108 



BELL SYSTEM TECH NIC A L JOVRNA L 



Here d^ij is a tensor of third rank and €,„„ one of second rank. The dmi) 

 constants are related to the eighteen ordinar}"- constants (/,/ by the equations 



du = d\n ; dn — d\oo ; dy 



di6 

 2 



"133 ; — "123 — "132 , — — "113 — "131 ; 



'^222 ; ^^23 — '^233 ', -Z '/223 



</o32 



-T- — dnu = ^231 ; ~r = 'A>i2 = fi'221 ; '/31 



2 



(})h) 



^34 _ , _ , ^35 



— "323 — "332 ; -^ 



</313 — dz 



' 2 



— "311 ; "32 — "322 ; "33 — "333 

 = "312 — "321 • 



The tensor equations (94) give a simple method of expressing the piezo- 

 electric equations in an alternate form which is useful for some purposes. 

 This involves relating the stress, strain, and displacement, rather than the 

 applied field strength as in (94;. To do this let us multii')ly through the 

 right hand equation of (94) by the tensor 47r,S,L, , obtaining 



A-K'Sl „ 5 „ = e J, ntimnEm + 47r(/ „ kt l^m n T k( 



(96) 



where /il,, is a icn:or of the "free" dielectric impermeability obtained from 

 the determinant. 



^L = (-1)' 



,.yJ. 



*r . 



whe e A is the determinant 



(97) 



(98) 



and Am,, the minor obtained from this by suppressing the wth row and nth 

 column. If we take the i)roduct el„ /i„.„ for the three values of w, we have 

 as multiijliers of E\ , Eo , E^ , respectively 



€11 Pn + €12 Pl2 + €13 Pl3 = 1 

 €21 P2I + €22 P22 + €23 P23 == 1 

 €31 P31 -r €32 P32 "T" €33 P33 — 1- 



(99): 



Bui by virtue of equations (97) and (98) it is obvious that the value of 

 each term of (99) is unity. Hence we have 



E„ — Aw0mn 5„ — (47r dnkt iSmn) 'i\t 



(100) 



i 



