Page 250 - Introduction to Continuum Mechanics

234 The Elastic Solid

Again, neglecting small quantities of higher order, we have

Thus, one can replace the equations of motion

with

In Eq. (5.5.7) all displacement components are regarded as functions of the spatial coordinates
and the equations simply state that for infinitesimal motions, there is no need to make the
distinction between the spatial coordinatesXj and the material coordinates^-. In the following
sections in part A and B of this chapter, all displacement components will be expressed as
functions of the spatial coordinates.
A displacement field «,- is said to describe a possible motion in an elastic medium with small
deformation if it satisfies Eq. (5.5.7). When a displacement field u\ = «/ (jcj, Jt2, £3, t ) is given,
to make sure that it is a possible motion, we can first compute the strain field E^ from
Eq. (3.7.10), i.e.,

and then the corresponding elastic stress field T^ from Eq. (5.3.6a), i.e.,

The substitution of «/ and T^ in Eq. (5.5.7) will then verify whether or not the given motion is
possible. If the motion is found to be possible, the surface tractions, on the boundary of the
body, needed to maintain the motion are given by Eq. (4.9.1), i.e.,

On the other hand, if the boundary conditions are prescribed (e.g., certain boundaries of the
body must remain fixed at all times and other boundaries must remain traction-free at all times,
etc.) then, in order that #/ be the solution to the problem, it must meet the prescribed conditions
on the boundary.