MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 31 



The elongation per unit length per degree in the direction (6, 4>) is 

 /cos 6 sin (f, sin 6 sin (p, cos (p^ 



Msifi — 



cos y sin ^ • 

 sin 6 sin (p • 



cos = (p 



whence 



iA/g^ = Ai cos" sin" (/? + /1 2 sin" 5 sin (^ + Az cos (^ (9.3) 



The strain can easily be extended to a function of / and f as follows: 



A/ = tAl + t'Bl (9.4) 



Applying the prefactor a to both sides and putting the idemfactor in 

 between A and / and between B and / in the form I = a~ a we have: 



aAl = t(aAa~ )al + t {aBoT )al or 



M' ^ tA'V + rB'l' where 



A' = aAa 



B' = aBa~' 



(9.5) 



SECTION 10 



Temperature Variation of the Isothermal Elastic Modulii and 

 Stress Variation of the Temperature Expansion Coefficients, 



We can write the isothermal elastic modulus matrix at temperature 

 d -{- tas 



S' = S'" + tH (10.1) 



and the coefficient of temperature expansion at constant stress X as 



A = A° -\- LX .(10.2) 



Let us take a unit cube of crystal about the cycle indicated in the table; 

 starting with the cube in the unstressed unstrained state at absolute tem- 

 perature 6: 



If we sum the strain changes in this cycle to zero we have 



H = L 



