106 BELL SYSTEM TECH NIC A L JOURNA L 



In a similar manner it can be shown that the elastic constants of (22) 

 correspond to the tensor elastic constants djui according to the relations 



C\\ = fun ; Cl2 = <'1122 = C22II ; Cl3 — Ca33 = f33n ', ^14 = ^1123 = ^132 = 



Cnn = C32U ; Cib = diw = ^1131 = ^'isu = Cun ', Cu = fiii2 = Cn2i = <^i2ii = 



^2111 ', C22 — <^2222 ', C2Z — <^2233 — ^3322 ', ^24 = ^2223 = ^2232 = ^2322 = ^3222 ', 

 C2b — ^2213 = <"2231 = '"1322 = ^3122 ', <^26 = <^2212 = <^2221 = <'l222 = ^2122 ', C33 = 

 C3333 ; C34 = ^3323 = ^3332 = ^2333 = ^3233 ', ^36 = 3313 = ^3331 = '^1333 — ^3133 J 



(87) 



^36 = ^3312 — C3321 — C1233 — ^2133 ', ^44 — ^2323 — ^2332 — ^3223 — f3232 y ^46 — 

 ^2313 — ^2331 = ^3213 = <^3231 = 1323 = 1332 = ^3132 = ^3123 ', ''46 = ^2312 = 

 £"2321 — C32I2 = C322I — ^1223 = C1232 = C2I23 — C2132 ; C55 = C1313 = C1331 = 

 f3U3 = ^^3131 ; Cb6 — fl312 = 0321 = ^3112 = C3121 = fl213 = ^1231 = ^2113 = 

 £"2131 ) f 66 = f 1212 = <"1221 = ^2112 — <^2121 • 



Hence there are only 21 independent constants of the 81 djkf constants 

 which are determined from the ordinarily elastic constants c,/ by replacing 



1 by 11 ; 2 by 22; 3 by 33; 4 by 23; 5 by 13; 6 by 12 (88) 



and taking all possible permutations of these constants by interchanging 

 them in pairs. 



The inverse elastic equations (26) can be written in the simplified form 



Sij = SijkfTk(. (89x 



By expanding these equations and comparing with equations (26) we can 

 establish the relationships 



_ _ Su _ _ _ _ 



Sn = ^1111 ; -^12 = 51122 — -^2211 ; -^13 — 51133 — -^3311 ; "y — -^1123 — -51132 — -52311 — 



■^16 _ _ ■^16 _ _ _ _ 



•^3211 ; -W — -51113 — -51131 — -51311 — -53111 ', -y — -5lll2 — ■51121 — -51211 — -52111 ; 



