434 RICE ART. K 



— pdv in the law for a fluid medium 



8e = tdr] — p8v. 



The natural method of procedure would be to consider the 

 movements of the points of application of the external forces 

 involved in the change of strain and, combining these with 

 the forces themselves, to determine the work of the external 

 forces; this work, if there is no exchange of heat, will be equal to 

 the change in internal energy. Unfortunately this method 

 involves the use of certain general theorems of mathematical 

 analysis which may be unfamihar to some readers and the 

 writer will therefore make shift with a more elementary, if less 

 rigorous, method. 



We revert to our picture of an element of volume surrounding 

 the point P in the state of strain determined by the values en, 

 ... 633 of the strain-coefficients (see Fig. 6). The element is 

 assumed to be strictly rectangular in this state (although not 

 necessarily so in the state of reference); its sides are parallel 

 to the axes OX, OY, OZ and have the elementary lengths h, k, I 

 respectively. We conceive that this medium receives a further 

 strain to the condition determined by en + Sen, etc., and this 

 involves infinitesimal elongations and shears in the rectangular 

 element. We now imagine the element to be isolated and to 

 experience the same movements under a set of external forces 

 which are equal to the forces which we assume to exist across its 

 faces when in situ. The work of these hypothetical forces we 

 take to be the increase in strain-energy of the element. In the 

 circumstances of the case en, ^22, 633 are near to unity in value, 

 so that in comparison with them en — 1, 622 — 1, 633 — 1, 623, 

 ^32, esi, ei3, ei2, 621, as we noted earlier, are small. The rectangular 

 element has had its side h elongated by a fraction 5fi. The 

 matter surrounding the element is exerting on it forces across the 

 kl faces equal to klXx. Hence work is done which can be 

 calculated by conceiving one of the kl faces fixed and the other 

 moving a distance h8fi in the direction of the force klXx. 

 (The shearing forces klYx and klZx across these faces are at 

 right angles to the elongation and so this movement involves no 

 work on their part.) This work is hklXx8fi, and this is therefore 



