STRAINED ELASTIC SOLIDS 



491 



quadric surfaces (see Article B of this volume) . For there we 

 have mentioned, with references to sources, the fact that it can 

 be proved that 



«i + ^2 + fls = ai + 0:2 + as, 



a2«3 + «3«1 + CLlCli — Cli — CI5 — Qq 



= azas + mai + aia2 — a^ — a^ — a^, 



ai 

 as 



as 



Thus we see that there are three fairly simple functions which 

 enjoy the property referred to in the enunciation; and of course 

 any given function of these three functions will also have the 

 property. Thus the strain energy of an isotropic body per unit 

 volume must be expressible in terms of the three functions writ- 

 ten above on either side of the equality sign. These functions 

 are in fact E, F, G of the text. The upshot of the argument is 

 that, while for any material the strain-energy per unit volume is 

 a function of the strain-functions ai, a^, aa, ^4, ob, a 6, it can be 

 shown that for isotropic material the function has a special form, 

 being a function of three special functions of the strain-func- 

 tions. Gibbs' own argument, based, as we stated, on the sen- 

 tence from page 205 quoted above, assumes that the strain- 

 energy is solely dependent on n, 7-2, rs (and temperature), and 

 of course by reason of [439] these are functions of E, F, G. As 

 he himself remarks on page 209, although we could regard the 

 strain-energy per unit volume as a function of n, ro, rs "it will 

 be more simple to regard €f' as a function of r]v' and the quan- 

 tities E, F,G." It seems therefore to the writer not out of place 

 to have put the argument on grounds which do not directly in- 

 volve the principal elongations and which appeal to general ideas 

 of isotropy. The argument outlined above does not apply to an 

 aeolotropic (anisotropic) body. We cannot afford space to go 

 into this further but must refer the reader to standard texts on 

 elasticity or to Goranson's book* on this matter. For one thing, 



* See p. 433 of this article. 



