no BELL SYSTEM TECHNICAL JOURNAL 



As before \vc find that the tensor product of cijk( Si,k( is unity for all values 

 of k and (. Hence equation (105) can be written in the form 



Tu(= clu(Si,- e„.uE„, (106) 



where Cmk( is the sum 



CmkC = d,„ij cljkl (107) 



surrn ed for all values of the dummy indices / and 7. If we substitute the 

 equation (106) in the last equation of (94) we lind 



s 



bn=^-PEm + er^^Sij (108) 



where e"™,, the clamped dielectric constant is related to the free dielectric 

 constant emn by the equation 



ein ^ tin- MdnUtemkt]. (109) 



Expressed in two index piezoelectric constants involving the strains ^u- • -Svi 

 and stresses Tw • • T12 the relation between the two and three index piezo- 

 electric constants is given by the equation 



en = ^ni ; ^12 = ^122 ; ^13 = ^133 ; ^14 — ^123 = ^132 ; ^15 = ^U3 = ^131 



e\e = «U2 = em ; ^21 = ^211; ^22 = ^222 ; ^23 = ^233 ; ^24 = ^223 = ^232 



e25 = ^213 — ^231 ; ^26 = ^212 = ^221 ", ^31 = ^3U ; <'32 = <'322 ', «33 = ^333 



^34 = ^323 = ^332 ', ^35 = ^313 = ^331 i ^36 = €312 = ^321 • 



(110) 



Finally, the fourth form for expressing the piezoelectric relation is the 

 one given by equation (53). Expressed in tensor form, these equations 

 become 



TkC = c'^]k(S,j — h„ktb„ ; Em = 47r^'l„ bn — hmijSij (111) 



In this equation the three index piezoelectric constants of equation (HI) are 

 related to the two index constants of equation (53) as the e constants of 

 (110). These equations can also be derived directly from (106) and (108) 

 by eliminating Em. from the two equations. This substitution yields the 

 additional relations 



h„k( = -^T^e-rnkf (imn \ ^ikf = cfjkf + C„,k( I'mrj = C^ijkl 



(112) 



+ 47r emk( Cnij 0mn 



where 



i3L = (-i)^"'*"'a:;Va'' 



