MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 25 



If the scalar W is to be unafifected by a transformation a we must have 

 W = CcX unaffected. If we write 



W = eca^'aX = eca'^X' 



we have 



W' = W = e'c X' 



if 



/ -1 



when 



e' = a7^ ec (8.5) 



This substantiates our previous statement. 



Relation Between Stress and Strain 



If the strain in an elastic body is proportioned to the stress we may write: 



€i = SuXi + ■5*12X2 + • • • Sv^Xf, 

 62 = S21X1 + • ■ • • • • 



Where the S's are elastic modulii. In matrix notation: 



e = SX (8.6) 



Conversely X = S~^e or if S^ = C 



X = Ce (8.7) 



The C's are called elastic constants to distinguish them from the modulii S. 

 As e = SX, ou'e = a^^ SoT^aX, and since a^ e = e\ (the representation 

 of e on a new axis system related to the old one through the matrix a) and 

 aX is X' , then we may write {a^ e) = (aj Sa' )(aX) as: 



e' = S'X' where S' = aj'Sa'' (8.8) 



Similarly operating on X = Ce we find 



X' = CV where C = aCac (8.9) 



The energy required to cause the strain e is 



W = f Xrder ^^-Xrer = ^ SrsXrXs (8.10) 



whence, if W is a perfect dififerential, 



s^^ = d'W ^ d'W ^^^^ ^g^^j) 



dXr dXs dXg dXr 



