STRAINED ELASTIC SOLIDS 433 



vanish except (say) Yz{or Zy). In this case, for an isotropic 

 body, /i = /2 = /s = and also /s = /e = 0. Only fi is finite 

 and for the case of Hooke's law varies directly as Yz. The 

 quotient of Yz by fi is called the "modulus of rigidity," or 

 simply the "rigidity" of the material. Of course one should 

 bear in mind that the strains must be small if the physical facts 

 are to be consistent with these definitions 



We thus see that a given system of external forces on a body 

 involves a determinate set of stress-constituents when the 

 body is in equilibrium under the forces, and these in their turn 

 by reason of the stress-strain relations (hnear or otherwise) 

 determine a definite condition of strain. Infinitesimal va- 

 riations in the external forces change the stress infinitesimally 

 to Zi + dXi, etc. in the new state of equilibrium, and the strain 

 coefficients are altered to/i + dfi, etc., where Xi + dXi, etc. are 

 connected with /i + dfi, etc. by the same six equations as 

 before. Actually we can conceive that "in the neighborhood" 

 of a given state of equilibrium involving a definite condition of 

 strain there are an infinite number of other states, which are not 

 necessarily equilibrium states, characterized by values /i + 

 8fi, etc. of the coefficients where the 8fr are entirely arbitrary, 

 so that /i + dfi, etc. are not connected with the external forces 

 by means of the stress-strain relations. For further information 

 on these matters the reader is referred to standard texts on 

 elasticity and to R. W. Goranson's "Thermodynamic Rela- 

 tions in Multi-component Srjstems" (Carnegie Institution of 

 Washington, Pub. No. 408, 1930).* 



Our ultimate object in what has preceded is to lead up to 

 the expression which represents the change in the energy of 

 strain when the condition of strain has been altered by a change 

 from a state of equilibrium to a neighboring state. This must 

 be included in the expression for the total change of energy 

 when we are formulating the first and second laws of thermo- 

 dynamics. It is in fact the expression which is to replace 



* The reader must be careful to remember that the author's symbol- 

 ism, which has been chosen to diverge as little as possible from that of 

 Gibbs, differs in some details from that used in these references. 



