STRAINED ELASTIC SOLIDS 489 



In these equations X and m represent the two elastic constants, 8 is 

 the sum of en, 622, 633 being known as the "dilatation." (623, ^32, 

 esi, ei3, 612, 621 as well as Yz, Zx, Xy are zero.) The various 

 moduli can be expressed in terms of X and n. (In fact /x hap- 

 pens to be the modulus of rigidity itself.) 



Indeed the idea of isotropy may be broadly indicated by 

 reverting to an illustration which we gave in a rather vague 

 form at the outset of our exposition. Imagine a system of 

 forces to be exerted on a body, spfierical in shape, at definite 

 points of the body. These will produce a system of strains and 

 stresses. In a given element there will be a common triad of 

 principal directions. Now conceive the body to be rotated 

 round its center to another orientation, but conceive also that 

 the same forces as before are acting, not at the same points in 

 the body, but at the same points in the frame of reference, i.e., 

 points with the same coordinates with respect to the axes of 

 reference, which we regard as fixed. Exactly the same system 

 of stresses and strains will be produced as before. This does 

 not mean that the element referred to above (i.e., the element 

 occupying the same situation in the body) will be strained just as 

 before; but the element of the body occupying the same situa- 

 tion in the frame of reference will experience the same strains and 

 stresses as were experienced previously by the element originally 

 in that situation, with the same orientation for the principal 

 axes. (It must be carefully borne in mind that this is true for 

 isotropic bodies only; in fact it constitutes a definition of isotropy 

 in elastic properties.) The energy of the spherical body after 

 the rotation is the same as before. This gives us the key to the 

 situation. Such a rotation would be equivalent mathematically 

 to referring a strained body first to any axes of reference (not 

 necessarily principal axes of stress or strain) and then referring 

 to another set; equivalent in fact to what the mathematician 

 calls a "transformation of axes." The values of the strain- 

 coefficients and strain-functions will change. In the first set 

 of axes OX', OY', OZ', ai, o^, az, at, a^, ae are the strain-functions 

 and ^', r]', f ' the local coordinates. The elongation-ellipsoid is 



ax^" + a,-n'^ + az^'^ + 2a,r]'^' -{- 2a,^'i' + 2ae^'r,' = k\ 



