PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



103 



In general the transformation equation of a tensor of the ;zth rank can be 

 written 



xi 



OXj^ OXj., a.V/„ 



(76) 



4. Application of Tensor Notation to the Elastic, Piezoelectric 

 AND Dielectric Equations of a Crystal 



Let us consider the stress components of equation (7) 



T T T 



^ XX ^ xy •* J2 



T T T 



^ yx ^ yy •'2/2 



T,x T,y r,, 



from which equation (8) is derived 



■i xy I yx ] ^ xz -i zi , ^ yz •* zj/ 



and designate them in the manner shown by equation (77) to correspond 

 with tensor notations 



(77) 



by virtue of the relations of (8). We wish to show now that the set of 9 

 elements of the equation constitutes a tensor, and by virtue of the relations 

 of (8) a symmetrical tensor. 



The transformation of the stress components to a new set of axes x', y', z' 

 has been shown bv Love to take the form 



T^x = fl T^j, + rn\Tyy ~\- nlT,, + lliMiT^y + 2(iUiTj,z + ImiUiTy, 



(78) 



Txy = (ifiTjcx + fnitnoTyy-'r nin2T,,-{- (Awo + limi)T^y + (A«2 + hnifT^^ 



+ {mini + niniiiTy^ 



where A to 113, are the direction cosines between the axes as specified by 

 equation (61). Noting that from (72) 





«3 = 



dXj 

 dx3 



the first of these equations can be put in the form 

 ^ See "Theory of Elasticity," Love, Page 80. 



