On the Energy of Deformation of an Elastic Solid.

In the theory of elasticity as at present accepted' it is assumed that the energy of deformation is a function of the strain components and the various types of crystalline media are distinguished from one another and from an isotropic medium by the form of this function. It is the object of the present note to show that the mere assumption that the energy of deformation is a function of the strain components implies that it is a function of the three strain invariants so that the medium is necessarily isotropic elastically. For a crystalline medium, therefore, it is impossible that the energy of deformation can be a function of the strain components alone; it will involve, probably, the space rates of change of these components. If the initial and final rectangular Cartesian coordinates of a particle of the medium be denoted by (xo, yo, zo) and (x, y, z), respectively, it follows that ds2 ds02 = 2 [Eldx2 + E2dy2 + E3dz2 + 2E4dydz + 2Egzdx + 2E6dxdy], where if U = x xo, V = y yo, W = z zo denote the disau 1 abu\2 v2 W T2placement components, E1 = x -2 ~ Zlx + ax + t )J , with 1 a3w b\ 1 -a au similar expressions for F2 and E3, and E4 = 2 + 2 + avaov awaw+ b ] ,with similar expressions for E5 and E6. The sixqtities E1... E6 are known as the strain components and they reduce, for au an infinitesimal deformation, to the simpler expressions E1 = -, etc; E4 = 1/aw avv + etc. Now in order to use the energy principle we must con-