Page 249 - Introduction to Continuum Mechanics

Equations of the Infinitesimal Theory of Elasticity 233

We shall consider only the case of small motions, that is, motions such that every particle
is always in a small neighborhood of the natural state. More specifically, ifXj denotes the
position in the natural state of a typical particle, we assume that

and that the magnitude of the components of the displacement gradient du/dXj, is also very
small
Since

therefore, the velocity component
r»~ /.a,, \

where v,- are the small velocity components associated with the small displacement com-
ponents. Neglecting the small quantities of higher order, we obtain the velocity component

and the acceleration component

Similar approximations are obtained for the other acceleration components. Thus,

Furthermore, since the differential volume dV is related to the initial volume dV 0 by the
equation [See Sect. 3.10]

therefore, the densities are related by

t We assume the existence of a state, called natural state, in which the body is unstressed