STRAINED ELASTIC SOLIDS 445 



must be taken into account will require a knowledge of how a 

 homogeneous substance when strained must be dealt with in 

 thermodynamical reasoning. The equation which is to replace 

 [11] is now easily derived in view of what has just been accom- 

 phshed in the previous parts of this exposition. Thus in [11] 

 c and r] are regarded as functions determined completely by 

 the state of the body. For a homogeneous fluid, we can regard 

 them as functions of its temperature and volume, or of its tem- 

 perature and pressure, and their differentials are connected by 

 the equation 



de = td-q — pdv. (39) 



If we consider this as applying to the matter within a unit of 

 volume, dv is actually the fraction of dilatation, essentially the 

 one strain-function which plays any part in the case of a fluid, 

 since the elongation in all directions is uniform and shears do 

 not exist. For a strained solid e and r? are still functions of the 

 state, and we can take as the variables the temperature and the 

 strain-coefficients. There are nine of the latter, but we have 

 seen that six quantities are sufficient. In equations (9) we 

 have defined six such quantities d, 62, ... ee, and later in (23) 

 and (24) we have seen that they are quantities which are 

 entirely independent of the choice of the axes in the strained 

 state, (of course, their particular values depend on what axes 

 we choose for OX', OY', OZ', the axes to which the unstrained 

 state is referred; in particular we can choose axes so that 

 64, 65, 6 6 vanish — the principal axes of the strain which are not 

 sheared but merely rotated). For our immediate purpose it 

 is more convenient to take the quantities /i, ... /e as our 

 "thermodynamical variables," where /] = ei' — 1, ...... .; 



f^ = 64/(62^3)% ....... As we know, /i then represents the 



fraction of elongation parallel to OX', etc., and fi represents 

 the shear of lines parallel to OY', OZ', etc. 



For a fluid body —p8v represents the change of internal 

 energy of strain (compression) when the (unit) volume ex- 

 periences a dilatation whose fraction is 8v. Similarly, when the 

 strain-functions /i, ... /e are altered, the energy of strain of unit 

 volume of the strained material alters by Xi8fi . . . + XeS/e. 



