172 



BELL SYSTEM TECHNICAL JOURNAL 



the stresses in terms of the independent variables, the strains and the elec- 

 tric displacements. Up to the second derivatives, these are 



_ dEm o, . dEm » 

 do ,7 dOn 



+ 



Tkt = 



+ 



1 

 dTrd 



d'Em O O r ^^ E,m r- ^ r d" Em 



.00,7 do gr OOijOOn OOnOOo 



+ 





(44) 



1 r rn-^ 



2!L55iy55 





dSij d8n 



d'Tkf 



8ndo 



+ 



For the electro-optic and photoelastic cases, the two tensors of interest are 



dbn d8o 

 d^Em 



d'U 



_ d En 



dSkid8nd8o dSk(d8o 



= Airmkinc 



d'U 



(45) 



= (4Tr) fmno. 



d8n d8o d8m d8n d8o 

 For the first partial derivatives, we have the values 



dTjd 

 dSij 



CijkC J 



dTjd 

 d8n 



dEm 



d8n 



d^U 



_ dEn 



dSki d8n dSkC 



'iTrl3mn 



= -hnkl 



(46) 



where cljkt are the elastic stiffnesses measured at constant electric displace- 

 ment, hnki are the piezoelectric constants that relate the open circuit voltages 

 to the strains, and ^mn are the impermeability constants measured for con- 

 stant strain. 



With these substitutions and neglecting the other second partial deriva- 

 tives, we have, from (44), 



■tLm — •^m ij 'J ij I •'>' n 



Tkt = c^ijklSij + D 



s 



+ 



_hoke . mklon D 

 17 2 



4 



(47) 



This equation shows that there is a relation between the change in the im- 

 permeability constant due to stress in the first equation, and the electro- 

 strictive constant in the second equation through the tensor nijjmn ■ These 



