PIEZOELECTRIC CRYSTALS IN TENSOR FORM 

 If, however, we take the strain components as 



105 



c _ c _ ^^ 



■'11 — 'In — TT 1 



ax 



S,2 



dr) 

 By ' 



c _ c _ ^f 



O33 — 'Jjz — :r- 

 dz 



2 \dx dx/ ' 



(83) 



Si-i — Siy-> — 



1 (dj 

 dy 



+ 



dr,\ 

 dzj 



the nine components 



(83) 



will form a tensor of the second rank, as can be sh(jwn by the transformation 

 equations of (82). 



The generaUzed Hooke's hiw given by equation {22) becomes 



'/'.-.= 



CijkfSkt 



(84) 



CijkC is a fourth rank tensor. The right hand side of the equation being 

 the product of a fourth rank tensor by a second rank tensor is a sixth rank 

 tensor, but since it has been contracted twice by having k and ^ in both 

 terms the resultant of the right hand side is a second rank tensor. Since 

 dm is a tensor of the fourth rank it will, in general, have 81 terms, but on 

 account of the symmetry of the T , j and Sic( tensors, there are many equiva- 

 lences between the resulting elastic constants. These equivalences can be 

 determined by expanding the terms of (84) and comparing with the equiva- 

 lent expressions of (22). For example 



+ ^1121621 -f- ril22'S'22 + ("1123»^23 

 + <"n3 Al + <"1132-S'32 + CU33'S33 • 



(85) 



Comparing this equation with the tirst of (22) noting that Su — S21 = 

 — ', etc., we have 



t'UU — C\\ ; ('1112 — ("1121 — '"in ; <"1133 — '"iS ', f-'llU 

 ^^1122 = fl2 ; f'll23 = t'll32 = 6"l4 • 



t-1131 



(86) 



