n one crtesn spce s ectors n nother, bt LSO ectors from the dsplcement ector spce to the force ector spce (s we st sw). Hgher order tensors flfll the sme role bt wth tensors nsted of ectors The dergence of tensor redces ts order b one. The grdent of tensor ncreses t order b one. The Crl of tensor elds tensor of the sme order.