PIEZOELECTRIC CRYSTALS IN TENSOR FORM 107 



-^44 ^45 



^3312 — •^3321 — -^1233 — -52133 ', -J — ■^2323 — -^2332 — -^3223 — -^3232 ', — — ^2313 = 



_ _ _ _ _ _ -^46 _ 



■^2331 — •^3213 — -^3231 — -^1323 — -^1332 — .^3123 — -^3132 ', -J — -^2312 — -^2321 = 



(90 B) 



_ _ _„_ _ ■>55_ _ 



^3212 — ■^3221 — -^1223 — J1232 — -^2123 — -^2132 ; ~J — ■^1313 — -^1331 — •^3113 = 



•^56 _ _ . 



•^3131 ; -J" ~ "^^^12 ~ "^13-1 ~ "^3112 — •^3121 — -^1213 — -^1231 — ■^2113 — •^2131 ', 



•^66 _ _ _ _ 



-; '■ •^1212 — •^1221 — -^2112 — 52121 • 



4 



Here again the SijkC elastic constants are determined from the ordinary 

 elastic constants 5,y by replacing 



1 by 11, 2 by 22, 3 by 33, 4 by 23, 5 by 13, 6 by 12. 



However for any number 4, 5, or 6 the elastic compliance Sij has to be di- 

 vided by two to equal the corresponding SijkC compliance, and if 4, 5 or 

 6 occurs twice, the divisor has to be 4. 



The isothermal elastic compliance of equations (39) can be expressed 

 in tensor form 



Si,^slk(T,c + a,,dQ (91) 



1 where as before a,; is a tensor of the second rank having the relations to 

 the ordinary coefficients of expansion 



Oil = «ii ; 02 = "22 ; "3 = «33 ,* y = ^23 i 



oib ae 



The heat temperature equation of (35) is written in the simple form 



I dQ = + akt Tut e + pCp de. (92) 



' . . . 



ii By eliminating dO from (92) and substituting in (91) the adiabatic constants 



!i are given in the simple form 



SijkC = SijkC - —^ — . (93) 



The combination elastic and piezoelectric equations (60) can be written 

 in the tensor form 



T 

 Sii = S^jkCTkC + d^ijEm ; hr, = ~ Eyn + dnkCTkC- (94) 



4ir 



