PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



109 



Since the dummy index n is summed for the values 1, 2, and 3, we can set 

 the value of the terms in brackets equal to 



and equation (100) becomes 



Em = 47r (3mn 5„ — gmkC Tkl . 



Substituting this equation in the first equations of (94) we have 

 where 



Si,k( = Sijkf. — d„ni gmkl = Sijkt — 4:X[j8„„ d nkt dmij\. 



(101) 

 (102) 

 (103) 



By substituting in the various values of i,j, k and ^ corresponding to the 21 

 elastic constants, the difference between the constant displacement and 

 constant potential elastic constants can be calculated. If equations (102) 

 and (103) are expressed in terms of the Si,- ■ -, S^ strains and Ti,- ■ •, T^ 

 stresses, the gnij constants are related to the gij constants as are the corre- 

 sponding dij constants to the (/„,/ constants of equation (95). 



Another variation of the piezoelectric equations which is sometimes em- 

 ployed is one for which the stresses are expressed in terms of the strains 

 and field strength. This form can be derived directly from equations (9-i) 

 by multiplying both sides of the first equation by the tensor c^jkC for the 

 elastic constants, where these are defined in terms of the corresponding 

 s^j elastic compliances by the equation 



4 = (-i)^'"^^a:;/a 



(104) 



where A is the determinant 



A^ = 



515 525 535 545 555 556 



516 526 536 546 566 566 



and A*y in the minor obtained by suppressing the /th row and^'th column. 

 Carrying out the tensor multiplication we have 



Cijkt Sij = djkt Sijkt Tkf + dmij c-jkC E„ 



(105) 



