Physical Properties as Tensors

Transcription

1 Phsicl Proerties s Tensors Proerties re either isotroic or nisotroic. Consier roert such s the ielectric suscetibilit, tht reltes the olrition (P) cuse b n electric fiel () in ielectric mteril. In isotroic mterils, the olrition inuce b the electric fiel tkes the sme irection s the electric fiel lie. In nisotroic mterils, the olrition hs constnt irection within the crstl structure n hence the electric fiel lie n the olrition irection re not the sme. P ( ) χ r O e O o χ e is the ielectric suscetibilit

2 In isotroic mterils, the reltionshi between the olrition n the electric fiel is vectoril reltionshi, exresse b P oχ In n nisotroic mteril, the ioles forme uring the ielectric olrition re generll not rllel with the electric fiel. Then consiering x, n coorinte sstem the following exression hols: P P P x (,, ) P ( P, P, P ) χ χ χ x xx x x x x x χ χ χ x χ χ χ x x Then, the ielectric suscetibilit of n nisotroic mteril cn be escribe s secon rnk tensor. Pi χij j (i,j,, ) D D D D

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4 The rnk of the tensors etermines the numbers of the tensor comonents. The number of the comonents of the 0,,,,4,5,6-rnk tensors re,, 9, 7, 8, 4, 79. However, certin smmetries consierbl reuce the number of the ineenent comonents. These m be intrinsic smmetries, inherent in the hsicl roert, or crstl smmetries.

5 In some instnces the intrinsic smmetries follow from the efinition of the hsicl roert in question. Thus for exmle in the cse of elsticit it follows from the smmetr of the stress n eformtion tensors tht lso the fourth-rnk tensor of the secon-orer elstic stiffnesses [c ijkl ] (see Tble ) is smmetric with resect to the (ij) n (kl) ermuttions. This w the number of the ineenent coefficients of the fourth-rnk elstic tensor ecreses from 8 to 6. Further on from the smmetr of the eformtion tensor follows the smmetr of the [ ijk ] ieo-electric tensor (see Tble ) with resect to the commutbilit of the j n k suffixes, which mens tht the ieoelectric tensor hs not more thn 8 ineenent comonents. Neumnns Princile the smmetr elements of n hsicl roert of crstl must inclue ll the smmetr elements of the oint grou of the crstl. Accoring to Neumnns rincile the tensor reresenting n hsicl roert shoul be invrint with regr to ever smmetr oertion of the given crstl clss. The conition of invrince reuces the number of the ineenent tensor comonents, since it signifies reltionshis between the tensor comonents.

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8 xmle: Let us consier the form of the roelectric tensor in the crstl clss of the trigonl sstem ssuming tht the x xis of the coorinte sstem is the three-fol rottion xis. Consulting the tbles, the coorinte trnsformtion relte to the smmetr oertion cn be escribe with the following mtrix Tking into consiertion the smmetr of the crstl n the invrince of the olr tensor: Bse on these equtions 0 n cn iffer from ero. Consequentl the form of the roelectric tensor of crstl clss of the trigonl sstem will be [0; 0; ].

9 The ieoelectric chrge constnt, (), olrition generte er unit of mechnicl stress (irect), or, mechnicl strin exerience b ieoelectric er unit of electric fiel lie (converse). The ieoelectric voltge constnt, (g), is the electric fiel generte b ieoelectric mteril er unit of mechnicl stress lie or, lterntivel, is the mechnicl strin exerience b ieoelectric mteril er unit of electric islcement lie. Definitions of frequent use g P g D Inuce _ Polrition Mechnicl _ tress _ lie mechnicl _ strin _ obtine electric _ fiel _ lie inuce _ electric _ fiel mechnicl _ stress _ lie mechnicl _ strin obtine electric _ islcement _ lie g r O

10 P P P Inuce _ Polrition i ijk jk Mechnicl _ tress _ lie jk Then, the olrition in the i irection (xis), is relte to the stress jk b the coefficient ijk. xx x x P P P The 7 ijk re the ieoelectric mouli n form thir rnk tensor. As ijk ikj, there re u to 8 ineenent mouli. x [ P ] [ ][ ] x [ ] xx x x

11 P P P P P

12 mechnicl _ strin _ obtine electric _ fiel _ lie jk i ijk i In this effect strin rises s result of n lie electric fiel. The mouli re the sme s for the irect effect.

13 [ ] [ ] [ ] T [ ]

14 mmetr Proerties of the Pieoelectric Tensor

15 Trnsformtion xis A vector P with comonents P, P n P with resect to the reference xis X, X n X will trnsform to P with comonents P, P n P with resect to the reference xis X, X n X Where ij is the cosine of the ngle between xis X i new n X j ol

16 x x x x x x xx T [ T ] [ T ] x x x x xmle: Consier sstem of stresses given in the reference coorintes X, X n X b: [ ] MP Wht will be the stresses t the coorinte sstem X, X n X, when the cosine ngles between the two sstems re given b the mtrix [ ] MP [T ]

17 Tensor Mtrix The reltionshi between stresses ( n rnk tensor) in element t ifferent orienttions is given b the exression: [ ] [ ] Α Α x x xx x x xx [ ] Α [ ] Α

18 [ ] Α

19 Coule qutions Accoring to our knowlege of mechnics of mterils ( is the comlince mtrix obtine t constnt electric fiel usull ero): For ielectric mterils, the reltionshi between the mechnicl strin obtine n the electric fiel lie is given b the ieoelectric chrge constnt x γ γ x γ x mechnicl _ strin _ obtine electric _ fiel _ lie [ ] [ ] T [ ] The strins ue to the lie electric fiel n lie mechnicl lo. T [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] x x x

20 [ ] [ ] [ ] [ ] [ ] T x x x

21 D D O D D O If stresses re lie to ieoelectric mteril, the resulting electric islcement is given b the eqution O P r O [ P ] [ ][ ] In the bsence of stress the electric islcement rouce b n lie electric fiel is [ D] [ ] [ ] P P P T Where is the ermittivit mtrix x mesure t constnt stress (usull ero) The electric islcement ue to both the lie stress n n electric fiel is: [ D] [ ] [ ] [ ][ ] T

22 [ ] [ ] [ ] [ ][ ] D T D D D

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