454 



RICE 



ART. K 



introduced in equations (31) as purely mechanical conceptions. 

 By means of equations (53) or (54) we can express the stress- 

 constituents as functions of the temperature and the strains; thus 



Xr = 



dfr 



(68) 



Now suppose the body experiences a small variation of strain 

 at constant temperature; the variations in the stresses are given 

 by the six equations 



where 



8Xt = Crl5/i . . . + Credfe, 



dXrjt, f) ] 

 _ d'Ht, f) 



dfr dfs 



(69) 



(70) 



Equation (69) replaces (31). The elastic constants are of 

 course functions of the temperature and the strains. If the xp 

 function is quadratic in the strains, the quantities Crs are inde- 

 pendent of the strains, and this leads to the generalized Hooke's 

 law referred to earlier. In any case equation (70) shows that 

 Cra = Csr aud that at the most there are only 21 elastic con- 

 stants. For an isotropic material, we have as before essentially 

 only two, the bulk modulus or elasticity of volume, defined as 

 before, and the modulus of rigidity given by any one of the 

 differential coefficients 



or 



a/4 

 a¥M), 



dX,{t, f) ^ 

 a/5 



aVO/), 

 a/52 



aXeO/), 

 a/e 



aVO/), 

 a/e^ 



(71) 



which are equal for such a substance. 



For those interested to pursue these matters further, a short 

 chapter on the thermodynamics of strain will be found in 

 Poynting & Thomsons' Properties of Matter. For a very full 



